O'Nan Moonshine
O'Nan Moonshine
Michael H. Mertens (joint work with J. Duncan and K. Ono)
Universität zu Köln 06 September, 2017 Simons Foundation, New York
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
5
Traces of singular moduli
Arithmetic applications M. H. Mertens
(U. Köln)
O'Nan Moonshine
2 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
2 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
3 / 48
Classi cation of nite simple groups
Theorem A nite simple group of
26
G
either belongs to one of
8
in nite families or is one
sporadic simple groups,
Source: wikipedia
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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The Monster group
M
Some properties of the Monster The largest of the 26 sporadic nite simple groups
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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The Monster group
M
Some properties of the Monster The largest of the 26 sporadic nite simple groups
#M = 246 ·320 ·59 ·76 ·112 ·133 ·17·19·23·29·31·41·47·59·71 ≈ 8.08·1053
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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The Monster group
M
Some properties of the Monster The largest of the 26 sporadic nite simple groups
#M = 246 ·320 ·59 ·76 ·112 ·133 ·17·19·23·29·31·41·47·59·71 ≈ 8.08·1053 194 conjugacy classes, hence 194 irreducible representations (over with characters
M. H. Mertens
(U. Köln)
C)
χ1 , ..., χ194
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Modular forms and functions
Reminder:
SL2 (R)
M. H. Mertens
(U. Köln)
H via aτ + b a b , τ 7→ . c d cτ + d
acts on the upper half-plane
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Modular forms and functions
Reminder:
SL2 (R)
acts on the upper half-plane
H
via
aτ + b a b , τ 7→ . c d cτ + d De nition Let
Γ ≤ SL2 (R)
vol(Γ \ H) < ∞. modular function for Γ
be a discrete subgroup such that
meromorphic function
b f :H→C
is called a
A if
f (γτ ) = f (τ ) for all
γ ∈ Γ, τ ∈ H
M. H. Mertens
(U. Köln)
(+growth condition at the boundary).
O'Nan Moonshine
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Modular forms and functions Reminder:
SL2 (R)
acts on the upper half-plane
H
via
aτ + b a b , τ 7→ . c d cτ + d De nition Let
Γ ≤ SL2 (R)
be a discrete subgroup such that
b holomorphic function f : H → C form of weight k for Γ if
vol(Γ \ H) < ∞.
A
is called a (weakly holomorphic) modular
f (γτ ) = (cτ + d)k f (τ ) for all
γ= k 2
Im(τ ) f (τ ) M. H. Mertens
a b c d
∈ Γ, τ ∈ H
is bounded on
(U. Köln)
(+growth condition at the boundary). If
H,
we call
f
a cusp form.
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Hauptmoduln
Facts 1
The quotient of genus
g.
Γ\H
can be compacti ed to a Riemann surface
Modular functions for
Γ
X(Γ)
de ne meromorphic functions on
X(Γ).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Hauptmoduln
Facts 1
The quotient of genus
g.
Γ\H
can be compacti ed to a Riemann surface
Modular functions for
Γ
X(Γ)
de ne meromorphic functions on
X(Γ). 2
X(Γ) is isomorphic to an g . In particular if g = 0, it
The eld of meromorphic functions on algebraic extension of isomorphic to
M. H. Mertens
C(x)
of degree
is
C(x).
(U. Köln)
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Hauptmoduln
Facts 1
The quotient of genus
g.
Γ\H
can be compacti ed to a Riemann surface
Modular functions for
Γ
X(Γ)
de ne meromorphic functions on
X(Γ). 2
X(Γ) is isomorphic to an g . In particular if g = 0, it
The eld of meromorphic functions on algebraic extension of isomorphic to
C(x)
of degree
is
C(x).
De nition Let
Γ
be as above such that
X(Γ)
has genus
0
(+ mild extra conditions).
A suitably normalized generator for the eld of modular functions for called the Hauptmodul for
M. H. Mertens
(U. Köln)
Γ
is
Γ.
O'Nan Moonshine
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Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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The Jack Daniels problem
For
N ∈N
let
Γ0 (N ) :=
γ ∈ SL2 (Z) : γ ≡
∗ ∗ 0 ∗
(mod N )
and
Γ0 (p)+ := NSL2 (R) (Γ0 (p))
M. H. Mertens
(U. Köln)
O'Nan Moonshine
(p
prime).
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The Jack Daniels problem
For
N ∈N
let
Γ0 (N ) :=
γ ∈ SL2 (Z) : γ ≡
∗ ∗ 0 ∗
(mod N )
and
Γ0 (p)+ := NSL2 (R) (Γ0 (p))
(p
prime).
Theorem (A. Ogg)
p prime, the Riemann surface X(Γ0 (p)+ ) has genus p ∈ {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}. For
M. H. Mertens
(U. Köln)
O'Nan Moonshine
zero if and only if
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The Jack Daniels problem
For
N ∈N
let
Γ0 (N ) :=
γ ∈ SL2 (Z) : γ ≡
∗ ∗ 0 ∗
(mod N )
and
Γ0 (p)+ := NSL2 (R) (Γ0 (p))
(p
prime).
Theorem (A. Ogg) For
p
p
prime, the Riemann surface
divides
X(Γ0 (p)+ )
has genus zero if and only if
#M.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
9 / 48
The Jack Daniels problem
For
N ∈N
let
Γ0 (N ) :=
γ ∈ SL2 (Z) : γ ≡
∗ ∗ 0 ∗
(mod N )
and
Γ0 (p)+ := NSL2 (R) (Γ0 (p))
(p
prime).
Theorem (A. Ogg) For
p
p
prime, the Riemann surface
divides
X(Γ0 (p)+ )
has genus zero if and only if
#M.
Question: Why is this so?
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Monstrous Moonshine I Dimensions of irreducible representations:
χ1 (1) = 1,
M. H. Mertens
χ2 (1) = 196 883,
(U. Köln)
χ3 (1) = 21 296 876,
O'Nan Moonshine
χ4 (1) = 842 609 326.
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Monstrous Moonshine I Dimensions of irreducible representations:
χ1 (1) = 1,
χ2 (1) = 196 883,
Hauptmodul for
SL2 (Z)
J(τ ) = j(τ ) − 744 = =
∞ X
(q
χ3 (1) = 21 296 876,
χ4 (1) = 842 609 326.
:= e2πiτ ):
E4 (τ )3 − 744 ∆(τ )
jn q n = q −1 + 196 884q + 21 493 760q 2 + 864 299 970q 3 + O(q 4 ).
n=−1
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Monstrous Moonshine I Dimensions of irreducible representations:
χ1 (1) = 1,
χ2 (1) = 196883,
Hauptmodul for
SL2 (Z)
J(τ ) = j(τ ) − 744 = =
∞ X
(q
χ3 (1) = 21 296 876,
χ4 (1) = 842 609 326.
:= e2πiτ ):
E4 (τ )3 − 744 ∆(τ )
jn q n = q −1 + 196 884q + 21 493 760q 2 + 864 299 970q 3 + O(q 4 ).
n=−1 Observation (J. McKay & J. G. Thompson, 1979)
j1 = χ1 (1) + χ2 (1). M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Monstrous Moonshine I Dimensions of irreducible representations:
χ1 (1) = 1,
χ2 (1) = 196883,
Hauptmodul for
SL2 (Z)
J(τ ) = j(τ ) − 744 = =
∞ X
(q
χ3 (1) = 21 296 876,
χ4 (1) = 842 609 326.
:= e2πiτ ):
E4 (τ )3 − 744 ∆(τ )
jn q n = q −1 + 196 884q + 21 493 760q 2 + 864 299 970q 3 + O(q 4 ).
n=−1 Observation (J. McKay & J. G. Thompson, 1979)
j2 = χ1 (1) + χ2 (1) + χ3 (1) M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Monstrous Moonshine I Dimensions of irreducible representations:
χ1 (1) = 1,
χ2 (1) = 196883,
Hauptmodul for
SL2 (Z)
J(τ ) = j(τ ) − 744 = =
∞ X
(q
χ3 (1) = 21 296 876,
χ4 (1) = 842 609 326.
:= e2πiτ ):
E4 (τ )3 − 744 ∆(τ )
jn q n = q −1 + 196 884q + 21 493 760q 2 + 864 299 970q 3 + O(q 4 ).
n=−1
Observation (J. McKay & J. G. Thompson, 1979)
j3 = 2χ1 (1) + 2χ2 (1) + χ3 (1) + χ4 (1) M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Monstrous Moonshine II
Values of irreducible characters at other conjugacy classes.
χ1 (2A) = 1,
M. H. Mertens
χ2 (2A) = 4 371,
(U. Köln)
χ3 (2A) = 91 884,
O'Nan Moonshine
χ4 (2A) = 1 139 374.
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Monstrous Moonshine II
Values of irreducible characters at other conjugacy classes.
χ1 (2A) = 1,
χ2 (2A) = 4 371,
Hauptmodul for
J2+ (τ ) = =
∞ X
χ3 (2A) = 91 884,
χ4 (2A) = 1 139 374.
Γ0 (2)+ : 24 η(τ )24 12 η(2τ ) + 2 + 24 η(2τ )24 η(τ )24
αn q n = q −1 + 4 372q + 96 256q 2 + 1 240 002q 3 + O(q 4 ).
n=−1
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Monstrous Moonshine II Values of irreducible characters at other conjugacy classes.
χ1 (2A) = 1,
χ2 (2A) = 4 371,
Hauptmodul for
J2+ (τ ) = =
∞ X
χ3 (2A) = 91 884,
χ4 (2A) = 1 139 374.
Γ0 (2)+ : 24 η(τ )24 12 η(2τ ) + 2 + 24 η(2τ )24 η(τ )24
αn q n = q −1 + 4 372q + 96 256q 2 + 1 240 002q 3 + O(q 4 ).
n=−1
Observation (J. H. Conway & S. P. Norton, 1979)
α1 = χ1 (2A) + χ2 (2A).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Monstrous Moonshine II Values of irreducible characters at other conjugacy classes.
χ1 (2A) = 1,
χ2 (2A) = 4 371,
Hauptmodul for
J2+ (τ ) = =
∞ X
χ3 (2A) = 91 884,
χ4 (2A) = 1 139 374.
Γ0 (2)+ : 24 η(τ )24 12 η(2τ ) + 2 + 24 η(2τ )24 η(τ )24
αn q n = q −1 + 4 372q + 96 256q 2 + 1 240 002q 3 + O(q 4 ).
n=−1
Observation (J. H. Conway & S. P. Norton, 1979)
α2 = χ1 (2A) + χ2 (2A) + χ3 (2A)
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Monstrous Moonshine II Values of irreducible characters at other conjugacy classes.
χ1 (2A) = 1,
χ2 (2A) = 4 371,
Hauptmodul for
J2+ (τ ) = =
∞ X
χ3 (2A) = 91 884,
χ4 (2A) = 1 139 374.
Γ0 (2)+ : 24 η(τ )24 12 η(2τ ) + 2 + 24 η(2τ )24 η(τ )24
αn q n = q −1 + 4 372q + 96 256q 2 + 1 240 002q 3 + O(q 4 ).
n=−1
Observation (J. H. Conway & S. P. Norton, 1979)
α3 = 2χ1 (2A) + 2χ2 (2A) + χ3 (2A) + χ4 (2A) M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Monstrous Moonshine II
Observation (J. H. Conway & S. P. Norton, 1979) There are
194
Hauptmoduln whose coe cients agree, as those of
with character values of
M. H. Mertens
(U. Köln)
J
above,
M.
O'Nan Moonshine
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Monstrous Moonshine II
Observation (J. H. Conway & S. P. Norton, 1979) There are
194
Hauptmoduln whose coe cients agree, as those of
with character values of
J
above,
M.
De nition For a nite group
G
graded components
let
Vn
V =
L
n Vn be a graded
G-module,
are nite-dimensional. Then for each
where all
g∈G
we call
the power series
Tg (q) =
X
tr(g|Vn )q n
n the McKay-Thompson series of
M. H. Mertens
(U. Köln)
g
with respect to
O'Nan Moonshine
V.
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Monstrous Moonshine III
Monstrous Moonshine Conjecture There is an in nite dimensional graded representation
V\
of
M
whose
McKay-Thompson series are the 194 Hauptmoduln found by ConwayNorton.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Monstrous Moonshine III
Monstrous Moonshine Conjecture There is an in nite dimensional graded representation
V\
of
M
whose
McKay-Thompson series are the 194 Hauptmoduln found by ConwayNorton.
Theorem (AtkinFongSmith, The Moonshine module
M. H. Mertens
(U. Köln)
V\
∼
1985)
exists (abstract existence proof ).
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Monstrous Moonshine III
Monstrous Moonshine Conjecture There is an in nite dimensional graded representation
V\
of
M
whose
McKay-Thompson series are the 194 Hauptmoduln found by ConwayNorton.
