Arithmetic Gauge Fields in Diophantine Geometry
Arithmetic Gauge Fields in Diophantine Geometry Minhyong Kim
Simons Foundation, March, 2018
I. Background: Arithmetic of Algebraic Curves
Arithmetic of algebraic curves
Arithmetic of algebraic curves X /Q, an algebraic curve of genus g . For example, given by a homogeneous equation f (x, y , z) = 0 of degree d with rational coefficients, where g = (d − 1)(d − 2)/2.
Arithmetic of algebraic curves X /Q, an algebraic curve of genus g . For example, given by a homogeneous equation f (x, y , z) = 0 of degree d with rational coefficients, where g = (d − 1)(d − 2)/2.
Interested in the set X (Q) of rational solutions.
Arithmetic of algebraic curves X /Q, an algebraic curve of genus g . For example, given by a homogeneous equation f (x, y , z) = 0 of degree d with rational coefficients, where g = (d − 1)(d − 2)/2.
Interested in the set X (Q) of rational solutions. Structure is quite different depending on the three cases: g = 0, spherical geometry (positive curvature); g = 1, flat geometry (zero curvature); g ≥ 2, hyperbolic geometry (negative curvature).
Arithmetic of algebraic curves
g = 0 (d=1,2)
Arithmetic of algebraic curves
g = 0 (d=1,2) X (Q) is empty (e.g., x 2 + y 2 + z 2 = 0); or X (Q) is infinite and can be ‘parametrised’ starting from one solution.
Arithmetic of algebraic curves
g = 0 (d=1,2) X (Q) is empty (e.g., x 2 + y 2 + z 2 = 0); or X (Q) is infinite and can be ‘parametrised’ starting from one solution. For example, for x 2 + y 2 − z 2 = 0, get solutions (t02 − t12 : 2t0 t1 : t02 + t12 ) starting from the solution (0 : 1 : 1).
Arithmetic of algebraic curves Also, have an algorithm to determine whether or not X (Q) 6= φ.
Arithmetic of algebraic curves Also, have an algorithm to determine whether or not X (Q) 6= φ. The Hasse principle : X (Q) 6= φ ⇔ X (Qv ) 6= φ, ∀v . Here, Qv is a completion of Q at some place v .
Arithmetic of algebraic curves Also, have an algorithm to determine whether or not X (Q) 6= φ. The Hasse principle : X (Q) 6= φ ⇔ X (Qv ) 6= φ, ∀v . Here, Qv is a completion of Q at some place v . Actually need only check at v = 2, v = ∞ and v dividing the coefficients of a defining equation.
Arithmetic of algebraic curves Also, have an algorithm to determine whether or not X (Q) 6= φ. The Hasse principle : X (Q) 6= φ ⇔ X (Qv ) 6= φ, ∀v . Here, Qv is a completion of Q at some place v . Actually need only check at v = 2, v = ∞ and v dividing the coefficients of a defining equation. For example, 37x 2 + 59y 2 − 67z 2 = 0 has a Q-solution if and only if it has a solution in each of R, Q2 , Q37 , Q59 , Q67 .
Arithmetic of algebraic curves Also, have an algorithm to determine whether or not X (Q) 6= φ. The Hasse principle : X (Q) 6= φ ⇔ X (Qv ) 6= φ, ∀v . Here, Qv is a completion of Q at some place v . Actually need only check at v = 2, v = ∞ and v dividing the coefficients of a defining equation. For example, 37x 2 + 59y 2 − 67z 2 = 0 has a Q-solution if and only if it has a solution in each of R, Q2 , Q37 , Q59 , Q67 . Question: Can the locus X (Q) ⊂
Y
X (Qv )
v
of rational points be somehow described?
Arithmetic of algebraic curves g = 1 (d=3).
Arithmetic of algebraic curves g = 1 (d=3). X (Q) = φ, non-empty finite, infinite, all are possible.
Arithmetic of algebraic curves g = 1 (d=3). X (Q) = φ, non-empty finite, infinite, all are possible. Hasse principle fails: 3x 3 + 4y 3 + 5z 3 = 0 has points in Qv for all v , but no rational points.
Arithmetic of algebraic curves g = 1 (d=3). X (Q) = φ, non-empty finite, infinite, all are possible. Hasse principle fails: 3x 3 + 4y 3 + 5z 3 = 0 has points in Qv for all v , but no rational points. If X (Q) 6= φ, then fixing a point O ∈ X (Q) gives X (Q) the structure of a finitely-generated abelian group: X (Q) ' X (Q)tor × Zr , for which O is the origin. Here, r is called the rank of the curve and X (Q)tor is a finite effectively computable abelian group.
Arithmetic of algebraic curves For example, for X := {y 2 = x 3 + ax + b} ∪ {∞} (a, b ∈ Z), (x, y ) ∈ X (Q)tor iff y 2 |(4a3 + 27b 2 ).
Arithmetic of algebraic curves For example, for X := {y 2 = x 3 + ax + b} ∪ {∞} (a, b ∈ Z), (x, y ) ∈ X (Q)tor iff y 2 |(4a3 + 27b 2 ).
A good deal more difficult fact (Mazur) is that the only possibilities for X (Q)tor are Z/NZ,
1 ≤ N ≤ 10 or N = 12;
Z/2Z × Z/2NZ, 1 ≤ N ≤ 4
Arithmetic of algebraic curves An outstanding problem is the algorithmic computation of the rank and a full set of generators for X (Q). This is the subject of the conjecture of Birch and Swinnerton-Dyer.
Arithmetic of algebraic curves An outstanding problem is the algorithmic computation of the rank and a full set of generators for X (Q). This is the subject of the conjecture of Birch and Swinnerton-Dyer. In practice, it is often possible to compute these. For example, for y 2 = x 3 − 2, Sage will give you r = 1 and the point (3, 5) as generator.
Arithmetic of algebraic curves An outstanding problem is the algorithmic computation of the rank and a full set of generators for X (Q). This is the subject of the conjecture of Birch and Swinnerton-Dyer. In practice, it is often possible to compute these. For example, for y 2 = x 3 − 2, Sage will give you r = 1 and the point (3, 5) as generator. Note that 2(3, 5) = (129/100, −383/1000) 3(3, 5) = (164323/29241, −66234835/5000211) 4(3, 5) = (2340922881/58675600, 113259286337279/449455096000)
Arithmetic of algebraic curves
g ≥ 2 (d ≥ 4)
Arithmetic of algebraic curves
g ≥ 2 (d ≥ 4) X (Q) is always finite (Faltings). However, *very* difficult to compute (cf. Fermat equations, ABC conjecure).
Arithmetic of algebraic curves
g ≥ 2 (d ≥ 4) X (Q) is always finite (Faltings). However, *very* difficult to compute (cf. Fermat equations, ABC conjecure). Effective Mordell conjecture: There is a terminating algorithm that takes X as input and produces X (Q) as output.
II. The Method of Principal Bundles
The method of principal bundles Given a field K with absolute Galois group GK = Gal(K¯ /K ), a sheaf of groups over K is a topological group R with a continuous action of GK by group automorphisms: GK × R
- R.
The method of principal bundles Given a field K with absolute Galois group GK = Gal(K¯ /K ), a sheaf of groups over K is a topological group R with a continuous action of GK by group automorphisms: GK × R
- R.
