HODGE THEORY OF KLOOSTERMAN
HODGE THEORY OF KLOOSTERMAN CONNECTIONS ´ JAVIER FRESAN
This talk is based on joint work in progress with Claude Sabbah and Jeng-Daw Yu. 1. L-functions of symmetric powers of Kloosterman sums Let p be a prime number and let ψ : Fp → C× be a non-trivial additive character, e.g. the one defined by x 7→ e2πix/p . By composition with the trace map, ψ induces a non-trivial additive character, still denoted by ψ, on every finite extension Fq of Fp (say inside a fixed algebraic closure). For each a ∈ F× q , the Kloosterman sum Kl2 (a; q) is the algebraic integer X Kl2 (a; q) = ψ(x + a/x). x∈F× q
1.1. `-adic interpretation. Let ` be a prime number distinct from p and let Q` be an algebraic closure of the field of `-adic numbers. We fix an embedding Q` ,→ C through which × we can see ψ as taking values in Q` . Let Lψ denote the associated Artin–Schreier sheaf on the affine line A1Fq and consider the diagram of varieties over Fp G2m
(1)
prod
sum
}
A1
Gm
where “sum” and “prod” stand for the sum and the product of the coordinates. Definition 1.1. The Kloosterman sheaf Kl2 is defined as Kl2 = R1 prod! (sum∗ Lψ ). Deligne [1] proves that Kl2 is a rank two `-adic local system on Gm,Fq , pure of weight one. By Grothendieck’s trace formula, for each a ∈ F× q , one has: tr(Froba | Kl2 ) = −Kl2 (a; q). √ In particular, there exist elements αa , βa ∈ Q` of absolute value q (when regarded as complex numbers through the fixed embedding) such that Kl2 (a; q) = αa + βa . 1.2. Moments. We can then form, for each integer k ≥ 1, the symmetric powers k KlSym (a; q) 2
=
k X i=0
1
αai βak−i
2
´ JAVIER FRESAN
which are the local traces of Frobenius of the `-adic local system Symk Kl2 , and the moments X k KlSym (a; q). mk2 (q) = 2 a∈F× q
Using the trace formula again, together with the fact that Symk Kl2 has no non-zero global sections, we can see mk2 (q) as the trace of Frobenius acting on the ´etale cohomology group Hc1 (Gm,Fq , Symk Kl2 ).
(2)
Remark 1.2. Note that mk2 (p) does not depend on the choice of the additive character ψ, which is the first indication that there could be a classical motive behind. 1.3. Local L-functions. The local L-function of the moments of symmetric powers of Kloosterman sums is defined as the generating series ! ∞ n X T Zk (T ; p) = exp mk2 (pn ) , n n=1
which is actually a polynomial with integer coefficients: the characteristic polynomial of Frobenius acting on (2). Robba [7] and Fu–Wan [5] computed its degree, at least when p 6= 2: h i k − k if k is even, 2 2p h i deg Zk (T ; p) = k+1 − k + 1 if k is odd. 2 2p 2 Remark 1.3. For p sufficiently big, the degree is [ k+1 2 ]. Moreover, there is a decomposition Zk (T ; p) = Pk (T ; p)Zek (T ; p), where Pk (T ; p) contains the so-called trivial factors. Fu-Wan give in loc.cit. an explicit expression for the trivial factors from which one can read that, for p big enough, ( k (1 − T )(1 − p 2 T ) if 4 divides k, Pk (T ; p) = (1 − T ) else. 1.4. Global L-functions. We now define a global L-function as the Euler product Y 1 Lk (s) = , Zek (p−s ; p) p prime
possibly modified at the bad primes p < k. This work arose as an attempt to understand experiments by Broadhurst and Roberts, who checked with amazing numerical precision that Lk (s) satisfies a functional equation. In what follows, we will focus on the odd symmetric powers, which are a bit easier to handle than the even ones. Let k = 2m + 1 be an odd integer ≥ 3. We define a “conductor” N as the product of the square-free parts of the odd integers smaller than or equal to k, that is N = 3s · 5s · · · ks . Broadhurst and Roberts consider the completed L-function m N sY s−j Λk (s) = Γ Lk (s). πm 2 j=1
Their computations provide strong evidence for the functional equation Λk (s) = ±Λk (k + 1 − s).