Theorem (AtkinFongSmith, The Moonshine module
V\
∼
1985)
exists (abstract existence proof ).
Theorem (R. E. Borcherds, 1992) The Moonshine module
V\
is a vertex operator algebra constructed by
FrenkelLepowskyMeurman, whose automorphism group is isomorphic to
M.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
14 / 48
The Umbral groups
There are 23 even unimodular lattices in dimension 24 with rootsystem of full rank, the Niemeier lattices. Examples:
A1 24 ,
M. H. Mertens
(U. Köln)
A2 12 .
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The Umbral groups
There are 23 even unimodular lattices in dimension 24 with rootsystem of full rank, the Niemeier lattices. Examples:
A1 24 ,
A2 12 .
For a Niemeier lattice
L,
its Umbral Group
GL
is de ned as
GL := Aut(L)/Weyl(L). Examples:
GA 1
M. H. Mertens
24
= M24 ,
(U. Köln)
G A2
12
= M12 .
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Umbral Moonshine I
Observation (EguchiOoguriTachikawa, 2010) Some dimensions of irreducible representations of
M24
are multiplicities of
superconformal algebra characters of the K3 elliptic genus, which are known to be coe cients of a (vector-valued) mock theta function.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Umbral Moonshine I
Observation (EguchiOoguriTachikawa, 2010) Some dimensions of irreducible representations of
M24
are multiplicities of
superconformal algebra characters of the K3 elliptic genus, which are known to be coe cients of a (vector-valued) mock theta function.
Theorem (T. Gannon, 2012) There is an in nite-dimensional graded
M24 -module
whose
McKay-Thompson series are speci c (vector-valued) mock theta functions.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Umbral Moonshine II
Umbral Moonshine Conjecture (ChengDuncanHarvey, 2012) For every Umbral Group
GL ,
there is an an in nite-dimensional graded
GL -module whose McKay-Thompson series are speci c (vector-valued) mock theta functions.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Umbral Moonshine II
Umbral Moonshine Conjecture (ChengDuncanHarvey, 2012) For every Umbral Group
GL ,
there is an an in nite-dimensional graded
GL -module whose McKay-Thompson series are speci c (vector-valued) mock theta functions.
Theorem (DuncanGri nOno, 2015) The Umbral Moonshine conjecture is true.
M. H. Mertens
(U. Köln)
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Thompson Moonshine
Conjecture (HarveyRayhaun, 2015) There is an in nite-dimensional graded T h-supermodule L + ⊕ W − has vanishing W = m≡0,1 (4) Wm , where Wm = Wm m m is even and vice versa, whose McKay-Thompson series X T[g] (τ ) = 2q −3 + str(g|Wm )q m
odd part if
m=0 m≡0,1 (4) are speci c weakly holomorphic weight
M. H. Mertens
(U. Köln)
1 2 modular forms.
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Thompson Moonshine
Conjecture (HarveyRayhaun, 2015) There is an in nite-dimensional graded T h-supermodule L + ⊕ W − has vanishing W = m≡0,1 (4) Wm , where Wm = Wm m m is even and vice versa, whose McKay-Thompson series X T[g] (τ ) = 2q −3 + str(g|Wm )q m
odd part if
m=0 m≡0,1 (4) are speci c weakly holomorphic weight
1 2 modular forms.
Theorem (Gri nM., 2016) The Thompson Moonshine Conjecture is true. Moreover, the occuring modular forms can be described systematically.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
19 / 48
Finite simple groups
Source: wikipedia M. H. Mertens
(U. Köln)
O'Nan Moonshine
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The O'Nan group Some properties of
ON
ON
One of the six pariah groups.
M. H. Mertens
(U. Köln)
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The O'Nan group Some properties of
ON
ON
One of the six pariah groups.
# ON = 460 815 505 920 = 29 · 34 · 5 · 73 · 11 · 19 · 31.
M. H. Mertens
(U. Köln)
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The O'Nan group Some properties of
ON
ON
One of the six pariah groups.
# ON = 460 815 505 920 = 29 · 34 · 5 · 73 · 11 · 19 · 31. 30
conjugacy classes, hence
with characters
M. H. Mertens
(U. Köln)
30
irreducible representations (over
C)
χ1 , ..., χ30 .
O'Nan Moonshine
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The O'Nan group Some properties of
ON
ON
One of the six pariah groups.
# ON = 460 815 505 920 = 29 · 34 · 5 · 73 · 11 · 19 · 31. 30
conjugacy classes, hence
with characters
30
irreducible representations (over
C)
χ1 , ..., χ30 .
Dimensions of irreducible representations:
χ1 (1) = 1,
M. H. Mertens
χ7 (1) = 26 752,
(U. Köln)
χ12 (1) = 58 311,
O'Nan Moonshine
χ18 (1) = 85 064.
21 / 48
The O'Nan group Some properties of
ON
ON
One of the six pariah groups.
# ON = 460 815 505 920 = 29 · 34 · 5 · 73 · 11 · 19 · 31. 30
conjugacy classes, hence
with characters
30
irreducible representations (over
C)
χ1 , ..., χ30 .
Dimensions of irreducible representations:
χ1 (1) = 1,
χ7 (1) = 26 752,
Zagier's basis of weight
− g4 (τ ) =
∞ X
3/2
χ12 (1) = 58 311,
χ18 (1) = 85 064.
forms:
an q n
n=−4
=−q
−4
!,+ + 2 + 26 752q 3 + 143 376q 4 + 8 288 256q 7 + O(q 8 ) ∈ M3/2 (Γ0 (4)).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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The O'Nan group Some properties of
ON
ON
One of the six pariah groups.
# ON = 460 815 505 920 = 29 · 34 · 5 · 73 · 11 · 19 · 31. 30
conjugacy classes, hence
with characters
30
irreducible representations (over
C)
χ1 , ..., χ30 .
Dimensions of irreducible representations:
χ1 (1) = 1,
χ7 (1) = 26 752,
Zagier's basis of weight
− g4 (τ ) =
∞ X
3/2
χ12 (1) = 58 311,
χ18 (1) = 85 064.
forms:
an q n
n=−4
=−q
−4
!,+ + 2 + 26 752q 3 + 143 376q 4 + 8 288 256q 7 + O(q 8 ) ∈ M3/2 (Γ0 (4)).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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The O'Nan group Some properties of
ON
ON
One of the six pariah groups.
# ON = 460 815 505 920 = 29 · 34 · 5 · 73 · 11 · 19 · 31. 30
conjugacy classes, hence
with characters
30
irreducible representations (over
C)
χ1 , ..., χ30 .
Dimensions of irreducible representations:
χ1 (1) = 1,
χ7 (1) = 26 752,
Zagier's basis of weight
− g4 (τ ) =
∞ X
3/2
χ12 (1) = 58 311,
χ18 (1) = 85 064.
forms:
an q n
n=−4
=−q
−4
!,+ + 2 + 26 752q 3 + 143 376q 4 + 8 288 256q 7 + O(q 8 ) ∈ M3/2 (Γ0 (4)).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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O'Nan Moonshine Theorem 1 (Duncan-M.-Ono, 2017) There is a (virtual) in nite-dimensional graded
W :=
M 0<m≡0,3
ON-module Wm
(mod 4)
whose associated McKay-Thompson series are speci c weight
3 2 (mock)
modular forms.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
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O'Nan Moonshine Theorem 1 (Duncan-M.-Ono, 2017) There is a (virtual) in nite-dimensional graded
W :=
M 0<m≡0,3
ON-module Wm
(mod 4)
whose associated McKay-Thompson series are speci c weight
3 2 (mock)
modular forms.
Remark We have that
1 dim W163 = (α2 + α − 393768), 2
where
l √ m α = eπ 163 = d262537412640768743.999999999999642...e. M. H. Mertens
(U. Köln)
O'Nan Moonshine
22 / 48
The relevant forms Proposition 1 (Duncan-M.-Ono, 2017) The following are true. 1
For every conjugacy class
[g]
of
ON
there is a unique mock modular
form
F[g] (τ ) = −q −4 + 2 +
X
a[g] (n)q n
n=1 of weight
3/2
for the group
Γ0 (4o(g))
satisfying the following
conditions:
M. H. Mertens
(U. Köln)
O'Nan Moonshine
23 / 48
The relevant forms Proposition 1 (Duncan-M.-Ono, 2017) The following are true. 1
For every conjugacy class
[g]
of
ON
there is a unique mock modular
form
F[g] (τ ) = −q −4 + 2 +
X
a[g] (n)q n
n=1 of weight
3/2
for the group
Γ0 (4o(g))
satisfying the following
conditions: 1
F[g] (τ ) lies (mod 4).
M. H. Mertens
(U. Köln)
in the Kohnen plus space, i.e.,
O'Nan Moonshine
a[g] (n) = 0
if
n ≡ 1, 2
23 / 48
The relevant forms Proposition 1 (Duncan-M.-Ono, 2017) The following are true. 1
For every conjugacy class
[g]
of
ON
there is a unique mock modular
form
F[g] (τ ) = −q −4 + 2 +
X
a[g] (n)q n
n=1 of weight
3/2
for the group
Γ0 (4o(g))
satisfying the following
conditions: 1
2
F[g] (τ ) lies in the Kohnen plus space, i.e., a[g] (n) = 0 if n ≡ 1, 2 (mod 4). F[g] (τ ) has a pole of order 4 at the cusp ∞ and vanishes at essentially all other cusps.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
23 / 48
The relevant forms Proposition 1 (Duncan-M.-Ono, 2017) The following are true. 1
For every conjugacy class
[g]
of
ON
there is a unique mock modular
form
F[g] (τ ) = −q −4 + 2 +
X
a[g] (n)q n
n=1 of weight
3/2
for the group
Γ0 (4o(g))
satisfying the following
conditions: 1
2
F[g] (τ ) lies in the Kohnen plus space, i.e., a[g] (n) = 0 if n ≡ 1, 2 (mod 4). F[g] (τ ) has a pole of order 4 at the cusp ∞ and vanishes at essentially all other cusps.
3
a[g] (3) = χ7 (g), and a[g] (4) = χ1 (g) + χ12 (g) + χ18 (g), a[g] (7) = more complicated . We have
M. H. Mertens
(U. Köln)
O'Nan Moonshine
and
23 / 48
The relevant forms Proposition 1 (Duncan-M.-Ono, 2017) The following are true. 1
For every conjugacy class
[g]
of
ON
there is a unique mock modular
form
F[g] (τ ) = −q −4 + 2 +
X
a[g] (n)q n
n=1 of weight
3/2
for the group
Γ0 (4o(g))
satisfying the following
conditions: 1
2
F[g] (τ ) lies in the Kohnen plus space, i.e., a[g] (n) = 0 if n ≡ 1, 2 (mod 4). F[g] (τ ) has a pole of order 4 at the cusp ∞ and vanishes at essentially all other cusps.
3
2
a[g] (3) = χ7 (g), and a[g] (4) = χ1 (g) + χ12 (g) + χ18 (g), a[g] (7) = more complicated . We have
The function
F[g] (τ ) above has integer Fourier o(g) 6= 16, then it is modular.
and
coe cients.
Furthermore, if M. H. Mertens
(U. Köln)
O'Nan Moonshine
23 / 48
Strategy of the proof Take
F[g] (τ ) = −q
−4
+2+
∞ X
a[g] (n)q n
n=1
M. H. Mertens
(U. Köln)
O'Nan Moonshine
24 / 48
Strategy of the proof Take
F[g] (τ ) = −q
−4
+2+
∞ X
a[g] (n)q n
n=1 De ne
C-valued
class function
ωn : ON → C, g 7→ a[g] (n).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
24 / 48
Strategy of the proof Take
F[g] (τ ) = −q
−4
+2+
∞ X
a[g] (n)q n
n=1 De ne
C-valued
class function
ωn : ON → C, g 7→ a[g] (n). Theorem 1 follows if we can show that
ωn =
30 X
mj (n)χj ,
j=1 with
mj (n) ∈ N0
M. H. Mertens
(U. Köln)
for all
j
and (su ciently large)
O'Nan Moonshine
n.
24 / 48
Strategy of the proof Take
F[g] (τ ) = −q
−4
+2+
∞ X
a[g] (n)q n
n=1 De ne
C-valued
class function
ωn : ON → C, g 7→ a[g] (n). Theorem 1 follows if we can show that
ωn =
30 X
mj (n)χj ,
j=1 with
mj (n) ∈ N0
for all
j
and (su ciently large)
n.