A principal R-bundle or R-torsor over K is a topological space P with compatible continuous actions of GK (left) and R (right, simply transitive): For g ∈ GK , z ∈ P, r ∈ R, g (zr ) = g (z)g (r ).
The method of principal bundles Choosing a point z ∈ P, for any g ∈ GK , there is a unique rg ∈ R such that g (z) = zrg .
The method of principal bundles Choosing a point z ∈ P, for any g ∈ GK , there is a unique rg ∈ R such that g (z) = zrg .
The map g 7→ rg is a 1-cocycle on GK with values in R, and defines a bijection Isomorphism classes of principal R-bundles over K ' H 1 (GK , R), where the last is continuous group cohomology.
The method of principal bundles Choosing a point z ∈ P, for any g ∈ GK , there is a unique rg ∈ R such that g (z) = zrg .
The map g 7→ rg is a 1-cocycle on GK with values in R, and defines a bijection Isomorphism classes of principal R-bundles over K ' H 1 (GK , R), where the last is continuous group cohomology. Note that P is trivial exactly when there is a fixed point z ∈ P GK .
The method of principal bundles
Example: ˆ Z(1) := lim µn , ← − where µn ⊂ K¯ is the group of n-th roots of 1.
The method of principal bundles
Example: ˆ Z(1) := lim µn , ← − where µn ⊂ K¯ is the group of n-th roots of 1. Thus, ˆ Z(1) = {(ζn )}, m = ζ . As a group, Z(1) ˆ ˆ but there is a where ζn ∈ µn and ζnm ' Z, n continuous action of GK .
The method of principal bundles ˆ Given any x ∈ K ∗ , get principal Z(1)-bundle n P(x) = {(yn )n | ynn = x, ynm = yn .}
over K .
The method of principal bundles ˆ Given any x ∈ K ∗ , get principal Z(1)-bundle n P(x) = {(yn )n | ynn = x, ynm = yn .}
over K . P(x) is trivial iff x admits an n-th root in K for all n.
The method of principal bundles ˆ Given any x ∈ K ∗ , get principal Z(1)-bundle n P(x) = {(yn )n | ynn = x, ynm = yn .}
over K . P(x) is trivial iff x admits an n-th root in K for all n. Recall that ˆ' Z
Y
Zp .
p
Corresponding to this, have variant Zp (1) := lim µpn ← − for a fixed prime p, and similarly for principalQbundles. These are ˆ studied much more extensively than Z(1) = p Zp (1).
The method of principal bundles When K = Q, there are completions Qv and injections ¯ ¯ v /Qv ) ⊂ - G = Gal(Q/Q). Gv = Gal(Q
The method of principal bundles When K = Q, there are completions Qv and injections ¯ ¯ v /Qv ) ⊂ - G = Gal(Q/Q). Gv = Gal(Q Given any Q-group R and principal R-bundle P, can restrict the G -action to a Gv -action for each v , giving a group and a principal bundle over Qv .
The method of principal bundles When K = Q, there are completions Qv and injections ¯ ¯ v /Qv ) ⊂ - G = Gal(Q/Q). Gv = Gal(Q Given any Q-group R and principal R-bundle P, can restrict the G -action to a Gv -action for each v , giving a group and a principal bundle over Qv . At the level of isomorphism classes, this gives rise to the localisation map Y H 1 (Gv , R). loc : H 1 (G , R) v
The method of principal bundles When K = Q, there are completions Qv and injections ¯ ¯ v /Qv ) ⊂ - G = Gal(Q/Q). Gv = Gal(Q Given any Q-group R and principal R-bundle P, can restrict the G -action to a Gv -action for each v , giving a group and a principal bundle over Qv . At the level of isomorphism classes, this gives rise to the localisation map Y H 1 (Gv , R). loc : H 1 (G , R) v
The Tate-Shafarevich group of R X(Q, R) is the kernel of loc. Thus, these are the principal R-bundles over Q that are ‘locally trivial’, that is, have a Gv -fixed point for each v .
The method of principal bundles: elliptic curves
The method of principal bundles: elliptic curves E , elliptic curve over Q.
The method of principal bundles: elliptic curves E , elliptic curve over Q. ¯ We let G = Gal(Q/Q) act on the exact sequence 0
- E [p]
- E
p
- E
- 0.
The method of principal bundles: elliptic curves E , elliptic curve over Q. ¯ We let G = Gal(Q/Q) act on the exact sequence 0
- E [p]
- E
p
- 0.
- E
Get long exact sequence 0
- E (Q)[p]
- H 1 (G , E [p])
p
- E (Q)
- E (Q) p
- H 1 (G , E )
- H 1 (G , E ),
from which we get the short exact sequence 0
- E (Q)/pE (Q)
⊂
- H 1 (G , E [p])
i
- H 1 (G , E )[p]
- 0.
The method of principal bundles: elliptic curves The central problem in the theory of elliptic curves is the identification of the image E (Q)/pE (Q) ⊂ - H 1 (G , E [p]).
The method of principal bundles: elliptic curves The central problem in the theory of elliptic curves is the identification of the image E (Q)/pE (Q) ⊂ - H 1 (G , E [p]).
To this end, define the p-Selmer group Sel(Q, E [p]) to be the inverse image of X(Q, E )[p] under the map i, so that it fits into an exact sequence 0
- E (Q)/pE (Q)
- Sel(Q, E [p])
- X(Q, E )[p]
- 0.
The method of principal bundles: elliptic curves
0- E (Q)/pE (Q) ⊂ - H 1 (G , E [p])
- H 1 (G , E )[p]
- 0
? ? ? 0 E (Qv )/pE (Qv ) ⊂- H 1 (Gv , E [p]) - H 1 (Gv , E )[p] - 0
The method of principal bundles: elliptic curves
0- E (Q)/pE (Q) ⊂ - H 1 (G , E [p])
- H 1 (G , E )[p]
- 0
? ? ? 0 E (Qv )/pE (Qv ) ⊂- H 1 (Gv , E [p]) - H 1 (Gv , E )[p] - 0
That is, Sel(Q, E [p]) consists of the E [p]-principal bundles that locally come from E (Qv ) for each v .
The method of principal bundles: elliptic curves The key point is that the p-Selmer group is a finite-dimensional Fp -vector space that is effectively computable, and this already gives us a bound on the Mordell-Weil group of E : E (Q)/pE (Q) ⊂ Sel(Q, E [p]).
The method of principal bundles: elliptic curves The key point is that the p-Selmer group is a finite-dimensional Fp -vector space that is effectively computable, and this already gives us a bound on the Mordell-Weil group of E : E (Q)/pE (Q) ⊂ Sel(Q, E [p]). This is then refined by way of the diagram 0
- E (Q)/p n E (Q) - Sel(Q, E [p n ]) - X(Q, E )[p n ]
- 0
0
? - E (Q)/pE (Q)
- 0
for increasing values of n.
? - Sel(Q, E [p])
? - X(Q, E )[p]
The method of principal bundles: elliptic curves
If we assume X is finite, we get lim X(Q, E )[p n ] = 0, ← − and hence, n Im(E (Q)/pE (Q)) = ∩∞ n=1 Im[Sel(Q, E [p ]] ⊂ Sel(Q, E [p]).
We get thereby, a cohomological expression for the group E (Q)/pE (Q) that can be used to compute its structure precisely.
The method of principal bundles: elliptic curves The idea is to compute the image Im[Sel(Q, E [p n ])] ⊂ Sel(Q, E [p]) for each n and compute in parallel the image of E (Q)≤n in Sel(Q, E [p]).