(3)
HODGE THEORY OF KLOOSTERMAN CONNECTIONS
3
1.5. Serre’s recipe. Let us recall Serre’s classical recipe [8] for the gamma factors at infinity in the conjectural functional equation for the L-function of a motive over Q. We first consider the following variants of the gamma function: ΓR (s) = π −s/2 Γ(s/2), 1 ΓC (s) = (2π)−s Γ(s) = ΓR (s)ΓR (s + 1). 2 If H is a real Hodge structure of weight w, we define w,w w,w Y p,w−p 2 2 2 2 ΓC (s − p)h ΓR (s − w/2)h+ ΓR (s − w/2 + 1)h− , ΓH (s) = p<w/2 w w
,
where hp,w−p is the dimension of H p,q and h±2 2 are the dimensions of the ±(−1)w/2 -eigenspaces for the action of complex conjugation on H w/2,w/2 . 1.6. Expectation. All the above suggests that, for each odd integer k ≥ 3, the “motive” 1 k 1 k M = Im Hc (Gm , Sym Kl2 ) −→ H (Gm , Sym Kl2 ) has Hodge types (2, k−1), (4, k−3), . . . , (k−1, 2) with all Hodge numbers equal to 1. Moreover, w/2,w/2 if k is congruent to 3 modulo 4, then complex conjugation should act as −Id on MC . 1.7. The work of Yun. By a beautiful result of Yun [9], M is indeed a usual motive. More precisely, Yun constructs a smooth projective variety X of dimension k − 1 over Q, an orthogonal Q` -vector space M` of dimension (k − 1)/2, and a Galois representation ρ` : Gal(Q/Q) −→ O(M` ) k+1 which appears as a subquotient of H´ek−1 t (XQ , Q` )( 2 ) and has the property that, for all prime numbers p 6= ` satisfying p > k or p = 2, the representation ρ` is unramified at p and
tr(Frobp | M` ) = −p−
k+1 2
(mk2 (p) + 1).
Remark 1.4. Yun also proves that this Galois representation is modular for k = 5, 7, which allows one to actually prove the functional equation (3) in those cases. 2. Hodge theory of Kloosterman connections The geometry behind Yun’s constructions is rather intricate. For the purpose of computing the Hodge numbers or relating the special values of the L-function to periods, it is more convenient to change gears and work in the framework of exponential motives, as introduced in my joint work with Peter Jossen [4]. 2.1. The Kloosterman connection. We now work over the field of complex numbers. Let x and t denote the coordinates on G2m and consider the diagram G2m π
}
Gm
f
!
A1 ,
4
´ JAVIER FRESAN
where f : G2m → A1 is the function x 7→ x + t/x and π : G2m → Gm is the projection to the t coordinate. After an obvious change of variables, we obtain the same diagram as in (1). The role of the Artin–Schreier sheaf is replaced by the exponential. Let E f = (OG2m , d − df ) denote the rank one connection whose local system of horizontal sections is spanned by ef . Definition 2.1. The Kloosterman connection is defined as Kl2 = π+ E f , where π+ stands for the direct image of D-modules. This is a priori a complex of D-modules, but one easily proves that it is concentrated in degree zero and has no singularity on Gm . Our goal is to study the cohomology of the symmetric powers of Kl2 : 1 HdR (Gm , Symk Kl2 ).