Idea of Thompson can reduce this to a nite computation.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
24 / 48
Strategy of the proof Take
F[g] (τ ) = −q
−4
+2+
∞ X
a[g] (n)q n
n=1 De ne
C-valued
class function
ωn : ON → C, g 7→ a[g] (n). Theorem 1 follows if we can show that
ωn =
30 X
mj (n)χj ,
j=1 with
mj (n) ∈ N0
for all
j
and (su ciently large)
n.
Idea of Thompson can reduce this to a nite computation.
BUT: There is a dierence between ' nite' and 'feasible'. M. H. Mertens
(U. Köln)
O'Nan Moonshine
24 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
25 / 48
Construction
Let
a b 2 ΓK,K 2 (N ) := ∈ Γ0 (N ) : |c| < K, |d| < K . c d
M. H. Mertens
(U. Köln)
O'Nan Moonshine
26 / 48
Construction
Let
a b 2 ΓK,K 2 (N ) := ∈ Γ0 (N ) : |c| < K, |d| < K . c d
De nition For
µ ∈ Z, k ∈ 12 Z,
and
ψ
a multiplier system for
Γ0 (N )
of weight
k,
we
de ne the Rademacher sum
[µ]
Rψ,k (τ ) := lim
X
K→∞
M. H. Mertens
(U. Köln)
ψ(γ)(q µ |k γ).
γ∈Γ∞ \ΓK,K 2 (N )
O'Nan Moonshine
26 / 48
Construction
Let
a b 2 ΓK,K 2 (N ) := ∈ Γ0 (N ) : |c| < K, |d| < K . c d
De nition For
µ ∈ Z, k ∈ 12 Z,
and
ψ
a multiplier system for
Γ0 (N )
of weight
k,
we
de ne the Rademacher sum
[µ]
Rψ,k (τ ) := lim
X
K→∞
ψ(γ)(q µ |k γ).
γ∈Γ∞ \ΓK,K 2 (N )
Low-weight analogue of Poincaré series.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
26 / 48
Construction
Let
a b 2 ΓK,K 2 (N ) := ∈ Γ0 (N ) : |c| < K, |d| < K . c d
De nition For
µ ∈ Z, k ∈ 12 Z,
and
ψ
a multiplier system for
Γ0 (N )
of weight
k,
we
de ne the Rademacher sum
[µ]
Rψ,k (τ ) := lim
X
K→∞
ψ(γ)(q µ |k γ).
γ∈Γ∞ \ΓK,K 2 (N )
Low-weight analogue of Poincaré series. Converges for
M. H. Mertens
k ≥ 1,
(U. Köln)
with regularization sometimes for
O'Nan Moonshine
k < 1.
26 / 48
Properties of Rademacher sums Facts Let
µ < 0. [µ]
Rψ,k
is a weight
k
mock modular form for
[−µ] with shadow R ψ,2−k
M. H. Mertens
(U. Köln)
Γ0 (N )
with multiplier
ψ
∈ M2−k (Γ0 (N ), ψ).
O'Nan Moonshine
27 / 48
Properties of Rademacher sums Facts Let
µ < 0. [µ]
Rψ,k
k mock modular form for Γ0 (N ) with multiplier ψ [−µ] with shadow R ∈ M2−k (Γ0 (N ), ψ). ψ,2−k [µ] Rψ,k has a pole of order |µ| at ∞ and vanishes at all other cusps. is a weight
M. H. Mertens
(U. Köln)
O'Nan Moonshine
27 / 48
Properties of Rademacher sums Facts Let
µ < 0. [µ]
Rψ,k
k mock modular form for Γ0 (N ) with multiplier ψ [−µ] with shadow R ∈ M2−k (Γ0 (N ), ψ). ψ,2−k [µ] Rψ,k has a pole of order |µ| at ∞ and vanishes at all other cusps. is a weight
De nition
f : H → C is called a mock modular form for Γ0 (N ) of multiplier ψ if there is a modular form g ∈ M2−k (Γ0 (N ), ψ)
A hol. function weight
k
and
s.t.
Z
∞
fb(τ ) := f (τ ) + −τ transforms like a modular form.
g
g(z) dz (z + τ )k
is called the shadow of
f , fb is
the
corresponding harmonic Maaÿ form. M. H. Mertens
(U. Köln)
O'Nan Moonshine
27 / 48
Construction of
F[g]
Proof of Proposition 1. Let
[µ]
[µ]
2
2
Z 3 ,ψ = R 3 ,ψ | pr, the projection to the Kohnen plus space.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
28 / 48
Construction of
F[g]
Proof of Proposition 1. Let
[µ]
[µ]
2
2
Z 3 ,ψ = R 3 ,ψ | pr, the projection to the Kohnen plus space.
[−4]
[0]
2
2
−Z 3 ,1 + 2Z 3 ,1
M. H. Mertens
(U. Köln)
satis es conditions 1 and 2 of Proposition 1
O'Nan Moonshine
28 / 48
Construction of
F[g]
Proof of Proposition 1. Let
[µ]
[µ]
2
2
Z 3 ,ψ = R 3 ,ψ | pr, the projection to the Kohnen plus space.
[−4]
[0]
2
2
−Z 3 ,1 + 2Z 3 ,1
satis es conditions 1 and 2 of Proposition 1
Bruinier-Funke pairing yields that 1 and 2 determine a mock modular form uniquely up to addition of cusp forms, so choose cusp forms where possible to ensure 3.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
28 / 48
Construction of
F[g]
Proof of Proposition 1. Let
[µ]
[µ]
2
2
Z 3 ,ψ = R 3 ,ψ | pr, the projection to the Kohnen plus space.
[−4]
[0]
2
2
−Z 3 ,1 + 2Z 3 ,1
satis es conditions 1 and 2 of Proposition 1
Bruinier-Funke pairing yields that 1 and 2 determine a mock modular form uniquely up to addition of cusp forms, so choose cusp forms where possible to ensure 3. Check that coe cients are integers and that (almost) all are modular using again Bruinier-Funke.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
28 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
29 / 48
Integrality of multiplicities I
Let
Fχj (τ ) :=
X 1 χj (g)F[g] (τ ) # ON g∈ON
M. H. Mertens
(U. Köln)
O'Nan Moonshine
30 / 48
Integrality of multiplicities I
Let
Fχj (τ ) :=
X X 1 Schur mj (n)q n . χj (g)F[g] (τ ) = −q −4 + 2 + # ON n=1
g∈ON
Proposition 1 yields: level
Nχj
M. H. Mertens
Fχj
is a mock modular form of weight
3/2
of
with rational coe cients and controllable shadow
(U. Köln)
O'Nan Moonshine
30 / 48
Integrality of multiplicities I
Let
Fχj (τ ) :=
X X 1 Schur mj (n)q n . χj (g)F[g] (τ ) = −q −4 + 2 + # ON n=1
g∈ON
Proposition 1 yields: level
Nχj
Fχj
is a mock modular form of weight
3/2
of
with rational coe cients and controllable shadow
Checking integrality naively by Sturm bound not feasible (Nχ1
= 10 884 720)
M. H. Mertens
(U. Köln)
O'Nan Moonshine
30 / 48
Integrality of the multiplicities II
Proposition 2
Fχj
have all integer Fourier coe cients.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
31 / 48
Integrality of the multiplicities II
Proposition 2
Fχj
have all integer Fourier coe cients.
Proof. The
F[g]
satisfy numerous congruences modulo powers of
(proved by Sturm bound argument,
M. H. Mertens
(U. Köln)
< 250
O'Nan Moonshine
p | # ON
coe cients to be checked).
31 / 48
Integrality of the multiplicities II
Proposition 2
Fχj
have all integer Fourier coe cients.
Proof. The
F[g]
satisfy numerous congruences modulo powers of
(proved by Sturm bound argument, One can then verify directly that
< 250
mj (n)
p | # ON
coe cients to be checked).
are
p-integral
for all
p | # ON,
hence by Proposition 1, the claim follows.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
31 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
32 / 48
Basic strategy
Fact Given convergence, Rademacher sums have a Fourier expansion whose coe cients are given in terms of in nite sums of Kloosterman sums
Kψ (m, n, c) =
md+nd ∗ ∗ ψ e2πi c c d ∗
X d (c)
times
I -Bessel
M. H. Mertens
functions.
(U. Köln)
O'Nan Moonshine
33 / 48
Basic strategy
Fact Given convergence, Rademacher sums have a Fourier expansion whose coe cients are given in terms of in nite sums of Kloosterman sums
Kψ (m, n, c) =
md+nd ∗ ∗ ψ e2πi c c d ∗
X d (c)
times
I -Bessel
functions.
By the triangle inequality we have
mj (n) ≥
X | str(g|Wn )| | str(1|Wn )| − |χj (g)|. # ON #CON (g) [g]6=1A
M. H. Mertens
(U. Köln)
O'Nan Moonshine
33 / 48
Basic strategy
Fact Given convergence, Rademacher sums have a Fourier expansion whose coe cients are given in terms of in nite sums of Kloosterman sums
Kψ (m, n, c) =
md+nd ∗ ∗ ψ e2πi c c d ∗
X d (c)
times
I -Bessel
functions.
By the triangle inequality we have
mj (n) ≥
X | str(g|Wn )| | str(1|Wn )| − |χj (g)|. # ON #CON (g) [g]6=1A
Show that from a certain point on, the rst term dominates. M. H. Mertens
(U. Köln)
O'Nan Moonshine
33 / 48
Positivity of the multiplicities Proposition 3 The multiplicities
M. H. Mertens
mj (n)
(U. Köln)
are all non-negative for
O'Nan Moonshine
n 6= 7, 8, 12.
34 / 48
Positivity of the multiplicities Proposition 3 The multiplicities
mj (n)
are all non-negative for
n 6= 7, 8, 12.
Ingredients of the proof. Careful, explicit estimates for Selberg-Kloosterman zeta functions.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
34 / 48
Positivity of the multiplicities Proposition 3 The multiplicities
mj (n)
are all non-negative for
n 6= 7, 8, 12.
Ingredients of the proof. Careful, explicit estimates for Selberg-Kloosterman zeta functions. Write Kloosterman sums as sums over a sparse set, i.e. equivalence classes of binary quadratic forms.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
34 / 48
Positivity of the multiplicities Proposition 3 The multiplicities
mj (n)
are all non-negative for
n 6= 7, 8, 12.
Ingredients of the proof. Careful, explicit estimates for Selberg-Kloosterman zeta functions. Write Kloosterman sums as sums over a sparse set, i.e. equivalence classes of binary quadratic forms. Explicit estimates for coe cients of weight 3/2 cusp forms
⇒
M. H. Mertens
(U. Köln)
mj (n) ≥ 0
O'Nan Moonshine
for
n ≥ 109.
34 / 48
Positivity of the multiplicities Proposition 3 The multiplicities
mj (n)
are all non-negative for
n 6= 7, 8, 12.
Ingredients of the proof. Careful, explicit estimates for Selberg-Kloosterman zeta functions. Write Kloosterman sums as sums over a sparse set, i.e. equivalence classes of binary quadratic forms. Explicit estimates for coe cients of weight 3/2 cusp forms
⇒
mj (n) ≥ 0
for
n ≥ 109.
Check rest by inspection.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
34 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
35 / 48
Singular moduli Fact
Q = [a, b, c] be a quadratic form of discriminant D < 0 and τQ ∈ H, 2 such that aτQ + bτQ + c = 0. Then J(τQ ) is a real-algebraic integer of degree h(D).
Let
E.g.:
√
−163 = −262 537 412 640 768 744 J 2 √ √ 1 + −15 192 513 + 85 995 5 J =− . 2 2
M. H. Mertens
1+
(U. Köln)
O'Nan Moonshine
36 / 48
Singular moduli Fact
Q = [a, b, c] be a quadratic form of discriminant D < 0 and τQ ∈ H, 2 such that aτQ + bτQ + c = 0. Then J(τQ ) is a real-algebraic integer of degree h(D).
Let
E.g.:
√
−163 = −262 537 412 640 768 744 J 2 √ √ 1 + −15 192 513 + 85 995 5 J =− . 2 2
1+
Play an important role in explicit class eld theory (Kronecker's
th problem)
Jugendtraum, Hilbert's 12
M. H. Mertens
(U. Köln)
O'Nan Moonshine
36 / 48
Singular moduli Fact
Q = [a, b, c] be a quadratic form of discriminant D < 0 and τQ ∈ H, 2 such that aτQ + bτQ + c = 0. Then J(τQ ) is a real-algebraic integer of degree h(D).