The method of principal bundles: elliptic curves The idea is to compute the image Im[Sel(Q, E [p n ])] ⊂ Sel(Q, E [p]) for each n and compute in parallel the image of E (Q)≤n in Sel(Q, E [p]). Here, E (Q)≤n consists of the point in E (Q) of height ≤ n. The height h(x, y ) of a point (x, y ) ∈ E (Q) is defined as follows. Write (x, y ) = (s/r , t/r ) for coprime integers s, t, r . Then h(x, y ) := log sup{|s|, |t|, |r |}.
The set E (Q)≤n is finite and can also be effectively computed.
The method of principal bundles: elliptic curves
Thus, we have an inclusion Im(E (Q)≤n ) ⊂ Im(Sel(Q, E [p n ]) ⊂ Sel(Q, E [p]). Assuming X is finite, we get Im(E (Q)≤n ) = Im(Sel(Q, E [p n ]) for n sufficiently large, at which point we can conclude that E (Q)/pE (Q) = Im(Sel(Q, E [p n ]).
The method of principal bundles: elliptic curves
This is a conditional algorithm for computing the rank, which is used by all the existing computer packages. With slightly more care, it also gives a set of generators for the group E (Q). In this sense, we eventually arrive at a conditional algorithm for ‘completely determining’ E (Q).
The method of principal bundles: elliptic curves
This is a conditional algorithm for computing the rank, which is used by all the existing computer packages. With slightly more care, it also gives a set of generators for the group E (Q). In this sense, we eventually arrive at a conditional algorithm for ‘completely determining’ E (Q). The essence of BSD is to remove the ‘conditional’ aspect.
III. The method of principal bundles II: The non-abelian case
The method of principal bundles II: The non-abelian case To generalise, focus on the sequence of maps ···
- E [p 3 ]
p
- E [p 2 ]
p
- E [p]
of which we take the inverse limit to get the p-adic Tate module of E: Tp E := lim E [p n ]. ← − This is a free Zp -module of rank 2.
The method of principal bundles II: The non-abelian case To generalise, focus on the sequence of maps ···
- E [p 3 ]
p
- E [p 2 ]
p
- E [p]
of which we take the inverse limit to get the p-adic Tate module of E: Tp E := lim E [p n ]. ← − This is a free Zp -module of rank 2. The previous finite Kummer theory maps can be packaged into E (Q)
- lim H 1 (G , E [p n ]) = H 1 (G , Tp E ).
← −
The method of principal bundles II: The non-abelian case The key abstraction is that Tp E ' π1p (E , O), where π1p refers to the pro-p fundamental group.
The method of principal bundles II: The non-abelian case The key abstraction is that Tp E ' π1p (E , O), where π1p refers to the pro-p fundamental group. The pro-finite completion of a group A is Aˆ := lim A/N, ← − where N ⊂ A runs over the normal subgroups of finite index.
The method of principal bundles II: The non-abelian case The key abstraction is that Tp E ' π1p (E , O), where π1p refers to the pro-p fundamental group. The pro-finite completion of a group A is Aˆ := lim A/N, ← − where N ⊂ A runs over the normal subgroups of finite index. The pro-p completion, is Ap := lim A/N ← − where N ⊂ A runs over normal subgroups of index a power of p.
The method of principal bundles II: The non-abelian case When A = π1 (X , b) for a variety X defined over Q and b ∈ X (Q), ¯ then both π ˆ1 (X , b) and π1p (X , b) admit actions of G = Gal(Q/Q).
The method of principal bundles II: The non-abelian case When A = π1 (X , b) for a variety X defined over Q and b ∈ X (Q), ¯ then both π ˆ1 (X , b) and π1p (X , b) admit actions of G = Gal(Q/Q). To see this, consider the universal pro-finite cover X˜ := (Xi
- X )i∈I ,
- X runs over a cofinal system of finite covering where Xi spaces of X .
The method of principal bundles II: The non-abelian case When A = π1 (X , b) for a variety X defined over Q and b ∈ X (Q), ¯ then both π ˆ1 (X , b) and π1p (X , b) admit actions of G = Gal(Q/Q). To see this, consider the universal pro-finite cover X˜ := (Xi
- X )i∈I ,
- X runs over a cofinal system of finite covering where Xi spaces of X .
Then the choice of a basepoint lift b˜ ∈ X˜ determines an isomorphism π ˆ1 (X , b) = X˜b from the fundamental group to the fibre of X˜ over b.
The method of principal bundles II: The non-abelian case
The rationality of b allows us to define the whole system over Q, and gives us the G -action on X˜b .
The method of principal bundles II: The non-abelian case
The rationality of b allows us to define the whole system over Q, and gives us the G -action on X˜b . Also, we get the profinite homotopy classes of paths π ˆ1 (X ; b, x) := X˜x .
The method of principal bundles II: The non-abelian case
The rationality of b allows us to define the whole system over Q, and gives us the G -action on X˜b . Also, we get the profinite homotopy classes of paths π ˆ1 (X ; b, x) := X˜x .
Get pro-p versions π1p (X , b) and π1p (X ; b, x) by considering coverings of p-power degree.
The method of principal bundles II: The non-abelian case
Examples: For X = Gm , g G m := (Gm
(·)n
- Gm ).
The method of principal bundles II: The non-abelian case
Examples: For X = Gm , g G m := (Gm
(·)n
- Gm ).
Thus, m ˆ π ˆ1 (Gm , 1) = {(ζn ) | ζmn = ζn } =: Z(1).
The method of principal bundles II: The non-abelian case
Examples: For X = Gm , g G m := (Gm
(·)n
- Gm ).
Thus, m ˆ π ˆ1 (Gm , 1) = {(ζn ) | ζmn = ζn } =: Z(1).
Similarly, π1p (Gm , 1) = Zp (1)
The method of principal bundles II: The non-abelian case
For X = E , E˜ = (E and π ˆ1 (E , O) ' (E
n
- E)
n
- E )O = Tˆ E
Also π1p (E , O) = Tp E .
The method of principal bundles II: The non-abelian case The map E (Q)
- H 1 (G , π p (E , O)) 1
is given by x 7→ [π1p (E ; O, x)] ' [(E
pn
- E )x ].
The method of principal bundles II: The non-abelian case The map - H 1 (G , π p (E , O)) 1
E (Q) is given by
x 7→ [π1p (E ; O, x)] ' [(E
pn
- E )x ].
For general X , this turns into j : X (Q)
- H 1 (G , π p (X , b)) 1
given by x 7→ π1p (X ; b, x)
The method of principal bundles II: The non-abelian case For each prime v , consider local versions jv : X (Qv )
- H 1 (Gv , π p (X , b)) 1
given by x 7→ π1p (X ; b, x) which are far more computable than the global map, together with the localisation map Y X (Qv ) X (Q) v
Q
j
?
?
H
1
(G , π1p (X , b))
v jv
-
Y v
H (Gv , π1p (X , b)) 1
The method of principal bundles II: The non-abelian case Actual applications use X (Q)
-
Y
X (Qv )
v
Q
j ? 1
H (G , U(X , b)) -
v jv
?
Y v
1
H (Gv , U(X , b))
The method of principal bundles II: The non-abelian case Actual applications use X (Q)
-
Y
X (Qv )
v
Q
j ? 1
H (G , U(X , b)) -
v jv
?