Remark 2.2 (A family of exponential motives). We think of Kl2 as the de Rham realisation of a family of exponential motives parametrised by Gm . Indeed, for a fixed value of t, H 1 (Gm , x + t/x)
(4)
is a rank two exponential motive in the sense of [4]. A basis of the de Rham realisation is given by the differential forms dx/x and dx. Besides, the rapid decay homology is spanned by a loop encircling 0 counterclockwise and by a second unbounded cycle which can be chosen to be (0, ∞) if t is a positive real number. Among the exponential periods of (4), we find the modified Bessel function of the first kind I ∞ X tn −x− xt dx e = 2πi x (n!)2 n=1
which, as a function of t, is a solution of the second order differential equation d2 u 1 du + − u = 0. dt2 t dt The object Kl2 is nothing but this differential equation seen as a connection on Gm . It has a regular singularity at t = 0 and an irregular singularity at infinity. 2.2. The irregular Hodge filtration. Let X be a smooth quasi-projective variety, together with a regular function f : X → A1 . After Deligne, Sabbah, Yu, Kontsevich, and Esnault– Sabbah–Yu, the de Rham cohomology n HdR (X, f ) = H n (X, (Ω•X , d − df ))
is equipped with an irregular Hodge filtration. A possible way to define it is as follows. We choose a smooth compactification X of X such that D = X \ X is a simple normal crossing divisor and such that the function f extends to a rational function: X f
1
A
/X
f
/ P1
HODGE THEORY OF KLOOSTERMAN CONNECTIONS
5
Let P = f ∗ (∞) be the pole divisor of f . For each α ∈ Q ∩ [0, 1), we define the sheaf p p+1 Ωpf (α) = {ω ∈ ΩX (log D) ⊗ OX ([αP ]) | df ∧ ω ∈ ΩX (log D) ⊗ OX ([αP ])},
where [·] denotes the integral part. Together with the differential d − df , the Ωpf (α) form a complex of coherent sheaves on X called the Kontsevich complex. A basic result is that it computes the de Rham cohomology n HdR (X, f ) = H n (X, (Ω•f (α), d − df )).
As in usual Hodge theory, we can then consider the bˆete filtration of this complex. By a theorem of Esnault–Sabbah–Yu [3], the natural map n • n H n (X, (Ω•≥p f (α), d − df )) −→ H (X, (Ωf (α), d − df )) = HdR (X, f )
is injective, which makes the following definition reasonable: n (X, f ) is given by Definition 2.3. The irregular Hodge filtration on HdR n n n (X, Ω•≥p (X, f ) = Im HdR F p−α HdR (X, f ) . (α)) −→ H dR f
The significant values of α are the rational numbers with denominator the multiplicities of the irreducible components of Pred , as one can already see in the following example: Example 2.4. If X = A1 and f = xn , the differentials dx, xdx, . . . , xn−2 dx form a basis of the first de Rham cohomology and one has 1
2
1 (A1 , xn ) = F 0 = F n ⊃ F n ⊃ · · · ⊃ F HdR
n−1 n
⊃ {0}
j
with F 1− n spanned by dx, . . . , xj−1 dx. This corresponds to Deligne’s intuition [2] that the special value of the gamma function Γ(j/n) has Hodge type (1 − j/n, j/n). 2.3. Main results. Returning to the symmetric powers of the Kloosterman connection, we 1 consider the function fk : Gk+1 m → A defined by fk =
k X i=1
k X 1 xi + t . xi i=1
In order to equip the cohomology of Symk Kl2 with the irregular Hodge filtration, we view it as a direct factor 1 1 HdR (Gm , Symk Kl2 ) ⊂ HdR (Gm , Kl2⊗k ) and observe that, by the K¨ unneth formula and the degeneration of the Leray spectral sequence associated with the projection π, the rightmost term is isomorphic to k+1 HdR (Gk+1 m , fk ).