Let
E.g.:
√
−163 = −262 537 412 640 768 744 J 2 √ √ 1 + −15 192 513 + 85 995 5 J =− . 2 2
1+
Play an important role in explicit class eld theory (Kronecker's
th problem)
Jugendtraum, Hilbert's 12
Similar results are true for Hauptmoduln of genus 0 congruence subgroups and other modular functions M. H. Mertens
(U. Köln)
O'Nan Moonshine
36 / 48
Traces
De nition For a function
f : H → C,
a discriminant
(N )
−D < 0
N ∈N
de ne
f (τQ ) , ω (N ) (Q)
X
TrD (f ) :=
and
(N )
Q∈Q−D /Γ0 (N ) where
(N )
Q−D = {[a, b, c] : b2 − 4ac = −D ω (N ) (Q)
M. H. Mertens
=
1 2
and
N | a},
· # StabΓ0 (N ) (Q).
(U. Köln)
O'Nan Moonshine
37 / 48
Generating functions Theorem (D. Zagier)
− q −1 + 2 +
X
(1)
TrD (J)q D
D≡0,3 (4)
=−q
−1
M. H. Mertens
!,+ + 2 − 248q 3 + 492q 4 − 4119q 7 + O(q 8 ) ∈ M3/2 (Γ0 (4)).
(U. Köln)
O'Nan Moonshine
38 / 48
Generating functions Theorem (D. Zagier)
− q −1 + 2 +
X
(1)
TrD (J)q D
D≡0,3 (4)
=−q
−1
!,+ + 2 − 248q 3 + 492q 4 − 4119q 7 + O(q 8 ) ∈ M3/2 (Γ0 (4)).
Can be extended to more general modular functions with vanishing constant terms (Bruinier-Funke, Miller-Pixton,...)
M. H. Mertens
(U. Köln)
O'Nan Moonshine
38 / 48
Generating functions Theorem (D. Zagier)
X
− q −1 + 2 +
(1)
TrD (J)q D
D≡0,3 (4)
=−q
−1
!,+ + 2 − 248q 3 + 492q 4 − 4119q 7 + O(q 8 ) ∈ M3/2 (Γ0 (4)).
Can be extended to more general modular functions with vanishing constant terms (Bruinier-Funke, Miller-Pixton,...)
De nition/Theorem For
N ∈ N,
we call
(N )
H (N ) (D) := TrD (1)
number and set
the generalized Hurwitz class
[Γ(1) : Γ0 (N )] X (N ) + H (D)q D . 12 D mock modular form of level 4N .
H (N ) (τ ) := − H (N )
is a weight 3/2
M. H. Mertens
(U. Köln)
O'Nan Moonshine
38 / 48
Traces and Rademacher series Proposition 5 Let
N ∈N
such that
(N )
X0 (N )
Tr4 (D) := where
J (N )
has genus
0
and
1 (N ) (N ) (N/d) TrD (J2 ) − TrD (J (N/d) ) , 2
denotes the Hauptmodul for
the unique modular function for
Γ0 (N )
Γ0 (N )
and
(N )
J2
= q −2 + O(q)
is
with this Fourier expansion at
d := gcd(N, 2). Then X (N ) T (N ) (τ ) := −q −4 + const + Tr4 (D)q D
in nity and no poles anywhere else and
we have
D>0
=
[−4],+ R 3 ,4o(g) (τ ) 2
for some rational numbers
c1
and
c2 c1 − H (N ) (τ ) + H (N/d) (τ ) 2 2 c2 .
In particular, the function
T (N )
has
integer Fourier coe cients. M. H. Mertens
(U. Köln)
O'Nan Moonshine
39 / 48
McKay-Thompson series and traces
N where genus(X0+ (N )) = 0.
There is an analogue of Proposition 5 for
genus(X0 (N )) > 0,
M. H. Mertens
(U. Köln)
but
O'Nan Moonshine
40 / 48
McKay-Thompson series and traces
N where genus(X0+ (N )) = 0.
There is an analogue of Proposition 5 for
genus(X0 (N )) > 0, ⇒
but
We can express character values of
ON
in terms of traces of
singular moduli, generalized class numbers and coe cients of cusp forms.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
40 / 48
McKay-Thompson series and traces
N where genus(X0+ (N )) = 0.
There is an analogue of Proposition 5 for
genus(X0 (N )) > 0, ⇒
but
We can express character values of
ON
in terms of traces of
singular moduli, generalized class numbers and coe cients of cusp forms.
Example
F1A = T (1) , F7AB = T (7) + 4H (1) − 4H (7) , 12 6 4 F11A = T (11,+) + H (1) − H (11) − G (11) , 5 5 5 where
+ G (11) (τ ) = q 3 − q 4 − q 11 + O(q 12 ) ∈ S3/2 (44).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
40 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
41 / 48
ON
knows
p-parts
in ideal class groups
Theorem 2 (DuncanM.Ono, 2017) Suppose that
−D < 0
is a fundamental discriminant. Then the following
are true:
M. H. Mertens
(U. Köln)
O'Nan Moonshine
42 / 48
ON
knows
p-parts
in ideal class groups
Theorem 2 (DuncanM.Ono, 2017) Suppose that
−D < 0
is a fundamental discriminant. Then the following
are true: 1
If
−D < −8
is even and
g2 ∈ ON
has order
2,
dim WD ≡ tr(g2 |WD ) ≡ −24H(D)
M. H. Mertens
(U. Köln)
O'Nan Moonshine
then
(mod 24 ).
42 / 48
ON
knows
p-parts
in ideal class groups
Theorem 2 (DuncanM.Ono, 2017) Suppose that
−D < 0
is a fundamental discriminant. Then the following
are true: 1
If
−D < −8
is even and
g2 ∈ ON
has order
2,
dim WD ≡ tr(g2 |WD ) ≡ −24H(D)
2
If
p ∈ {3, 5, 7},
−D p
= −1
and
gp ∈ ON
then
(mod 24 ).
has order
( −24H(D) (mod 32 ) dim WD ≡ tr(gp |WD ) ≡ −24H(D) (mod p)
M. H. Mertens
(U. Köln)
O'Nan Moonshine
p,
then
if if
p = 3, p = 5, 7.
42 / 48
The BSD-conjecture and Waldspurger's theorem I Conjecture (Birch and Swinnerton-Dyer) Let
E/Q
be an elliptic curve. Then we have that
Q #X(E) · Reg(E) ` c` (E) L(r) (E, 1) = , r!ΩE (#E(Q)tors )2 where
r
denotes the order of vanishing of
the MordellWeil rank of
M. H. Mertens
(U. Köln)
L(E, s)
at
s = 1,
which equals
E.
O'Nan Moonshine
43 / 48
The BSD-conjecture and Waldspurger's theorem I Conjecture (Birch and Swinnerton-Dyer) Let
E/Q
be an elliptic curve. Then we have that
Q #X(E) · Reg(E) ` c` (E) L(r) (E, 1) = , r!ΩE (#E(Q)tors )2 where
r
denotes the order of vanishing of
the MordellWeil rank of
L(E, s)
at
s = 1,
which equals
E.
Theorem (Waldspurger, Kohnen) Let
N ∈N
be odd and square-free,
F ∈ S2k (Γ0 (N ))
be a newform and
2
the image of
f
under the Shimura correspondence. For a
suitable fundamental discriminant
hf, f i = M. H. Mertens
+ f ∈ Sk+ 1 (Γ0 (4N ))
(U. Köln)
D
we have
hF, F iπ k 1
2ω(N ) (k − 1)!|D|k− 2 L(F, D; k) O'Nan Moonshine
· |bf (|D|)|2 . 43 / 48
Quadratic twists
Connection through Modularity Theorem:
Lemma Let
−D < 0
be a suitable fundamental discriminant and
curve of odd, square-free conductor
N.
E/Q an elliptic 2 newform
Denote the weight
) and its Shintani lift by P E by FE n∈ S2 (N + fE (τ ) = ∞ b (n)q ∈ S n=3 E 3/2 (4N ). If E(−D) denotes twist of E by −D , we have associated to
the quadratic
L(E(−D), 1) = CE · |bE (D)|2 , ΩE(−D) where
CE
is a constant depending on
M. H. Mertens
(U. Köln)
E,
but not (really) on
O'Nan Moonshine
D.
44 / 48
ON
knows about Selmer and Tate-Shafarevich groups I
Theorem 3 (DuncanM.Ono, 2017)
= 11 or 19 and −D < 0 is a fundamental discriminant = −1, and gp ∈ ON has order p, then the following are
Assume BSD. If p for which
−D p
true.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
45 / 48
ON
knows about Selmer and Tate-Shafarevich groups I
Theorem 3 (DuncanM.Ono, 2017)
= 11 or 19 and −D < 0 is a fundamental discriminant = −1, and gp ∈ ON has order p, then the following are
Assume BSD. If p for which
−D p
true. 1
Sel(Ep (−D))[p] 6= {0}
if and only if
dim WD ≡ tr(gp |WD ) ≡ −24H(D)
M. H. Mertens
(U. Köln)
O'Nan Moonshine
(mod p).
45 / 48
ON
knows about Selmer and Tate-Shafarevich groups I
Theorem 3 (DuncanM.Ono, 2017)
= 11 or 19 and −D < 0 is a fundamental discriminant = −1, and gp ∈ ON has order p, then the following are
Assume BSD. If p for which
−D p
true. 1
Sel(Ep (−D))[p] 6= {0}
if and only if
dim WD ≡ tr(gp |WD ) ≡ −24H(D)
2
Suppose further that
L(Ep (−D), 1) 6= 0.
Then
(mod p).
p | #X(Ep (−D))
if
and only if
dim WD ≡ tr(gp |WD ) ≡ −24H(D) (mod p).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
45 / 48
ON
knows about Selmer and Tate-Shafarevich groups II
Theorem 4 (DuncanM.Ono, 2017) Let
N ∈ {14, 15}
−D < 0
and write
N = p0 p, p0 < p,
δp := p−1 2 . If −D = −1, then p
and let
is a fundamental discriminant for which
the
following are true.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
46 / 48
ON
knows about Selmer and Tate-Shafarevich groups II
Theorem 4 (DuncanM.Ono, 2017) Let
N ∈ {14, 15}
−D < 0
and write
N = p0 p, p0 < p,
δp := p−1 2 . If −D = −1, then p
and let
is a fundamental discriminant for which
the
following are true. 1
Sel(EN (−D))[p] 6= {0}
if and only if 0
tr(gp0 |WD ) ≡ tr(gN |WD ) ≡ δp · (H(D) − δp H (p ) (D))
M. H. Mertens
(U. Köln)
O'Nan Moonshine
(mod p).
46 / 48
ON
knows about Selmer and Tate-Shafarevich groups II
Theorem 4 (DuncanM.Ono, 2017) Let
N ∈ {14, 15}
−D < 0
and write
N = p0 p, p0 < p,
δp := p−1 2 . If −D = −1, then p
and let
is a fundamental discriminant for which
the
following are true. 1
Sel(EN (−D))[p] 6= {0}
if and only if 0
tr(gp0 |WD ) ≡ tr(gN |WD ) ≡ δp · (H(D) − δp H (p ) (D))
2
(mod p).
L(EN (−D), 1) 6= 0. p | #X(EN (−D)) if and only if
Suppose further that Then
0
tr(gp0 |WD ) ≡ tr(gN |WD ) ≡ δp · (H(D) − δp H (p ) (D))
M. H. Mertens
(U. Köln)
O'Nan Moonshine
(mod p).
46 / 48
Example
D
tr2 (D)
H14 (D)
Diff 14 (D)
rk(E14 (−D))
#X(E14 (−D))
15
-96256
-30
3
0
1
23
-1746944
-45
0
2
1
39
-165...168
-60
4
0
1
71
-156...880
-105
4
0
1 1
79
-669...192
-75
3
0
239
-619...040
-225
0
2
1
2671
-163...664
-345
0
0
49
Table: Examples for the curve
H14 (D) := δ7 (H(D) − δ7 H (2) (D)),
E14
tr2 (D) := tr(g2 |WD ),
Diff 14 (D) := H14 (D) − tr2 (D) M. H. Mertens
(U. Köln)
O'Nan Moonshine
47 / 48
Thank you for your attention.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
48 / 48
Michael H. Mertens (joint work with J. Duncan and K. Ono)
Universität zu Köln 06 September, 2017 Simons Foundation, New York
M. H. Mertens
(U. Köln)
O'Nan Moonshine
1 / 48
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
5
Traces of singular moduli
Arithmetic applications M. H. Mertens
(U. Köln)
O'Nan Moonshine
2 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
2 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
3 / 48
Classi cation of nite simple groups
Theorem A nite simple group of
26
G
either belongs to one of
8
in nite families or is one
sporadic simple groups,
Source: wikipedia
M. H. Mertens
(U. Köln)
O'Nan Moonshine
4 / 48
The Monster group
M
Some properties of the Monster The largest of the 26 sporadic nite simple groups
M. H. Mertens
(U. Köln)
O'Nan Moonshine
5 / 48
The Monster group
M
Some properties of the Monster The largest of the 26 sporadic nite simple groups
#M = 246 ·320 ·59 ·76 ·112 ·133 ·17·19·23·29·31·41·47·59·71 ≈ 8.08·1053
M. H. Mertens
(U. Köln)
O'Nan Moonshine
5 / 48
The Monster group
M
Some properties of the Monster The largest of the 26 sporadic nite simple groups
#M = 246 ·320 ·59 ·76 ·112 ·133 ·17·19·23·29·31·41·47·59·71 ≈ 8.08·1053 194 conjugacy classes, hence 194 irreducible representations (over with characters
M. H. Mertens
(U. Köln)
C)
χ1 , ..., χ194
O'Nan Moonshine
5 / 48
Modular forms and functions
Reminder:
SL2 (R)
M. H. Mertens
(U. Köln)
H via aτ + b a b , τ 7→ . c d cτ + d
acts on the upper half-plane
O'Nan Moonshine
6 / 48
Modular forms and functions
Reminder:
SL2 (R)
acts on the upper half-plane
H
via
aτ + b a b , τ 7→ . c d cτ + d De nition Let
Γ ≤ SL2 (R)
vol(Γ \ H) < ∞. modular function for Γ
be a discrete subgroup such that
meromorphic function
b f :H→C
is called a
A if
f (γτ ) = f (τ ) for all
γ ∈ Γ, τ ∈ H
M. H. Mertens
(U. Köln)
(+growth condition at the boundary).