Y
1
H (Gv , U(X , b))
v
The lower row of this diagram is an algebraic map of algebraic varieties over Qp , which can be computed in principle.
The method of principal bundles II: The non-abelian case Actual applications use X (Q)
-
Y
X (Qv )
v
Q
j
v jv
?
? 1
H (G , U(X , b)) -
Y
1
H (Gv , U(X , b))
v
The lower row of this diagram is an algebraic map of algebraic varieties over Qp , which can be computed in principle. If α is an algebraic function vanishing on the image, then Y α◦ jv v
gives a defining equation for X (Q) inside
Q
v
X (Qv ).
The method of principal bundles II: The non-abelian case Here, U(X , b) is the Qp -pro-unipotent of π1p (X , b). The group U(X , b) is defined to be the group like elements in the completed group algebra lim Qp [π1p (X , b)]/I n , ← − where I ⊂ Qp [π1p (X , b)] is the augmentation ideal. Thus, U(X , b) contains elements like ga where g ∈ π1p (X , b) and a ∈ Qp . Key point is that U(X , b) is a pro-algebraic group.
The method of principal bundles II: The non-abelian case The construction U(Γ) makes sense for any group Γ: U(Γ) = “Γ ⊗ Qp ”
The method of principal bundles II: The non-abelian case The construction U(Γ) makes sense for any group Γ: U(Γ) = “Γ ⊗ Qp ” Examples:
The method of principal bundles II: The non-abelian case The construction U(Γ) makes sense for any group Γ: U(Γ) = “Γ ⊗ Qp ” Examples: U(Zn ) = Qnp .
The method of principal bundles II: The non-abelian case The construction U(Γ) makes sense for any group Γ: U(Γ) = “Γ ⊗ Qp ” Examples: U(Zn ) = Qnp .
1 a b Γ = {0 1 c | a, b, c ∈ Z} 0 0 1 Then
1 a b U(Γ) = {0 1 c | a, b, c ∈ Qp }. 0 0 1
The method of principal bundles II: The non-abelian case
To make this concretely computable, we take the projection Y prp : X (Qv ) - X (Qp ) v
and try to compute ∩α prp (Z (α ◦
Y v
jv )) ⊂ X (Qp ).
IV. Computing Rational Points
Computing rational points [Dan-Cohen, Wewers] For X = P1 \ {0, 1, ∞}, X (Z[1/2]) ⊂ {D2 (z) = 0} ∩ {D4 (z) = 0},
Computing rational points [Dan-Cohen, Wewers] For X = P1 \ {0, 1, ∞}, X (Z[1/2]) ⊂ {D2 (z) = 0} ∩ {D4 (z) = 0}, where D2 (z) = `2 (z) + (1/2) log(z) log(1 − z), D4 (z) = ζ(3)`4 (z) + (8/7)[log3 2/24 + `4 (1/2)/ log 2] log(z)`3 (z) +[(4/21)(log3 2/24 + `4 (1/2)/ log 2) + ζ(3)/24] log3 (z) log(1 − z), and `k (z) =
∞ X zn n=1
nk
.
Computing rational points [Dan-Cohen, Wewers] For X = P1 \ {0, 1, ∞}, X (Z[1/2]) ⊂ {D2 (z) = 0} ∩ {D4 (z) = 0}, where D2 (z) = `2 (z) + (1/2) log(z) log(1 − z), D4 (z) = ζ(3)`4 (z) + (8/7)[log3 2/24 + `4 (1/2)/ log 2] log(z)`3 (z) +[(4/21)(log3 2/24 + `4 (1/2)/ log 2) + ζ(3)/24] log3 (z) log(1 − z), and `k (z) =
∞ X zn n=1
nk
.
Numerically, this appears to be an equality.
Computing rational points
Some qualitative results: (Coates and K.) ax n + by n = cz n for n ≥ 4 has only finitely many rational points.
Computing rational points
Some qualitative results: (Coates and K.) ax n + by n = cz n for n ≥ 4 has only finitely many rational points. Standard structural conjectures on mixed motives ⇒Faltings theorem over Q.
Computing rational points A recent result on modular curves by Balakrishnan, Dogra, Mueller, Tuitmann, Vonk (arXiv 1711.05846, ‘Explicit Chabauty-Kim theory for the split modular curve of level 13’):
Computing rational points A recent result on modular curves by Balakrishnan, Dogra, Mueller, Tuitmann, Vonk (arXiv 1711.05846, ‘Explicit Chabauty-Kim theory for the split modular curve of level 13’): Xs+ (N) = X (N)/Cs+ (N), where X (N) the the compactification of the moduli space of pairs (E , φ : E [N] ' (Z/N)2 ), and Cs+ (N) ⊂ GL2 (Z/N) is the normaliser of a split Cartan subgroup.
Computing rational points A recent result on modular curves by Balakrishnan, Dogra, Mueller, Tuitmann, Vonk (arXiv 1711.05846, ‘Explicit Chabauty-Kim theory for the split modular curve of level 13’): Xs+ (N) = X (N)/Cs+ (N), where X (N) the the compactification of the moduli space of pairs (E , φ : E [N] ' (Z/N)2 ), and Cs+ (N) ⊂ GL2 (Z/N) is the normaliser of a split Cartan subgroup. Bilu-Parent-Rebolledo had shown that Xs+ (p)(Q) consists entirely of cusps and CM points for all primes p > 7, p 6= 13. They called p = 13 the ‘cursed level’.
Computing rational points
Theorem (BDMTV) The modular curve Xs+ (13) has exactly 7 rational points, consisting of the cusp and 6 CM points.
Computing rational points
Theorem (BDMTV) The modular curve Xs+ (13) has exactly 7 rational points, consisting of the cusp and 6 CM points. This concludes an important chapter of a conjecture of Serre: There is an absolute constant A such that G
- Aut(E [p])
is surjective for all non-CM elliptic curves E /Q and primes p > A.
V. Rational points and critical points
Rational points and critical points
Actually, interested in Im(H 1 (G , U)) ∩
Y v
Hf1 (Gv , U) ⊂
Y
H 1 (Gv , U),
v
where Hf1 (Gv , U) ⊂ H 1 (Gv , U) is a subvariety defined by some Hodge-theoretic conditions.
Rational points and critical points
Actually, interested in Im(H 1 (G , U)) ∩
Y v
Hf1 (Gv , U) ⊂
Y
H 1 (Gv , U),
v
where Hf1 (Gv , U) ⊂ H 1 (Gv , U) is a subvariety defined by some Hodge-theoretic conditions. In order to apply symplectic techniques, replace U by T ∗ (1)U := (LieU)∗ (1) o U.
Rational points and critical points Then Y
H 1 (Gv , T ∗ (1)U)
v
is a symplectic variety and Im(H 1 (G , T ∗ (1)U)),
Y v
are Lagrangian subvarieties.
Hf1 (Gv , T ∗ (1)U)
Rational points and critical points Then Y
H 1 (Gv , T ∗ (1)U)
v
is a symplectic variety and Im(H 1 (G , T ∗ (1)U)),
Y
Hf1 (Gv , T ∗ (1)U)
v
are Lagrangian subvarieties. Thus, the derived intersection Im(H 1 (G , U)) ∩
Y
Hf1 (Gv , U)
v
has a [−1]-shifted symplectic structure. Should be the critical set of a function.