(5)
Thanks to the analogue of the Grothendieck–Ogg–Shafarevich formula, the dimension of the cohomology is equal to k+1 1 k k dim HdR (Gm , Sym Kl2 ) = Irr∞ (Sym Kl2 ) = . 2
´ JAVIER FRESAN
6
Note that the result agrees with the degree of the characteristic polynomial of Frobenius for a big enough prime number p (Remark 1.3). Moreover, the classes of ⊗k dx dt k−1 j t , j = 0, . . . , x t 2 form a basis of the first cohomology. Our first result is that this basis is adapted to the irregular Hodge filtration induced from (5). Theorem 2.5. Assume k is odd. Then * + ⊗k dx dt k + 1 − p 1 F p HdR (Gm , Symk Kl2 ) = tj . |0≤j≤ x t 2 In particular, the jumps of the irregular Hodge filtration occur at the even integers p = 2r, for r = 1, . . . , (k + 1)/2 all with multiplicity 1. The theorem gives Hodge types (2, k − 1), (4, k − 3), . . . , (k − 1, 2), (k + 1, 0). The last one corresponds to (dx/x)⊗k dt/t which does not belong to the image of the compactly supported cohomology, so that the result fully agrees with the expectaction of 1.6. 2.4. A Kloosterman motive. The second main result is an elementary construction of a motive over Q whose de Rham realisation is 1 1 (Gm , Symk Kl2 ) . (Gm , Symk Kl2 ) −→ HdR Im HdR,c It is first constructed as an object of the category of exponential motives M exp (Q), then one shows that it actually lies in the full subcategory of usual motives M (Q). In a nutshell, one proceeds as follows: (a) As above, one realises H 1 (Gm , Symk Kl2 ) as the factor of the exponential motive H 1 (Gm , Kl2⊗k ) = H k+1 (Gk+1 m , fk ) cut off by the alternating projector. (b) Consider the function ϕk = y1 + · · · + yk + y1−1 + · · · + yk−1 . Passing to the degree two ´etale cover Gm → Gm that sends t to s = t2 and making the change of variables yi = sxi , one can recover (a) from the exponential motive H k+1 (Gk+1 m , sϕk ) and the action of the group Z/2Z coming from the cover. (c) One replaces (b) with H k+1 (A1 × Gkm , sϕk ) and proves that the latter is a classical motive, essentially the middle cohomology of the zero locus of ϕk . A concrete consequence of the construction of the Kloosterman motive is the following lower bound for the p-adic valuations of the coefficients of the characteristic polynomial of Frobenius, which sharpens a result of Haessig [6]: Corollary 2.6 (Newton above Hodge). Let k be integer and p > k a prime number. Pan odd m Write the local L-function of 1.3 as Zk (T ; p) = cm T . Then ordp cm ≥ m(m − 1).
HODGE THEORY OF KLOOSTERMAN CONNECTIONS
0
1
2
3
7
4
1 Figure 1. The irregular Hodge polygon of HdR,c (Gm , Symk Kl2 )
1 (Gm , Symk Kl2 ) has vertices (m, m(m − 1)), as depicted Since the Hodge polygon of HdR,c in Figure 1, this corollary can be thought of as a “Newton above Hodge” statement.
References [1] P. Deligne. Applications de la formule des traces aux sommes trigonom´etriques. In Cohomologie ´etale (S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 4 21 ). Lectures Notes in Math. 569. Springer-Verlag, 1977. [2] P. Deligne, B. Malgrange, and J-P. Ramis. Singularit´es irr´eguli`eres. Correspondance et documents. Documents Math´ematiques 5. Soci´et´e Math´ematique de France, Paris, 2007. [3] H. Esnault, C. Sabbah, and J-D. Yu. E1 degeneration of the irregular Hodge filtration. With an appendix by Morihiko Saito. J. reine und angew. Math. 729 (2017), 171–227. ´ n and P. Jossen. Exponential motives. Preprint. [4] J. Fresa [5] L. Fu and D. Wan. L-functions for symmetric products of Kloosterman sums. J. reine und angew. Math. 589 (2005), 79–103. [6] C. D. Haessig. L-functions of symmetric powers of Kloosterman sums (unit root L-functions and p-adic estimates). Math. Ann. 369 (2017), no. 1-2, 17–47. [7] P. Robba. Symmetric powers of the p-adic Bessel equation. J. reine und angew. Math. 366 (1986), 194–220. [8] J.-P. Serre. Facteurs locaux des fonctions zˆeta des variet´es alg´ebriques (d´efinitions et conjectures). S´eminaire Delange–Pisot–Poitou. 11e ann´ee: 1969/70. Th´eorie des nombres. Fasc. 1: Expos´e 1 a ` 15; Fasc. 2: Expos´e 16 a ` 24, 15 pp., Secr´etariat Math., Paris, 1970. [9] Z. Yun. Galois representations attached to moments of Kloosterman sums and conjectures of Evans. Appendix B by Christelle Vincent. Compos. Math. 151 (2015), no. 1, 68–120.