O'Nan Moonshine
6 / 48
Modular forms and functions Reminder:
SL2 (R)
acts on the upper half-plane
H
via
aτ + b a b , τ 7→ . c d cτ + d De nition Let
Γ ≤ SL2 (R)
be a discrete subgroup such that
b holomorphic function f : H → C form of weight k for Γ if
vol(Γ \ H) < ∞.
A
is called a (weakly holomorphic) modular
f (γτ ) = (cτ + d)k f (τ ) for all
γ= k 2
Im(τ ) f (τ ) M. H. Mertens
a b c d
∈ Γ, τ ∈ H
is bounded on
(U. Köln)
(+growth condition at the boundary). If
H,
we call
f
a cusp form.
O'Nan Moonshine
6 / 48
Hauptmoduln
Facts 1
The quotient of genus
g.
Γ\H
can be compacti ed to a Riemann surface
Modular functions for
Γ
X(Γ)
de ne meromorphic functions on
X(Γ).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
7 / 48
Hauptmoduln
Facts 1
The quotient of genus
g.
Γ\H
can be compacti ed to a Riemann surface
Modular functions for
Γ
X(Γ)
de ne meromorphic functions on
X(Γ). 2
X(Γ) is isomorphic to an g . In particular if g = 0, it
The eld of meromorphic functions on algebraic extension of isomorphic to
M. H. Mertens
C(x)
of degree
is
C(x).
(U. Köln)
O'Nan Moonshine
7 / 48
Hauptmoduln
Facts 1
The quotient of genus
g.
Γ\H
can be compacti ed to a Riemann surface
Modular functions for
Γ
X(Γ)
de ne meromorphic functions on
X(Γ). 2
X(Γ) is isomorphic to an g . In particular if g = 0, it
The eld of meromorphic functions on algebraic extension of isomorphic to
C(x)
of degree
is
C(x).
De nition Let
Γ
be as above such that
X(Γ)
has genus
0
(+ mild extra conditions).
A suitably normalized generator for the eld of modular functions for called the Hauptmodul for
M. H. Mertens
(U. Köln)
Γ
is
Γ.
O'Nan Moonshine
7 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
8 / 48
The Jack Daniels problem
For
N ∈N
let
Γ0 (N ) :=
γ ∈ SL2 (Z) : γ ≡
∗ ∗ 0 ∗
(mod N )
and
Γ0 (p)+ := NSL2 (R) (Γ0 (p))
M. H. Mertens
(U. Köln)
O'Nan Moonshine
(p
prime).
9 / 48
The Jack Daniels problem
For
N ∈N
let
Γ0 (N ) :=
γ ∈ SL2 (Z) : γ ≡
∗ ∗ 0 ∗
(mod N )
and
Γ0 (p)+ := NSL2 (R) (Γ0 (p))
(p
prime).
Theorem (A. Ogg)
p prime, the Riemann surface X(Γ0 (p)+ ) has genus p ∈ {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}. For
M. H. Mertens
(U. Köln)
O'Nan Moonshine
zero if and only if
9 / 48
The Jack Daniels problem
For
N ∈N
let
Γ0 (N ) :=
γ ∈ SL2 (Z) : γ ≡
∗ ∗ 0 ∗
(mod N )
and
Γ0 (p)+ := NSL2 (R) (Γ0 (p))
(p
prime).
Theorem (A. Ogg) For
p
p
prime, the Riemann surface
divides
X(Γ0 (p)+ )
has genus zero if and only if
#M.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
9 / 48
The Jack Daniels problem
For
N ∈N
let
Γ0 (N ) :=
γ ∈ SL2 (Z) : γ ≡
∗ ∗ 0 ∗
(mod N )
and
Γ0 (p)+ := NSL2 (R) (Γ0 (p))
(p
prime).
Theorem (A. Ogg) For
p
p
prime, the Riemann surface
divides
X(Γ0 (p)+ )
has genus zero if and only if
#M.
Question: Why is this so?
M. H. Mertens
(U. Köln)
O'Nan Moonshine
9 / 48
Monstrous Moonshine I Dimensions of irreducible representations:
χ1 (1) = 1,
M. H. Mertens
χ2 (1) = 196 883,
(U. Köln)
χ3 (1) = 21 296 876,
O'Nan Moonshine
χ4 (1) = 842 609 326.
10 / 48
Monstrous Moonshine I Dimensions of irreducible representations:
χ1 (1) = 1,
χ2 (1) = 196 883,
Hauptmodul for
SL2 (Z)
J(τ ) = j(τ ) − 744 = =
∞ X
(q
χ3 (1) = 21 296 876,
χ4 (1) = 842 609 326.
:= e2πiτ ):
E4 (τ )3 − 744 ∆(τ )
jn q n = q −1 + 196 884q + 21 493 760q 2 + 864 299 970q 3 + O(q 4 ).
n=−1
M. H. Mertens
(U. Köln)
O'Nan Moonshine
10 / 48
Monstrous Moonshine I Dimensions of irreducible representations:
χ1 (1) = 1,
χ2 (1) = 196883,
Hauptmodul for
SL2 (Z)
J(τ ) = j(τ ) − 744 = =
∞ X
(q
χ3 (1) = 21 296 876,
χ4 (1) = 842 609 326.
:= e2πiτ ):
E4 (τ )3 − 744 ∆(τ )
jn q n = q −1 + 196 884q + 21 493 760q 2 + 864 299 970q 3 + O(q 4 ).
n=−1 Observation (J. McKay & J. G. Thompson, 1979)
j1 = χ1 (1) + χ2 (1). M. H. Mertens
(U. Köln)
O'Nan Moonshine
10 / 48
Monstrous Moonshine I Dimensions of irreducible representations:
χ1 (1) = 1,
χ2 (1) = 196883,
Hauptmodul for
SL2 (Z)
J(τ ) = j(τ ) − 744 = =
∞ X
(q
χ3 (1) = 21 296 876,
χ4 (1) = 842 609 326.
:= e2πiτ ):
E4 (τ )3 − 744 ∆(τ )
jn q n = q −1 + 196 884q + 21 493 760q 2 + 864 299 970q 3 + O(q 4 ).
n=−1 Observation (J. McKay & J. G. Thompson, 1979)
j2 = χ1 (1) + χ2 (1) + χ3 (1) M. H. Mertens
(U. Köln)
O'Nan Moonshine
10 / 48
Monstrous Moonshine I Dimensions of irreducible representations:
χ1 (1) = 1,
χ2 (1) = 196883,
Hauptmodul for
SL2 (Z)
J(τ ) = j(τ ) − 744 = =
∞ X
(q
χ3 (1) = 21 296 876,
χ4 (1) = 842 609 326.
:= e2πiτ ):
E4 (τ )3 − 744 ∆(τ )
jn q n = q −1 + 196 884q + 21 493 760q 2 + 864 299 970q 3 + O(q 4 ).
n=−1
Observation (J. McKay & J. G. Thompson, 1979)
j3 = 2χ1 (1) + 2χ2 (1) + χ3 (1) + χ4 (1) M. H. Mertens
(U. Köln)
O'Nan Moonshine
10 / 48
Monstrous Moonshine II
Values of irreducible characters at other conjugacy classes.
χ1 (2A) = 1,
M. H. Mertens
χ2 (2A) = 4 371,
(U. Köln)
χ3 (2A) = 91 884,
O'Nan Moonshine
χ4 (2A) = 1 139 374.
11 / 48
Monstrous Moonshine II
Values of irreducible characters at other conjugacy classes.
χ1 (2A) = 1,
χ2 (2A) = 4 371,
Hauptmodul for
J2+ (τ ) = =
∞ X
χ3 (2A) = 91 884,
χ4 (2A) = 1 139 374.
Γ0 (2)+ : 24 η(τ )24 12 η(2τ ) + 2 + 24 η(2τ )24 η(τ )24
αn q n = q −1 + 4 372q + 96 256q 2 + 1 240 002q 3 + O(q 4 ).
n=−1
M. H. Mertens
(U. Köln)
O'Nan Moonshine
11 / 48
Monstrous Moonshine II Values of irreducible characters at other conjugacy classes.
χ1 (2A) = 1,
χ2 (2A) = 4 371,
Hauptmodul for
J2+ (τ ) = =
∞ X
χ3 (2A) = 91 884,
χ4 (2A) = 1 139 374.
Γ0 (2)+ : 24 η(τ )24 12 η(2τ ) + 2 + 24 η(2τ )24 η(τ )24
αn q n = q −1 + 4 372q + 96 256q 2 + 1 240 002q 3 + O(q 4 ).
n=−1
Observation (J. H. Conway & S. P. Norton, 1979)
α1 = χ1 (2A) + χ2 (2A).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
11 / 48
Monstrous Moonshine II Values of irreducible characters at other conjugacy classes.
χ1 (2A) = 1,
χ2 (2A) = 4 371,
Hauptmodul for
J2+ (τ ) = =
∞ X
χ3 (2A) = 91 884,
χ4 (2A) = 1 139 374.
Γ0 (2)+ : 24 η(τ )24 12 η(2τ ) + 2 + 24 η(2τ )24 η(τ )24
αn q n = q −1 + 4 372q + 96 256q 2 + 1 240 002q 3 + O(q 4 ).
n=−1
Observation (J. H. Conway & S. P. Norton, 1979)
α2 = χ1 (2A) + χ2 (2A) + χ3 (2A)
M. H. Mertens
(U. Köln)
O'Nan Moonshine
11 / 48
Monstrous Moonshine II Values of irreducible characters at other conjugacy classes.
χ1 (2A) = 1,
χ2 (2A) = 4 371,
Hauptmodul for
J2+ (τ ) = =
∞ X
χ3 (2A) = 91 884,
χ4 (2A) = 1 139 374.
Γ0 (2)+ : 24 η(τ )24 12 η(2τ ) + 2 + 24 η(2τ )24 η(τ )24
αn q n = q −1 + 4 372q + 96 256q 2 + 1 240 002q 3 + O(q 4 ).
n=−1
Observation (J. H. Conway & S. P. Norton, 1979)
α3 = 2χ1 (2A) + 2χ2 (2A) + χ3 (2A) + χ4 (2A) M. H. Mertens
(U. Köln)
O'Nan Moonshine
11 / 48
Monstrous Moonshine II
Observation (J. H. Conway & S. P. Norton, 1979) There are
194
Hauptmoduln whose coe cients agree, as those of
with character values of
M. H. Mertens
(U. Köln)
J
above,
M.
O'Nan Moonshine
12 / 48
Monstrous Moonshine II
Observation (J. H. Conway & S. P. Norton, 1979) There are
194
Hauptmoduln whose coe cients agree, as those of
with character values of
J
above,
M.
De nition For a nite group
G
graded components
let
Vn
V =
L
n Vn be a graded
G-module,
are nite-dimensional. Then for each
where all
g∈G
we call
the power series
Tg (q) =
X
tr(g|Vn )q n
n the McKay-Thompson series of
M. H. Mertens
(U. Köln)
g
with respect to
O'Nan Moonshine
V.
12 / 48
Monstrous Moonshine III
Monstrous Moonshine Conjecture There is an in nite dimensional graded representation
V\
of
M
whose
McKay-Thompson series are the 194 Hauptmoduln found by ConwayNorton.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
13 / 48
Monstrous Moonshine III
Monstrous Moonshine Conjecture There is an in nite dimensional graded representation
V\
of
M
whose
McKay-Thompson series are the 194 Hauptmoduln found by ConwayNorton.