VI. Concluding remark
Modularity is a mystery in number theory
Simons Foundation, March, 2018
I. Background: Arithmetic of Algebraic Curves
Arithmetic of algebraic curves
Arithmetic of algebraic curves X /Q, an algebraic curve of genus g . For example, given by a homogeneous equation f (x, y , z) = 0 of degree d with rational coefficients, where g = (d − 1)(d − 2)/2.
Arithmetic of algebraic curves X /Q, an algebraic curve of genus g . For example, given by a homogeneous equation f (x, y , z) = 0 of degree d with rational coefficients, where g = (d − 1)(d − 2)/2.
Interested in the set X (Q) of rational solutions.
Arithmetic of algebraic curves X /Q, an algebraic curve of genus g . For example, given by a homogeneous equation f (x, y , z) = 0 of degree d with rational coefficients, where g = (d − 1)(d − 2)/2.
Interested in the set X (Q) of rational solutions. Structure is quite different depending on the three cases: g = 0, spherical geometry (positive curvature); g = 1, flat geometry (zero curvature); g ≥ 2, hyperbolic geometry (negative curvature).
Arithmetic of algebraic curves
g = 0 (d=1,2)
Arithmetic of algebraic curves
g = 0 (d=1,2) X (Q) is empty (e.g., x 2 + y 2 + z 2 = 0); or X (Q) is infinite and can be ‘parametrised’ starting from one solution.
Arithmetic of algebraic curves
g = 0 (d=1,2) X (Q) is empty (e.g., x 2 + y 2 + z 2 = 0); or X (Q) is infinite and can be ‘parametrised’ starting from one solution. For example, for x 2 + y 2 − z 2 = 0, get solutions (t02 − t12 : 2t0 t1 : t02 + t12 ) starting from the solution (0 : 1 : 1).
Arithmetic of algebraic curves Also, have an algorithm to determine whether or not X (Q) 6= φ.
Arithmetic of algebraic curves Also, have an algorithm to determine whether or not X (Q) 6= φ. The Hasse principle : X (Q) 6= φ ⇔ X (Qv ) 6= φ, ∀v . Here, Qv is a completion of Q at some place v .
Arithmetic of algebraic curves Also, have an algorithm to determine whether or not X (Q) 6= φ. The Hasse principle : X (Q) 6= φ ⇔ X (Qv ) 6= φ, ∀v . Here, Qv is a completion of Q at some place v . Actually need only check at v = 2, v = ∞ and v dividing the coefficients of a defining equation.
Arithmetic of algebraic curves Also, have an algorithm to determine whether or not X (Q) 6= φ. The Hasse principle : X (Q) 6= φ ⇔ X (Qv ) 6= φ, ∀v . Here, Qv is a completion of Q at some place v . Actually need only check at v = 2, v = ∞ and v dividing the coefficients of a defining equation. For example, 37x 2 + 59y 2 − 67z 2 = 0 has a Q-solution if and only if it has a solution in each of R, Q2 , Q37 , Q59 , Q67 .
Arithmetic of algebraic curves Also, have an algorithm to determine whether or not X (Q) 6= φ. The Hasse principle : X (Q) 6= φ ⇔ X (Qv ) 6= φ, ∀v . Here, Qv is a completion of Q at some place v . Actually need only check at v = 2, v = ∞ and v dividing the coefficients of a defining equation. For example, 37x 2 + 59y 2 − 67z 2 = 0 has a Q-solution if and only if it has a solution in each of R, Q2 , Q37 , Q59 , Q67 . Question: Can the locus X (Q) ⊂
Y
X (Qv )
v
of rational points be somehow described?
Arithmetic of algebraic curves g = 1 (d=3).
Arithmetic of algebraic curves g = 1 (d=3). X (Q) = φ, non-empty finite, infinite, all are possible.
Arithmetic of algebraic curves g = 1 (d=3). X (Q) = φ, non-empty finite, infinite, all are possible. Hasse principle fails: 3x 3 + 4y 3 + 5z 3 = 0 has points in Qv for all v , but no rational points.
Arithmetic of algebraic curves g = 1 (d=3). X (Q) = φ, non-empty finite, infinite, all are possible. Hasse principle fails: 3x 3 + 4y 3 + 5z 3 = 0 has points in Qv for all v , but no rational points. If X (Q) 6= φ, then fixing a point O ∈ X (Q) gives X (Q) the structure of a finitely-generated abelian group: X (Q) ' X (Q)tor × Zr , for which O is the origin. Here, r is called the rank of the curve and X (Q)tor is a finite effectively computable abelian group.
Arithmetic of algebraic curves For example, for X := {y 2 = x 3 + ax + b} ∪ {∞} (a, b ∈ Z), (x, y ) ∈ X (Q)tor iff y 2 |(4a3 + 27b 2 ).
Arithmetic of algebraic curves For example, for X := {y 2 = x 3 + ax + b} ∪ {∞} (a, b ∈ Z), (x, y ) ∈ X (Q)tor iff y 2 |(4a3 + 27b 2 ).
A good deal more difficult fact (Mazur) is that the only possibilities for X (Q)tor are Z/NZ,
1 ≤ N ≤ 10 or N = 12;
Z/2Z × Z/2NZ, 1 ≤ N ≤ 4
Arithmetic of algebraic curves An outstanding problem is the algorithmic computation of the rank and a full set of generators for X (Q). This is the subject of the conjecture of Birch and Swinnerton-Dyer.
Arithmetic of algebraic curves An outstanding problem is the algorithmic computation of the rank and a full set of generators for X (Q). This is the subject of the conjecture of Birch and Swinnerton-Dyer. In practice, it is often possible to compute these. For example, for y 2 = x 3 − 2, Sage will give you r = 1 and the point (3, 5) as generator.
Arithmetic of algebraic curves An outstanding problem is the algorithmic computation of the rank and a full set of generators for X (Q). This is the subject of the conjecture of Birch and Swinnerton-Dyer. In practice, it is often possible to compute these. For example, for y 2 = x 3 − 2, Sage will give you r = 1 and the point (3, 5) as generator. Note that 2(3, 5) = (129/100, −383/1000) 3(3, 5) = (164323/29241, −66234835/5000211) 4(3, 5) = (2340922881/58675600, 113259286337279/449455096000)
Arithmetic of algebraic curves
g ≥ 2 (d ≥ 4)
Arithmetic of algebraic curves
g ≥ 2 (d ≥ 4) X (Q) is always finite (Faltings). However, *very* difficult to compute (cf. Fermat equations, ABC conjecure).
Arithmetic of algebraic curves
g ≥ 2 (d ≥ 4) X (Q) is always finite (Faltings). However, *very* difficult to compute (cf. Fermat equations, ABC conjecure). Effective Mordell conjecture: There is a terminating algorithm that takes X as input and produces X (Q) as output.
II. The Method of Principal Bundles
The method of principal bundles Given a field K with absolute Galois group GK = Gal(K¯ /K ), a sheaf of groups over K is a topological group R with a continuous action of GK by group automorphisms: GK × R
- R.
The method of principal bundles Given a field K with absolute Galois group GK = Gal(K¯ /K ), a sheaf of groups over K is a topological group R with a continuous action of GK by group automorphisms: GK × R
- R.
A principal R-bundle or R-torsor over K is a topological space P with compatible continuous actions of GK (left) and R (right, simply transitive): For g ∈ GK , z ∈ P, r ∈ R, g (zr ) = g (z)g (r ).