Theorem (AtkinFongSmith, The Moonshine module
M. H. Mertens
(U. Köln)
V\
∼
1985)
exists (abstract existence proof ).
O'Nan Moonshine
13 / 48
Monstrous Moonshine III
Monstrous Moonshine Conjecture There is an in nite dimensional graded representation
V\
of
M
whose
McKay-Thompson series are the 194 Hauptmoduln found by ConwayNorton.
Theorem (AtkinFongSmith, The Moonshine module
V\
∼
1985)
exists (abstract existence proof ).
Theorem (R. E. Borcherds, 1992) The Moonshine module
V\
is a vertex operator algebra constructed by
FrenkelLepowskyMeurman, whose automorphism group is isomorphic to
M.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
13 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
14 / 48
The Umbral groups
There are 23 even unimodular lattices in dimension 24 with rootsystem of full rank, the Niemeier lattices. Examples:
A1 24 ,
M. H. Mertens
(U. Köln)
A2 12 .
O'Nan Moonshine
15 / 48
The Umbral groups
There are 23 even unimodular lattices in dimension 24 with rootsystem of full rank, the Niemeier lattices. Examples:
A1 24 ,
A2 12 .
For a Niemeier lattice
L,
its Umbral Group
GL
is de ned as
GL := Aut(L)/Weyl(L). Examples:
GA 1
M. H. Mertens
24
= M24 ,
(U. Köln)
G A2
12
= M12 .
O'Nan Moonshine
15 / 48
Umbral Moonshine I
Observation (EguchiOoguriTachikawa, 2010) Some dimensions of irreducible representations of
M24
are multiplicities of
superconformal algebra characters of the K3 elliptic genus, which are known to be coe cients of a (vector-valued) mock theta function.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
16 / 48
Umbral Moonshine I
Observation (EguchiOoguriTachikawa, 2010) Some dimensions of irreducible representations of
M24
are multiplicities of
superconformal algebra characters of the K3 elliptic genus, which are known to be coe cients of a (vector-valued) mock theta function.
Theorem (T. Gannon, 2012) There is an in nite-dimensional graded
M24 -module
whose
McKay-Thompson series are speci c (vector-valued) mock theta functions.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
16 / 48
Umbral Moonshine II
Umbral Moonshine Conjecture (ChengDuncanHarvey, 2012) For every Umbral Group
GL ,
there is an an in nite-dimensional graded
GL -module whose McKay-Thompson series are speci c (vector-valued) mock theta functions.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
17 / 48
Umbral Moonshine II
Umbral Moonshine Conjecture (ChengDuncanHarvey, 2012) For every Umbral Group
GL ,
there is an an in nite-dimensional graded
GL -module whose McKay-Thompson series are speci c (vector-valued) mock theta functions.
Theorem (DuncanGri nOno, 2015) The Umbral Moonshine conjecture is true.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
17 / 48
Thompson Moonshine
Conjecture (HarveyRayhaun, 2015) There is an in nite-dimensional graded T h-supermodule L + ⊕ W − has vanishing W = m≡0,1 (4) Wm , where Wm = Wm m m is even and vice versa, whose McKay-Thompson series X T[g] (τ ) = 2q −3 + str(g|Wm )q m
odd part if
m=0 m≡0,1 (4) are speci c weakly holomorphic weight
M. H. Mertens
(U. Köln)
1 2 modular forms.
O'Nan Moonshine
18 / 48
Thompson Moonshine
Conjecture (HarveyRayhaun, 2015) There is an in nite-dimensional graded T h-supermodule L + ⊕ W − has vanishing W = m≡0,1 (4) Wm , where Wm = Wm m m is even and vice versa, whose McKay-Thompson series X T[g] (τ ) = 2q −3 + str(g|Wm )q m
odd part if
m=0 m≡0,1 (4) are speci c weakly holomorphic weight
1 2 modular forms.
Theorem (Gri nM., 2016) The Thompson Moonshine Conjecture is true. Moreover, the occuring modular forms can be described systematically.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
18 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
19 / 48
Finite simple groups
Source: wikipedia M. H. Mertens
(U. Köln)
O'Nan Moonshine
20 / 48
The O'Nan group Some properties of
ON
ON
One of the six pariah groups.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
21 / 48
The O'Nan group Some properties of
ON
ON
One of the six pariah groups.
# ON = 460 815 505 920 = 29 · 34 · 5 · 73 · 11 · 19 · 31.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
21 / 48
The O'Nan group Some properties of
ON
ON
One of the six pariah groups.
# ON = 460 815 505 920 = 29 · 34 · 5 · 73 · 11 · 19 · 31. 30
conjugacy classes, hence
with characters
M. H. Mertens
(U. Köln)
30
irreducible representations (over
C)
χ1 , ..., χ30 .
O'Nan Moonshine
21 / 48
The O'Nan group Some properties of
ON
ON
One of the six pariah groups.
# ON = 460 815 505 920 = 29 · 34 · 5 · 73 · 11 · 19 · 31. 30
conjugacy classes, hence
with characters
30
irreducible representations (over
C)
χ1 , ..., χ30 .
Dimensions of irreducible representations:
χ1 (1) = 1,
M. H. Mertens
χ7 (1) = 26 752,
(U. Köln)
χ12 (1) = 58 311,
O'Nan Moonshine
χ18 (1) = 85 064.
21 / 48
The O'Nan group Some properties of
ON
ON
One of the six pariah groups.
# ON = 460 815 505 920 = 29 · 34 · 5 · 73 · 11 · 19 · 31. 30
conjugacy classes, hence
with characters
30
irreducible representations (over
C)
χ1 , ..., χ30 .
Dimensions of irreducible representations:
χ1 (1) = 1,
χ7 (1) = 26 752,
Zagier's basis of weight
− g4 (τ ) =
∞ X
3/2
χ12 (1) = 58 311,
χ18 (1) = 85 064.
forms:
an q n
n=−4
=−q
−4
!,+ + 2 + 26 752q 3 + 143 376q 4 + 8 288 256q 7 + O(q 8 ) ∈ M3/2 (Γ0 (4)).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
21 / 48
The O'Nan group Some properties of
ON
ON
One of the six pariah groups.
# ON = 460 815 505 920 = 29 · 34 · 5 · 73 · 11 · 19 · 31. 30
conjugacy classes, hence
with characters
30
irreducible representations (over
C)
χ1 , ..., χ30 .
Dimensions of irreducible representations:
χ1 (1) = 1,
χ7 (1) = 26 752,
Zagier's basis of weight
− g4 (τ ) =
∞ X
3/2
χ12 (1) = 58 311,
χ18 (1) = 85 064.
forms:
an q n
n=−4
=−q
−4
!,+ + 2 + 26 752q 3 + 143 376q 4 + 8 288 256q 7 + O(q 8 ) ∈ M3/2 (Γ0 (4)).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
21 / 48
The O'Nan group Some properties of
ON
ON
One of the six pariah groups.
# ON = 460 815 505 920 = 29 · 34 · 5 · 73 · 11 · 19 · 31. 30
conjugacy classes, hence
with characters
30
irreducible representations (over
C)
χ1 , ..., χ30 .
Dimensions of irreducible representations:
χ1 (1) = 1,
χ7 (1) = 26 752,
Zagier's basis of weight
− g4 (τ ) =
∞ X
3/2
χ12 (1) = 58 311,
χ18 (1) = 85 064.
forms:
an q n
n=−4
=−q
−4
!,+ + 2 + 26 752q 3 + 143 376q 4 + 8 288 256q 7 + O(q 8 ) ∈ M3/2 (Γ0 (4)).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
21 / 48
O'Nan Moonshine Theorem 1 (Duncan-M.-Ono, 2017) There is a (virtual) in nite-dimensional graded
W :=
M 0<m≡0,3
ON-module Wm
(mod 4)
whose associated McKay-Thompson series are speci c weight
3 2 (mock)
modular forms.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
22 / 48
O'Nan Moonshine Theorem 1 (Duncan-M.-Ono, 2017) There is a (virtual) in nite-dimensional graded
W :=
M 0<m≡0,3
ON-module Wm
(mod 4)
whose associated McKay-Thompson series are speci c weight
3 2 (mock)
modular forms.
Remark We have that
1 dim W163 = (α2 + α − 393768), 2
where
l √ m α = eπ 163 = d262537412640768743.999999999999642...e. M. H. Mertens
(U. Köln)
O'Nan Moonshine
22 / 48
The relevant forms Proposition 1 (Duncan-M.-Ono, 2017) The following are true. 1
For every conjugacy class
[g]
of
ON
there is a unique mock modular
form
F[g] (τ ) = −q −4 + 2 +
X
a[g] (n)q n
n=1 of weight
3/2
for the group
Γ0 (4o(g))
satisfying the following
conditions:
M. H. Mertens
(U. Köln)
O'Nan Moonshine
23 / 48
The relevant forms Proposition 1 (Duncan-M.-Ono, 2017) The following are true. 1
For every conjugacy class
[g]
of
ON
there is a unique mock modular
form
F[g] (τ ) = −q −4 + 2 +
X
a[g] (n)q n
n=1 of weight
3/2
for the group
Γ0 (4o(g))
satisfying the following
conditions: 1
F[g] (τ ) lies (mod 4).
M. H. Mertens
(U. Köln)
in the Kohnen plus space, i.e.,
O'Nan Moonshine
a[g] (n) = 0
if
n ≡ 1, 2
23 / 48
The relevant forms Proposition 1 (Duncan-M.-Ono, 2017) The following are true. 1
For every conjugacy class
[g]
of
ON
there is a unique mock modular
form
F[g] (τ ) = −q −4 + 2 +
X
a[g] (n)q n
n=1 of weight
3/2
for the group
Γ0 (4o(g))
satisfying the following
conditions: 1
2
F[g] (τ ) lies in the Kohnen plus space, i.e., a[g] (n) = 0 if n ≡ 1, 2 (mod 4). F[g] (τ ) has a pole of order 4 at the cusp ∞ and vanishes at essentially all other cusps.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
23 / 48
The relevant forms Proposition 1 (Duncan-M.-Ono, 2017) The following are true. 1
For every conjugacy class
[g]
of
ON
there is a unique mock modular
form
F[g] (τ ) = −q −4 + 2 +
X
a[g] (n)q n
n=1 of weight
3/2
for the group
Γ0 (4o(g))
satisfying the following
conditions: 1
2
F[g] (τ ) lies in the Kohnen plus space, i.e., a[g] (n) = 0 if n ≡ 1, 2 (mod 4). F[g] (τ ) has a pole of order 4 at the cusp ∞ and vanishes at essentially all other cusps.
3
a[g] (3) = χ7 (g), and a[g] (4) = χ1 (g) + χ12 (g) + χ18 (g), a[g] (7) = more complicated . We have
M. H. Mertens
(U. Köln)
O'Nan Moonshine
and
23 / 48
The relevant forms Proposition 1 (Duncan-M.-Ono, 2017) The following are true. 1
For every conjugacy class
[g]
of
ON
there is a unique mock modular
form
F[g] (τ ) = −q −4 + 2 +
X
a[g] (n)q n
n=1 of weight
3/2
for the group
Γ0 (4o(g))
satisfying the following
conditions: 1
2
F[g] (τ ) lies in the Kohnen plus space, i.e., a[g] (n) = 0 if n ≡ 1, 2 (mod 4). F[g] (τ ) has a pole of order 4 at the cusp ∞ and vanishes at essentially all other cusps.
3
2
a[g] (3) = χ7 (g), and a[g] (4) = χ1 (g) + χ12 (g) + χ18 (g), a[g] (7) = more complicated . We have
The function
F[g] (τ ) above has integer Fourier o(g) 6= 16, then it is modular.
and
coe cients.
Furthermore, if M. H. Mertens
(U. Köln)
O'Nan Moonshine
23 / 48
Strategy of the proof Take
F[g] (τ ) = −q
−4
+2+
∞ X
a[g] (n)q n
n=1
M. H. Mertens
(U. Köln)
O'Nan Moonshine
24 / 48
Strategy of the proof Take
F[g] (τ ) = −q
−4
+2+
∞ X
a[g] (n)q n
n=1 De ne
C-valued
class function
ωn : ON → C, g 7→ a[g] (n).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
24 / 48
Strategy of the proof Take
F[g] (τ ) = −q
−4
+2+
∞ X
a[g] (n)q n
n=1 De ne
C-valued
class function
ωn : ON → C, g 7→ a[g] (n). Theorem 1 follows if we can show that
ωn =
30 X
mj (n)χj ,
j=1 with
mj (n) ∈ N0
M. H. Mertens
(U. Köln)
for all
j
and (su ciently large)
O'Nan Moonshine
n.
24 / 48
Strategy of the proof Take
F[g] (τ ) = −q
−4
+2+
∞ X
a[g] (n)q n
n=1 De ne
C-valued
class function
ωn : ON → C, g 7→ a[g] (n). Theorem 1 follows if we can show that
ωn =
30 X
mj (n)χj ,
j=1 with
mj (n) ∈ N0
for all
j
and (su ciently large)
n.
Idea of Thompson can reduce this to a nite computation.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
24 / 48
Strategy of the proof Take
F[g] (τ ) = −q
−4
+2+
∞ X
a[g] (n)q n
n=1 De ne
C-valued
class function
ωn : ON → C, g 7→ a[g] (n). Theorem 1 follows if we can show that
ωn =
30 X
mj (n)χj ,
j=1 with
mj (n) ∈ N0
for all
j
and (su ciently large)
n.
Idea of Thompson can reduce this to a nite computation.
BUT: There is a dierence between ' nite' and 'feasible'. M. H. Mertens
(U. Köln)
O'Nan Moonshine
24 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
25 / 48
Construction
Let
a b 2 ΓK,K 2 (N ) := ∈ Γ0 (N ) : |c| < K, |d| < K . c d
M. H. Mertens
(U. Köln)
O'Nan Moonshine
26 / 48
Construction
Let
a b 2 ΓK,K 2 (N ) := ∈ Γ0 (N ) : |c| < K, |d| < K . c d
De nition For
µ ∈ Z, k ∈ 12 Z,
and
ψ
a multiplier system for
Γ0 (N )
of weight
k,
we
de ne the Rademacher sum
[µ]
Rψ,k (τ ) := lim
X
K→∞
M. H. Mertens
(U. Köln)
ψ(γ)(q µ |k γ).
γ∈Γ∞ \ΓK,K 2 (N )
O'Nan Moonshine
26 / 48
Construction
Let
a b 2 ΓK,K 2 (N ) := ∈ Γ0 (N ) : |c| < K, |d| < K . c d
De nition For
µ ∈ Z, k ∈ 12 Z,
and
ψ
a multiplier system for
Γ0 (N )
of weight
k,
we
de ne the Rademacher sum
[µ]
Rψ,k (τ ) := lim
X
K→∞
ψ(γ)(q µ |k γ).
γ∈Γ∞ \ΓK,K 2 (N )
Low-weight analogue of Poincaré series.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
26 / 48
Construction
Let
a b 2 ΓK,K 2 (N ) := ∈ Γ0 (N ) : |c| < K, |d| < K . c d
De nition For
µ ∈ Z, k ∈ 12 Z,
and
ψ
a multiplier system for
Γ0 (N )
of weight
k,
we
de ne the Rademacher sum
[µ]
Rψ,k (τ ) := lim
X
K→∞
ψ(γ)(q µ |k γ).
γ∈Γ∞ \ΓK,K 2 (N )
Low-weight analogue of Poincaré series. Converges for
M. H. Mertens
k ≥ 1,
(U. Köln)
with regularization sometimes for
O'Nan Moonshine
k < 1.
26 / 48
Properties of Rademacher sums Facts Let
µ < 0. [µ]
Rψ,k
is a weight
k
mock modular form for
[−µ] with shadow R ψ,2−k
M. H. Mertens
(U. Köln)
Γ0 (N )
with multiplier
ψ
∈ M2−k (Γ0 (N ), ψ).
O'Nan Moonshine
27 / 48
Properties of Rademacher sums Facts Let
µ < 0. [µ]
Rψ,k
k mock modular form for Γ0 (N ) with multiplier ψ [−µ] with shadow R ∈ M2−k (Γ0 (N ), ψ). ψ,2−k [µ] Rψ,k has a pole of order |µ| at ∞ and vanishes at all other cusps. is a weight
M. H. Mertens
(U. Köln)
O'Nan Moonshine
27 / 48
Properties of Rademacher sums Facts Let
µ < 0. [µ]
Rψ,k
k mock modular form for Γ0 (N ) with multiplier ψ [−µ] with shadow R ∈ M2−k (Γ0 (N ), ψ). ψ,2−k [µ] Rψ,k has a pole of order |µ| at ∞ and vanishes at all other cusps. is a weight
De nition
f : H → C is called a mock modular form for Γ0 (N ) of multiplier ψ if there is a modular form g ∈ M2−k (Γ0 (N ), ψ)
A hol. function weight
k
and
s.t.
Z
∞
fb(τ ) := f (τ ) + −τ transforms like a modular form.
g
g(z) dz (z + τ )k
is called the shadow of
f , fb is
the
corresponding harmonic Maaÿ form. M. H. Mertens
(U. Köln)
O'Nan Moonshine
27 / 48
Construction of
F[g]
Proof of Proposition 1. Let
[µ]
[µ]
2
2
Z 3 ,ψ = R 3 ,ψ | pr, the projection to the Kohnen plus space.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
28 / 48
Construction of
F[g]
Proof of Proposition 1. Let
[µ]
[µ]
2
2
Z 3 ,ψ = R 3 ,ψ | pr, the projection to the Kohnen plus space.
[−4]
[0]
2
2
−Z 3 ,1 + 2Z 3 ,1
M. H. Mertens
(U. Köln)
satis es conditions 1 and 2 of Proposition 1
O'Nan Moonshine
28 / 48
Construction of
F[g]
Proof of Proposition 1. Let
[µ]
[µ]
2
2
Z 3 ,ψ = R 3 ,ψ | pr, the projection to the Kohnen plus space.
[−4]
[0]
2
2
−Z 3 ,1 + 2Z 3 ,1
satis es conditions 1 and 2 of Proposition 1
Bruinier-Funke pairing yields that 1 and 2 determine a mock modular form uniquely up to addition of cusp forms, so choose cusp forms where possible to ensure 3.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
28 / 48
Construction of
F[g]
Proof of Proposition 1. Let
[µ]
[µ]
2
2
Z 3 ,ψ = R 3 ,ψ | pr, the projection to the Kohnen plus space.
[−4]
[0]
2
2
−Z 3 ,1 + 2Z 3 ,1
satis es conditions 1 and 2 of Proposition 1
Bruinier-Funke pairing yields that 1 and 2 determine a mock modular form uniquely up to addition of cusp forms, so choose cusp forms where possible to ensure 3. Check that coe cients are integers and that (almost) all are modular using again Bruinier-Funke.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
28 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
29 / 48
Integrality of multiplicities I
Let
Fχj (τ ) :=
X 1 χj (g)F[g] (τ ) # ON g∈ON
M. H. Mertens
(U. Köln)
O'Nan Moonshine
30 / 48
Integrality of multiplicities I
Let
Fχj (τ ) :=
X X 1 Schur mj (n)q n . χj (g)F[g] (τ ) = −q −4 + 2 + # ON n=1
g∈ON
Proposition 1 yields: level
Nχj
M. H. Mertens
Fχj
is a mock modular form of weight
3/2
of
with rational coe cients and controllable shadow
(U. Köln)
O'Nan Moonshine
30 / 48
Integrality of multiplicities I
Let
Fχj (τ ) :=
X X 1 Schur mj (n)q n . χj (g)F[g] (τ ) = −q −4 + 2 + # ON n=1
g∈ON
Proposition 1 yields: level
Nχj
Fχj
is a mock modular form of weight
3/2
of
with rational coe cients and controllable shadow
Checking integrality naively by Sturm bound not feasible (Nχ1
= 10 884 720)
M. H. Mertens
(U. Köln)
O'Nan Moonshine
30 / 48
Integrality of the multiplicities II
Proposition 2
Fχj
have all integer Fourier coe cients.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
31 / 48
Integrality of the multiplicities II
Proposition 2
Fχj
have all integer Fourier coe cients.
Proof. The
F[g]
satisfy numerous congruences modulo powers of
(proved by Sturm bound argument,
M. H. Mertens
(U. Köln)
< 250
O'Nan Moonshine
p | # ON
coe cients to be checked).
31 / 48
Integrality of the multiplicities II
Proposition 2
Fχj
have all integer Fourier coe cients.
Proof. The
F[g]
satisfy numerous congruences modulo powers of
(proved by Sturm bound argument, One can then verify directly that
< 250
mj (n)
p | # ON
coe cients to be checked).
are
p-integral
for all
p | # ON,
hence by Proposition 1, the claim follows.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
31 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
32 / 48
Basic strategy
Fact Given convergence, Rademacher sums have a Fourier expansion whose coe cients are given in terms of in nite sums of Kloosterman sums
Kψ (m, n, c) =
md+nd ∗ ∗ ψ e2πi c c d ∗
X d (c)
times
I -Bessel
M. H. Mertens
functions.
(U. Köln)
O'Nan Moonshine
33 / 48
Basic strategy
Fact Given convergence, Rademacher sums have a Fourier expansion whose coe cients are given in terms of in nite sums of Kloosterman sums
Kψ (m, n, c) =
md+nd ∗ ∗ ψ e2πi c c d ∗
X d (c)
times
I -Bessel
functions.
By the triangle inequality we have
mj (n) ≥
X | str(g|Wn )| | str(1|Wn )| − |χj (g)|. # ON #CON (g) [g]6=1A
M. H. Mertens
(U. Köln)
O'Nan Moonshine
33 / 48
Basic strategy
Fact Given convergence, Rademacher sums have a Fourier expansion whose coe cients are given in terms of in nite sums of Kloosterman sums
Kψ (m, n, c) =
md+nd ∗ ∗ ψ e2πi c c d ∗
X d (c)
times
I -Bessel
functions.
By the triangle inequality we have
mj (n) ≥
X | str(g|Wn )| | str(1|Wn )| − |χj (g)|. # ON #CON (g) [g]6=1A
Show that from a certain point on, the rst term dominates. M. H. Mertens
(U. Köln)
O'Nan Moonshine
33 / 48
Positivity of the multiplicities Proposition 3 The multiplicities
M. H. Mertens
mj (n)
(U. Köln)
are all non-negative for
O'Nan Moonshine
n 6= 7, 8, 12.
34 / 48
Positivity of the multiplicities Proposition 3 The multiplicities
mj (n)
are all non-negative for
n 6= 7, 8, 12.
Ingredients of the proof. Careful, explicit estimates for Selberg-Kloosterman zeta functions.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
34 / 48
Positivity of the multiplicities Proposition 3 The multiplicities
mj (n)
are all non-negative for
n 6= 7, 8, 12.
Ingredients of the proof. Careful, explicit estimates for Selberg-Kloosterman zeta functions. Write Kloosterman sums as sums over a sparse set, i.e. equivalence classes of binary quadratic forms.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
34 / 48
Positivity of the multiplicities Proposition 3 The multiplicities
mj (n)
are all non-negative for
n 6= 7, 8, 12.
Ingredients of the proof. Careful, explicit estimates for Selberg-Kloosterman zeta functions. Write Kloosterman sums as sums over a sparse set, i.e. equivalence classes of binary quadratic forms. Explicit estimates for coe cients of weight 3/2 cusp forms
⇒
M. H. Mertens
(U. Köln)
mj (n) ≥ 0
O'Nan Moonshine
for
n ≥ 109.
34 / 48
Positivity of the multiplicities Proposition 3 The multiplicities
mj (n)
are all non-negative for
n 6= 7, 8, 12.
Ingredients of the proof. Careful, explicit estimates for Selberg-Kloosterman zeta functions. Write Kloosterman sums as sums over a sparse set, i.e. equivalence classes of binary quadratic forms. Explicit estimates for coe cients of weight 3/2 cusp forms
⇒
mj (n) ≥ 0
for
n ≥ 109.
Check rest by inspection.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
34 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
35 / 48
Singular moduli Fact
Q = [a, b, c] be a quadratic form of discriminant D < 0 and τQ ∈ H, 2 such that aτQ + bτQ + c = 0. Then J(τQ ) is a real-algebraic integer of degree h(D).
Let
E.g.:
√
−163 = −262 537 412 640 768 744 J 2 √ √ 1 + −15 192 513 + 85 995 5 J =− . 2 2
M. H. Mertens
1+
(U. Köln)
O'Nan Moonshine
36 / 48
Singular moduli Fact
Q = [a, b, c] be a quadratic form of discriminant D < 0 and τQ ∈ H, 2 such that aτQ + bτQ + c = 0. Then J(τQ ) is a real-algebraic integer of degree h(D).
Let
E.g.:
√
−163 = −262 537 412 640 768 744 J 2 √ √ 1 + −15 192 513 + 85 995 5 J =− . 2 2
1+
Play an important role in explicit class eld theory (Kronecker's
th problem)
Jugendtraum, Hilbert's 12
M. H. Mertens
(U. Köln)
O'Nan Moonshine
36 / 48
Singular moduli Fact
Q = [a, b, c] be a quadratic form of discriminant D < 0 and τQ ∈ H, 2 such that aτQ + bτQ + c = 0. Then J(τQ ) is a real-algebraic integer of degree h(D).