The method of principal bundles Choosing a point z ∈ P, for any g ∈ GK , there is a unique rg ∈ R such that g (z) = zrg .
The method of principal bundles Choosing a point z ∈ P, for any g ∈ GK , there is a unique rg ∈ R such that g (z) = zrg .
The map g 7→ rg is a 1-cocycle on GK with values in R, and defines a bijection Isomorphism classes of principal R-bundles over K ' H 1 (GK , R), where the last is continuous group cohomology.
The method of principal bundles Choosing a point z ∈ P, for any g ∈ GK , there is a unique rg ∈ R such that g (z) = zrg .
The map g 7→ rg is a 1-cocycle on GK with values in R, and defines a bijection Isomorphism classes of principal R-bundles over K ' H 1 (GK , R), where the last is continuous group cohomology. Note that P is trivial exactly when there is a fixed point z ∈ P GK .
The method of principal bundles
Example: ˆ Z(1) := lim µn , ← − where µn ⊂ K¯ is the group of n-th roots of 1.
The method of principal bundles
Example: ˆ Z(1) := lim µn , ← − where µn ⊂ K¯ is the group of n-th roots of 1. Thus, ˆ Z(1) = {(ζn )}, m = ζ . As a group, Z(1) ˆ ˆ but there is a where ζn ∈ µn and ζnm ' Z, n continuous action of GK .
The method of principal bundles ˆ Given any x ∈ K ∗ , get principal Z(1)-bundle n P(x) = {(yn )n | ynn = x, ynm = yn .}
over K .
The method of principal bundles ˆ Given any x ∈ K ∗ , get principal Z(1)-bundle n P(x) = {(yn )n | ynn = x, ynm = yn .}
over K . P(x) is trivial iff x admits an n-th root in K for all n.
The method of principal bundles ˆ Given any x ∈ K ∗ , get principal Z(1)-bundle n P(x) = {(yn )n | ynn = x, ynm = yn .}
over K . P(x) is trivial iff x admits an n-th root in K for all n. Recall that ˆ' Z
Y
Zp .
p
Corresponding to this, have variant Zp (1) := lim µpn ← − for a fixed prime p, and similarly for principalQbundles. These are ˆ studied much more extensively than Z(1) = p Zp (1).
The method of principal bundles When K = Q, there are completions Qv and injections ¯ ¯ v /Qv ) ⊂ - G = Gal(Q/Q). Gv = Gal(Q
The method of principal bundles When K = Q, there are completions Qv and injections ¯ ¯ v /Qv ) ⊂ - G = Gal(Q/Q). Gv = Gal(Q Given any Q-group R and principal R-bundle P, can restrict the G -action to a Gv -action for each v , giving a group and a principal bundle over Qv .
The method of principal bundles When K = Q, there are completions Qv and injections ¯ ¯ v /Qv ) ⊂ - G = Gal(Q/Q). Gv = Gal(Q Given any Q-group R and principal R-bundle P, can restrict the G -action to a Gv -action for each v , giving a group and a principal bundle over Qv . At the level of isomorphism classes, this gives rise to the localisation map Y H 1 (Gv , R). loc : H 1 (G , R) v
The method of principal bundles When K = Q, there are completions Qv and injections ¯ ¯ v /Qv ) ⊂ - G = Gal(Q/Q). Gv = Gal(Q Given any Q-group R and principal R-bundle P, can restrict the G -action to a Gv -action for each v , giving a group and a principal bundle over Qv . At the level of isomorphism classes, this gives rise to the localisation map Y H 1 (Gv , R). loc : H 1 (G , R) v
The Tate-Shafarevich group of R X(Q, R) is the kernel of loc. Thus, these are the principal R-bundles over Q that are ‘locally trivial’, that is, have a Gv -fixed point for each v .
The method of principal bundles: elliptic curves
The method of principal bundles: elliptic curves E , elliptic curve over Q.
The method of principal bundles: elliptic curves E , elliptic curve over Q. ¯ We let G = Gal(Q/Q) act on the exact sequence 0
- E [p]
- E
p
- E
- 0.
The method of principal bundles: elliptic curves E , elliptic curve over Q. ¯ We let G = Gal(Q/Q) act on the exact sequence 0
- E [p]
- E
p
- 0.
- E
Get long exact sequence 0
- E (Q)[p]
- H 1 (G , E [p])
p
- E (Q)
- E (Q) p
- H 1 (G , E )
- H 1 (G , E ),
from which we get the short exact sequence 0
- E (Q)/pE (Q)
⊂
- H 1 (G , E [p])
i
- H 1 (G , E )[p]
- 0.
The method of principal bundles: elliptic curves The central problem in the theory of elliptic curves is the identification of the image E (Q)/pE (Q) ⊂ - H 1 (G , E [p]).
The method of principal bundles: elliptic curves The central problem in the theory of elliptic curves is the identification of the image E (Q)/pE (Q) ⊂ - H 1 (G , E [p]).
To this end, define the p-Selmer group Sel(Q, E [p]) to be the inverse image of X(Q, E )[p] under the map i, so that it fits into an exact sequence 0
- E (Q)/pE (Q)
- Sel(Q, E [p])
- X(Q, E )[p]
- 0.
The method of principal bundles: elliptic curves
0- E (Q)/pE (Q) ⊂ - H 1 (G , E [p])
- H 1 (G , E )[p]
- 0
? ? ? 0 E (Qv )/pE (Qv ) ⊂- H 1 (Gv , E [p]) - H 1 (Gv , E )[p] - 0
The method of principal bundles: elliptic curves
0- E (Q)/pE (Q) ⊂ - H 1 (G , E [p])
- H 1 (G , E )[p]
- 0
? ? ? 0 E (Qv )/pE (Qv ) ⊂- H 1 (Gv , E [p]) - H 1 (Gv , E )[p] - 0
That is, Sel(Q, E [p]) consists of the E [p]-principal bundles that locally come from E (Qv ) for each v .
The method of principal bundles: elliptic curves The key point is that the p-Selmer group is a finite-dimensional Fp -vector space that is effectively computable, and this already gives us a bound on the Mordell-Weil group of E : E (Q)/pE (Q) ⊂ Sel(Q, E [p]).
The method of principal bundles: elliptic curves The key point is that the p-Selmer group is a finite-dimensional Fp -vector space that is effectively computable, and this already gives us a bound on the Mordell-Weil group of E : E (Q)/pE (Q) ⊂ Sel(Q, E [p]). This is then refined by way of the diagram 0
- E (Q)/p n E (Q) - Sel(Q, E [p n ]) - X(Q, E )[p n ]
- 0
0
? - E (Q)/pE (Q)
- 0
for increasing values of n.
? - Sel(Q, E [p])
? - X(Q, E )[p]
The method of principal bundles: elliptic curves
If we assume X is finite, we get lim X(Q, E )[p n ] = 0, ← − and hence, n Im(E (Q)/pE (Q)) = ∩∞ n=1 Im[Sel(Q, E [p ]] ⊂ Sel(Q, E [p]).
We get thereby, a cohomological expression for the group E (Q)/pE (Q) that can be used to compute its structure precisely.
The method of principal bundles: elliptic curves The idea is to compute the image Im[Sel(Q, E [p n ])] ⊂ Sel(Q, E [p]) for each n and compute in parallel the image of E (Q)≤n in Sel(Q, E [p]).
The method of principal bundles: elliptic curves The idea is to compute the image Im[Sel(Q, E [p n ])] ⊂ Sel(Q, E [p]) for each n and compute in parallel the image of E (Q)≤n in Sel(Q, E [p]). Here, E (Q)≤n consists of the point in E (Q) of height ≤ n. The height h(x, y ) of a point (x, y ) ∈ E (Q) is defined as follows. Write (x, y ) = (s/r , t/r ) for coprime integers s, t, r . Then h(x, y ) := log sup{|s|, |t|, |r |}.
The set E (Q)≤n is finite and can also be effectively computed.
The method of principal bundles: elliptic curves
Thus, we have an inclusion Im(E (Q)≤n ) ⊂ Im(Sel(Q, E [p n ]) ⊂ Sel(Q, E [p]). Assuming X is finite, we get Im(E (Q)≤n ) = Im(Sel(Q, E [p n ]) for n sufficiently large, at which point we can conclude that E (Q)/pE (Q) = Im(Sel(Q, E [p n ]).
The method of principal bundles: elliptic curves
This is a conditional algorithm for computing the rank, which is used by all the existing computer packages. With slightly more care, it also gives a set of generators for the group E (Q). In this sense, we eventually arrive at a conditional algorithm for ‘completely determining’ E (Q).
The method of principal bundles: elliptic curves
This is a conditional algorithm for computing the rank, which is used by all the existing computer packages. With slightly more care, it also gives a set of generators for the group E (Q). In this sense, we eventually arrive at a conditional algorithm for ‘completely determining’ E (Q). The essence of BSD is to remove the ‘conditional’ aspect.
III. The method of principal bundles II: The non-abelian case
The method of principal bundles II: The non-abelian case To generalise, focus on the sequence of maps ···
- E [p 3 ]
p
- E [p 2 ]
p
- E [p]
of which we take the inverse limit to get the p-adic Tate module of E: Tp E := lim E [p n ]. ← − This is a free Zp -module of rank 2.
The method of principal bundles II: The non-abelian case To generalise, focus on the sequence of maps ···
- E [p 3 ]
p
- E [p 2 ]
p
- E [p]
of which we take the inverse limit to get the p-adic Tate module of E: Tp E := lim E [p n ]. ← − This is a free Zp -module of rank 2. The previous finite Kummer theory maps can be packaged into E (Q)
- lim H 1 (G , E [p n ]) = H 1 (G , Tp E ).
← −
The method of principal bundles II: The non-abelian case The key abstraction is that Tp E ' π1p (E , O), where π1p refers to the pro-p fundamental group.
The method of principal bundles II: The non-abelian case The key abstraction is that Tp E ' π1p (E , O), where π1p refers to the pro-p fundamental group. The pro-finite completion of a group A is Aˆ := lim A/N, ← − where N ⊂ A runs over the normal subgroups of finite index.
The method of principal bundles II: The non-abelian case The key abstraction is that Tp E ' π1p (E , O), where π1p refers to the pro-p fundamental group. The pro-finite completion of a group A is Aˆ := lim A/N, ← − where N ⊂ A runs over the normal subgroups of finite index. The pro-p completion, is Ap := lim A/N ← − where N ⊂ A runs over normal subgroups of index a power of p.
The method of principal bundles II: The non-abelian case When A = π1 (X , b) for a variety X defined over Q and b ∈ X (Q), ¯ then both π ˆ1 (X , b) and π1p (X , b) admit actions of G = Gal(Q/Q).
The method of principal bundles II: The non-abelian case When A = π1 (X , b) for a variety X defined over Q and b ∈ X (Q), ¯ then both π ˆ1 (X , b) and π1p (X , b) admit actions of G = Gal(Q/Q). To see this, consider the universal pro-finite cover X˜ := (Xi
- X )i∈I ,
- X runs over a cofinal system of finite covering where Xi spaces of X .
The method of principal bundles II: The non-abelian case When A = π1 (X , b) for a variety X defined over Q and b ∈ X (Q), ¯ then both π ˆ1 (X , b) and π1p (X , b) admit actions of G = Gal(Q/Q). To see this, consider the universal pro-finite cover X˜ := (Xi
- X )i∈I ,
- X runs over a cofinal system of finite covering where Xi spaces of X .
Then the choice of a basepoint lift b˜ ∈ X˜ determines an isomorphism π ˆ1 (X , b) = X˜b from the fundamental group to the fibre of X˜ over b.
The method of principal bundles II: The non-abelian case
The rationality of b allows us to define the whole system over Q, and gives us the G -action on X˜b .
The method of principal bundles II: The non-abelian case
The rationality of b allows us to define the whole system over Q, and gives us the G -action on X˜b . Also, we get the profinite homotopy classes of paths π ˆ1 (X ; b, x) := X˜x .
The method of principal bundles II: The non-abelian case
The rationality of b allows us to define the whole system over Q, and gives us the G -action on X˜b . Also, we get the profinite homotopy classes of paths π ˆ1 (X ; b, x) := X˜x .
Get pro-p versions π1p (X , b) and π1p (X ; b, x) by considering coverings of p-power degree.
The method of principal bundles II: The non-abelian case
Examples: For X = Gm , g G m := (Gm
(·)n
- Gm ).
The method of principal bundles II: The non-abelian case
Examples: For X = Gm , g G m := (Gm
(·)n
- Gm ).
Thus, m ˆ π ˆ1 (Gm , 1) = {(ζn ) | ζmn = ζn } =: Z(1).
The method of principal bundles II: The non-abelian case
Examples: For X = Gm , g G m := (Gm
(·)n
- Gm ).
Thus, m ˆ π ˆ1 (Gm , 1) = {(ζn ) | ζmn = ζn } =: Z(1).
Similarly, π1p (Gm , 1) = Zp (1)
The method of principal bundles II: The non-abelian case
For X = E , E˜ = (E and π ˆ1 (E , O) ' (E
n
- E)
n
- E )O = Tˆ E
Also π1p (E , O) = Tp E .
The method of principal bundles II: The non-abelian case The map E (Q)
- H 1 (G , π p (E , O)) 1
is given by x 7→ [π1p (E ; O, x)] ' [(E
pn
- E )x ].
The method of principal bundles II: The non-abelian case The map - H 1 (G , π p (E , O)) 1
E (Q) is given by
x 7→ [π1p (E ; O, x)] ' [(E
pn
- E )x ].
For general X , this turns into j : X (Q)
- H 1 (G , π p (X , b)) 1
given by x 7→ π1p (X ; b, x)
The method of principal bundles II: The non-abelian case For each prime v , consider local versions jv : X (Qv )
- H 1 (Gv , π p (X , b)) 1
given by x 7→ π1p (X ; b, x) which are far more computable than the global map, together with the localisation map Y X (Qv ) X (Q) v
Q
j
?
?
H
1
(G , π1p (X , b))
v jv
-
Y v
H (Gv , π1p (X , b)) 1
The method of principal bundles II: The non-abelian case Actual applications use X (Q)
-
Y
X (Qv )
v
Q
j ? 1
H (G , U(X , b)) -
v jv
?
Y v
1
H (Gv , U(X , b))
The method of principal bundles II: The non-abelian case Actual applications use X (Q)
-
Y
X (Qv )
v
Q
j ? 1
H (G , U(X , b)) -
v jv
?