Let
E.g.:
√
−163 = −262 537 412 640 768 744 J 2 √ √ 1 + −15 192 513 + 85 995 5 J =− . 2 2
1+
Play an important role in explicit class eld theory (Kronecker's
th problem)
Jugendtraum, Hilbert's 12
Similar results are true for Hauptmoduln of genus 0 congruence subgroups and other modular functions M. H. Mertens
(U. Köln)
O'Nan Moonshine
36 / 48
Traces
De nition For a function
f : H → C,
a discriminant
(N )
−D < 0
N ∈N
de ne
f (τQ ) , ω (N ) (Q)
X
TrD (f ) :=
and
(N )
Q∈Q−D /Γ0 (N ) where
(N )
Q−D = {[a, b, c] : b2 − 4ac = −D ω (N ) (Q)
M. H. Mertens
=
1 2
and
N | a},
· # StabΓ0 (N ) (Q).
(U. Köln)
O'Nan Moonshine
37 / 48
Generating functions Theorem (D. Zagier)
− q −1 + 2 +
X
(1)
TrD (J)q D
D≡0,3 (4)
=−q
−1
M. H. Mertens
!,+ + 2 − 248q 3 + 492q 4 − 4119q 7 + O(q 8 ) ∈ M3/2 (Γ0 (4)).
(U. Köln)
O'Nan Moonshine
38 / 48
Generating functions Theorem (D. Zagier)
− q −1 + 2 +
X
(1)
TrD (J)q D
D≡0,3 (4)
=−q
−1
!,+ + 2 − 248q 3 + 492q 4 − 4119q 7 + O(q 8 ) ∈ M3/2 (Γ0 (4)).
Can be extended to more general modular functions with vanishing constant terms (Bruinier-Funke, Miller-Pixton,...)
M. H. Mertens
(U. Köln)
O'Nan Moonshine
38 / 48
Generating functions Theorem (D. Zagier)
X
− q −1 + 2 +
(1)
TrD (J)q D
D≡0,3 (4)
=−q
−1
!,+ + 2 − 248q 3 + 492q 4 − 4119q 7 + O(q 8 ) ∈ M3/2 (Γ0 (4)).
Can be extended to more general modular functions with vanishing constant terms (Bruinier-Funke, Miller-Pixton,...)
De nition/Theorem For
N ∈ N,
we call
(N )
H (N ) (D) := TrD (1)
number and set
the generalized Hurwitz class
[Γ(1) : Γ0 (N )] X (N ) + H (D)q D . 12 D mock modular form of level 4N .
H (N ) (τ ) := − H (N )
is a weight 3/2
M. H. Mertens
(U. Köln)
O'Nan Moonshine
38 / 48
Traces and Rademacher series Proposition 5 Let
N ∈N
such that
(N )
X0 (N )
Tr4 (D) := where
J (N )
has genus
0
and
1 (N ) (N ) (N/d) TrD (J2 ) − TrD (J (N/d) ) , 2
denotes the Hauptmodul for
the unique modular function for
Γ0 (N )
Γ0 (N )
and
(N )
J2
= q −2 + O(q)
is
with this Fourier expansion at
d := gcd(N, 2). Then X (N ) T (N ) (τ ) := −q −4 + const + Tr4 (D)q D
in nity and no poles anywhere else and
we have
D>0
=
[−4],+ R 3 ,4o(g) (τ ) 2
for some rational numbers
c1
and
c2 c1 − H (N ) (τ ) + H (N/d) (τ ) 2 2 c2 .
In particular, the function
T (N )
has
integer Fourier coe cients. M. H. Mertens
(U. Köln)
O'Nan Moonshine
39 / 48
McKay-Thompson series and traces
N where genus(X0+ (N )) = 0.
There is an analogue of Proposition 5 for
genus(X0 (N )) > 0,
M. H. Mertens
(U. Köln)
but
O'Nan Moonshine
40 / 48
McKay-Thompson series and traces
N where genus(X0+ (N )) = 0.
There is an analogue of Proposition 5 for
genus(X0 (N )) > 0, ⇒
but
We can express character values of
ON
in terms of traces of
singular moduli, generalized class numbers and coe cients of cusp forms.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
40 / 48
McKay-Thompson series and traces
N where genus(X0+ (N )) = 0.
There is an analogue of Proposition 5 for
genus(X0 (N )) > 0, ⇒
but
We can express character values of
ON
in terms of traces of
singular moduli, generalized class numbers and coe cients of cusp forms.
Example
F1A = T (1) , F7AB = T (7) + 4H (1) − 4H (7) , 12 6 4 F11A = T (11,+) + H (1) − H (11) − G (11) , 5 5 5 where
+ G (11) (τ ) = q 3 − q 4 − q 11 + O(q 12 ) ∈ S3/2 (44).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
40 / 48
Table of Contents
1
Monstrous Moonshine Preliminaries A connection between the Monster and modular functions
2
Other Moonshine
3
O'Nan Moonshine Rademacher sums Integrality Positivity
4
Traces of singular moduli
5
Arithmetic applications
M. H. Mertens
(U. Köln)
O'Nan Moonshine
41 / 48
ON
knows
p-parts
in ideal class groups
Theorem 2 (DuncanM.Ono, 2017) Suppose that
−D < 0
is a fundamental discriminant. Then the following
are true:
M. H. Mertens
(U. Köln)
O'Nan Moonshine
42 / 48
ON
knows
p-parts
in ideal class groups
Theorem 2 (DuncanM.Ono, 2017) Suppose that
−D < 0
is a fundamental discriminant. Then the following
are true: 1
If
−D < −8
is even and
g2 ∈ ON
has order
2,
dim WD ≡ tr(g2 |WD ) ≡ −24H(D)
M. H. Mertens
(U. Köln)
O'Nan Moonshine
then
(mod 24 ).
42 / 48
ON
knows
p-parts
in ideal class groups
Theorem 2 (DuncanM.Ono, 2017) Suppose that
−D < 0
is a fundamental discriminant. Then the following
are true: 1
If
−D < −8
is even and
g2 ∈ ON
has order
2,
dim WD ≡ tr(g2 |WD ) ≡ −24H(D)
2
If
p ∈ {3, 5, 7},
−D p
= −1
and
gp ∈ ON
then
(mod 24 ).
has order
( −24H(D) (mod 32 ) dim WD ≡ tr(gp |WD ) ≡ −24H(D) (mod p)
M. H. Mertens
(U. Köln)
O'Nan Moonshine
p,
then
if if
p = 3, p = 5, 7.
42 / 48
The BSD-conjecture and Waldspurger's theorem I Conjecture (Birch and Swinnerton-Dyer) Let
E/Q
be an elliptic curve. Then we have that
Q #X(E) · Reg(E) ` c` (E) L(r) (E, 1) = , r!ΩE (#E(Q)tors )2 where
r
denotes the order of vanishing of
the MordellWeil rank of
M. H. Mertens
(U. Köln)
L(E, s)
at
s = 1,
which equals
E.
O'Nan Moonshine
43 / 48
The BSD-conjecture and Waldspurger's theorem I Conjecture (Birch and Swinnerton-Dyer) Let
E/Q
be an elliptic curve. Then we have that
Q #X(E) · Reg(E) ` c` (E) L(r) (E, 1) = , r!ΩE (#E(Q)tors )2 where
r
denotes the order of vanishing of
the MordellWeil rank of
L(E, s)
at
s = 1,
which equals
E.
Theorem (Waldspurger, Kohnen) Let
N ∈N
be odd and square-free,
F ∈ S2k (Γ0 (N ))
be a newform and
2
the image of
f
under the Shimura correspondence. For a
suitable fundamental discriminant
hf, f i = M. H. Mertens
+ f ∈ Sk+ 1 (Γ0 (4N ))
(U. Köln)
D
we have
hF, F iπ k 1
2ω(N ) (k − 1)!|D|k− 2 L(F, D; k) O'Nan Moonshine
· |bf (|D|)|2 . 43 / 48
Quadratic twists
Connection through Modularity Theorem:
Lemma Let
−D < 0
be a suitable fundamental discriminant and
curve of odd, square-free conductor
N.
E/Q an elliptic 2 newform
Denote the weight
) and its Shintani lift by P E by FE n∈ S2 (N + fE (τ ) = ∞ b (n)q ∈ S n=3 E 3/2 (4N ). If E(−D) denotes twist of E by −D , we have associated to
the quadratic
L(E(−D), 1) = CE · |bE (D)|2 , ΩE(−D) where
CE
is a constant depending on
M. H. Mertens
(U. Köln)
E,
but not (really) on
O'Nan Moonshine
D.
44 / 48
ON
knows about Selmer and Tate-Shafarevich groups I
Theorem 3 (DuncanM.Ono, 2017)
= 11 or 19 and −D < 0 is a fundamental discriminant = −1, and gp ∈ ON has order p, then the following are
Assume BSD. If p for which
−D p
true.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
45 / 48
ON
knows about Selmer and Tate-Shafarevich groups I
Theorem 3 (DuncanM.Ono, 2017)
= 11 or 19 and −D < 0 is a fundamental discriminant = −1, and gp ∈ ON has order p, then the following are
Assume BSD. If p for which
−D p
true. 1
Sel(Ep (−D))[p] 6= {0}
if and only if
dim WD ≡ tr(gp |WD ) ≡ −24H(D)
M. H. Mertens
(U. Köln)
O'Nan Moonshine
(mod p).
45 / 48
ON
knows about Selmer and Tate-Shafarevich groups I
Theorem 3 (DuncanM.Ono, 2017)
= 11 or 19 and −D < 0 is a fundamental discriminant = −1, and gp ∈ ON has order p, then the following are
Assume BSD. If p for which
−D p
true. 1
Sel(Ep (−D))[p] 6= {0}
if and only if
dim WD ≡ tr(gp |WD ) ≡ −24H(D)
2
Suppose further that
L(Ep (−D), 1) 6= 0.
Then
(mod p).
p | #X(Ep (−D))
if
and only if
dim WD ≡ tr(gp |WD ) ≡ −24H(D) (mod p).
M. H. Mertens
(U. Köln)
O'Nan Moonshine
45 / 48
ON
knows about Selmer and Tate-Shafarevich groups II
Theorem 4 (DuncanM.Ono, 2017) Let
N ∈ {14, 15}
−D < 0
and write
N = p0 p, p0 < p,
δp := p−1 2 . If −D = −1, then p
and let
is a fundamental discriminant for which
the
following are true.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
46 / 48
ON
knows about Selmer and Tate-Shafarevich groups II
Theorem 4 (DuncanM.Ono, 2017) Let
N ∈ {14, 15}
−D < 0
and write
N = p0 p, p0 < p,
δp := p−1 2 . If −D = −1, then p
and let
is a fundamental discriminant for which
the
following are true. 1
Sel(EN (−D))[p] 6= {0}
if and only if 0
tr(gp0 |WD ) ≡ tr(gN |WD ) ≡ δp · (H(D) − δp H (p ) (D))
M. H. Mertens
(U. Köln)
O'Nan Moonshine
(mod p).
46 / 48
ON
knows about Selmer and Tate-Shafarevich groups II
Theorem 4 (DuncanM.Ono, 2017) Let
N ∈ {14, 15}
−D < 0
and write
N = p0 p, p0 < p,
δp := p−1 2 . If −D = −1, then p
and let
is a fundamental discriminant for which
the
following are true. 1
Sel(EN (−D))[p] 6= {0}
if and only if 0
tr(gp0 |WD ) ≡ tr(gN |WD ) ≡ δp · (H(D) − δp H (p ) (D))
2
(mod p).
L(EN (−D), 1) 6= 0. p | #X(EN (−D)) if and only if
Suppose further that Then
0
tr(gp0 |WD ) ≡ tr(gN |WD ) ≡ δp · (H(D) − δp H (p ) (D))
M. H. Mertens
(U. Köln)
O'Nan Moonshine
(mod p).
46 / 48
Example
D
tr2 (D)
H14 (D)
Diff 14 (D)
rk(E14 (−D))
#X(E14 (−D))
15
-96256
-30
3
0
1
23
-1746944
-45
0
2
1
39
-165...168
-60
4
0
1
71
-156...880
-105
4
0
1 1
79
-669...192
-75
3
0
239
-619...040
-225
0
2
1
2671
-163...664
-345
0
0
49
Table: Examples for the curve
H14 (D) := δ7 (H(D) − δ7 H (2) (D)),
E14
tr2 (D) := tr(g2 |WD ),
Diff 14 (D) := H14 (D) − tr2 (D) M. H. Mertens
(U. Köln)
O'Nan Moonshine
47 / 48
Thank you for your attention.
M. H. Mertens
(U. Köln)
O'Nan Moonshine
48 / 48