Y
1
H (Gv , U(X , b))
v
The lower row of this diagram is an algebraic map of algebraic varieties over Qp , which can be computed in principle.
The method of principal bundles II: The non-abelian case Actual applications use X (Q)
-
Y
X (Qv )
v
Q
j
v jv
?
? 1
H (G , U(X , b)) -
Y
1
H (Gv , U(X , b))
v
The lower row of this diagram is an algebraic map of algebraic varieties over Qp , which can be computed in principle. If α is an algebraic function vanishing on the image, then Y α◦ jv v
gives a defining equation for X (Q) inside
Q
v
X (Qv ).
The method of principal bundles II: The non-abelian case Here, U(X , b) is the Qp -pro-unipotent of π1p (X , b). The group U(X , b) is defined to be the group like elements in the completed group algebra lim Qp [π1p (X , b)]/I n , ← − where I ⊂ Qp [π1p (X , b)] is the augmentation ideal. Thus, U(X , b) contains elements like ga where g ∈ π1p (X , b) and a ∈ Qp . Key point is that U(X , b) is a pro-algebraic group.
The method of principal bundles II: The non-abelian case The construction U(Γ) makes sense for any group Γ: U(Γ) = “Γ ⊗ Qp ”
The method of principal bundles II: The non-abelian case The construction U(Γ) makes sense for any group Γ: U(Γ) = “Γ ⊗ Qp ” Examples:
The method of principal bundles II: The non-abelian case The construction U(Γ) makes sense for any group Γ: U(Γ) = “Γ ⊗ Qp ” Examples: U(Zn ) = Qnp .
The method of principal bundles II: The non-abelian case The construction U(Γ) makes sense for any group Γ: U(Γ) = “Γ ⊗ Qp ” Examples: U(Zn ) = Qnp .
1 a b Γ = {0 1 c | a, b, c ∈ Z} 0 0 1 Then
1 a b U(Γ) = {0 1 c | a, b, c ∈ Qp }. 0 0 1
The method of principal bundles II: The non-abelian case
To make this concretely computable, we take the projection Y prp : X (Qv ) - X (Qp ) v
and try to compute ∩α prp (Z (α ◦
Y v
jv )) ⊂ X (Qp ).
IV. Computing Rational Points
Computing rational points [Dan-Cohen, Wewers] For X = P1 \ {0, 1, ∞}, X (Z[1/2]) ⊂ {D2 (z) = 0} ∩ {D4 (z) = 0},
Computing rational points [Dan-Cohen, Wewers] For X = P1 \ {0, 1, ∞}, X (Z[1/2]) ⊂ {D2 (z) = 0} ∩ {D4 (z) = 0}, where D2 (z) = `2 (z) + (1/2) log(z) log(1 − z), D4 (z) = ζ(3)`4 (z) + (8/7)[log3 2/24 + `4 (1/2)/ log 2] log(z)`3 (z) +[(4/21)(log3 2/24 + `4 (1/2)/ log 2) + ζ(3)/24] log3 (z) log(1 − z), and `k (z) =
∞ X zn n=1
nk
.
Computing rational points [Dan-Cohen, Wewers] For X = P1 \ {0, 1, ∞}, X (Z[1/2]) ⊂ {D2 (z) = 0} ∩ {D4 (z) = 0}, where D2 (z) = `2 (z) + (1/2) log(z) log(1 − z), D4 (z) = ζ(3)`4 (z) + (8/7)[log3 2/24 + `4 (1/2)/ log 2] log(z)`3 (z) +[(4/21)(log3 2/24 + `4 (1/2)/ log 2) + ζ(3)/24] log3 (z) log(1 − z), and `k (z) =
∞ X zn n=1
nk
.
Numerically, this appears to be an equality.
Computing rational points
Some qualitative results: (Coates and K.) ax n + by n = cz n for n ≥ 4 has only finitely many rational points.
Computing rational points
Some qualitative results: (Coates and K.) ax n + by n = cz n for n ≥ 4 has only finitely many rational points. Standard structural conjectures on mixed motives ⇒Faltings theorem over Q.
Computing rational points A recent result on modular curves by Balakrishnan, Dogra, Mueller, Tuitmann, Vonk (arXiv 1711.05846, ‘Explicit Chabauty-Kim theory for the split modular curve of level 13’):
Computing rational points A recent result on modular curves by Balakrishnan, Dogra, Mueller, Tuitmann, Vonk (arXiv 1711.05846, ‘Explicit Chabauty-Kim theory for the split modular curve of level 13’): Xs+ (N) = X (N)/Cs+ (N), where X (N) the the compactification of the moduli space of pairs (E , φ : E [N] ' (Z/N)2 ), and Cs+ (N) ⊂ GL2 (Z/N) is the normaliser of a split Cartan subgroup.
Computing rational points A recent result on modular curves by Balakrishnan, Dogra, Mueller, Tuitmann, Vonk (arXiv 1711.05846, ‘Explicit Chabauty-Kim theory for the split modular curve of level 13’): Xs+ (N) = X (N)/Cs+ (N), where X (N) the the compactification of the moduli space of pairs (E , φ : E [N] ' (Z/N)2 ), and Cs+ (N) ⊂ GL2 (Z/N) is the normaliser of a split Cartan subgroup. Bilu-Parent-Rebolledo had shown that Xs+ (p)(Q) consists entirely of cusps and CM points for all primes p > 7, p 6= 13. They called p = 13 the ‘cursed level’.
Computing rational points
Theorem (BDMTV) The modular curve Xs+ (13) has exactly 7 rational points, consisting of the cusp and 6 CM points.
Computing rational points
Theorem (BDMTV) The modular curve Xs+ (13) has exactly 7 rational points, consisting of the cusp and 6 CM points. This concludes an important chapter of a conjecture of Serre: There is an absolute constant A such that G
- Aut(E [p])
is surjective for all non-CM elliptic curves E /Q and primes p > A.
V. Rational points and critical points
Rational points and critical points
Actually, interested in Im(H 1 (G , U)) ∩
Y v
Hf1 (Gv , U) ⊂
Y
H 1 (Gv , U),
v
where Hf1 (Gv , U) ⊂ H 1 (Gv , U) is a subvariety defined by some Hodge-theoretic conditions.
Rational points and critical points
Actually, interested in Im(H 1 (G , U)) ∩
Y v
Hf1 (Gv , U) ⊂
Y
H 1 (Gv , U),
v
where Hf1 (Gv , U) ⊂ H 1 (Gv , U) is a subvariety defined by some Hodge-theoretic conditions. In order to apply symplectic techniques, replace U by T ∗ (1)U := (LieU)∗ (1) o U.
Rational points and critical points Then Y
H 1 (Gv , T ∗ (1)U)
v
is a symplectic variety and Im(H 1 (G , T ∗ (1)U)),
Y v
are Lagrangian subvarieties.
Hf1 (Gv , T ∗ (1)U)
Rational points and critical points Then Y
H 1 (Gv , T ∗ (1)U)
v
is a symplectic variety and Im(H 1 (G , T ∗ (1)U)),
Y
Hf1 (Gv , T ∗ (1)U)
v
are Lagrangian subvarieties. Thus, the derived intersection Im(H 1 (G , U)) ∩
Y
Hf1 (Gv , U)
v
has a [−1]-shifted symplectic structure. Should be the critical set of a function.
VI. Concluding remark
Modularity is a mystery in number theory