8
´ JAVIER FRESAN
´ CMLS, Ecole Polytechnique, F-91128 Palaiseau, France E-mail address: [email protected]
This talk is based on joint work in progress with Claude Sabbah and Jeng-Daw Yu. 1. L-functions of symmetric powers of Kloosterman sums Let p be a prime number and let ψ : Fp → C× be a non-trivial additive character, e.g. the one defined by x 7→ e2πix/p . By composition with the trace map, ψ induces a non-trivial additive character, still denoted by ψ, on every finite extension Fq of Fp (say inside a fixed algebraic closure). For each a ∈ F× q , the Kloosterman sum Kl2 (a; q) is the algebraic integer X Kl2 (a; q) = ψ(x + a/x). x∈F× q
1.1. `-adic interpretation. Let ` be a prime number distinct from p and let Q` be an algebraic closure of the field of `-adic numbers. We fix an embedding Q` ,→ C through which × we can see ψ as taking values in Q` . Let Lψ denote the associated Artin–Schreier sheaf on the affine line A1Fq and consider the diagram of varieties over Fp G2m
(1)
prod
sum
}
A1
Gm
where “sum” and “prod” stand for the sum and the product of the coordinates. Definition 1.1. The Kloosterman sheaf Kl2 is defined as Kl2 = R1 prod! (sum∗ Lψ ). Deligne [1] proves that Kl2 is a rank two `-adic local system on Gm,Fq , pure of weight one. By Grothendieck’s trace formula, for each a ∈ F× q , one has: tr(Froba | Kl2 ) = −Kl2 (a; q). √ In particular, there exist elements αa , βa ∈ Q` of absolute value q (when regarded as complex numbers through the fixed embedding) such that Kl2 (a; q) = αa + βa . 1.2. Moments. We can then form, for each integer k ≥ 1, the symmetric powers k KlSym (a; q) 2
=
k X i=0
1
αai βak−i
2
´ JAVIER FRESAN
which are the local traces of Frobenius of the `-adic local system Symk Kl2 , and the moments X k KlSym (a; q). mk2 (q) = 2 a∈F× q
Using the trace formula again, together with the fact that Symk Kl2 has no non-zero global sections, we can see mk2 (q) as the trace of Frobenius acting on the ´etale cohomology group Hc1 (Gm,Fq , Symk Kl2 ).
(2)
Remark 1.2. Note that mk2 (p) does not depend on the choice of the additive character ψ, which is the first indication that there could be a classical motive behind. 1.3. Local L-functions. The local L-function of the moments of symmetric powers of Kloosterman sums is defined as the generating series ! ∞ n X T Zk (T ; p) = exp mk2 (pn ) , n n=1
which is actually a polynomial with integer coefficients: the characteristic polynomial of Frobenius acting on (2). Robba [7] and Fu–Wan [5] computed its degree, at least when p 6= 2: h i k − k if k is even, 2 2p h i deg Zk (T ; p) = k+1 − k + 1 if k is odd. 2 2p 2 Remark 1.3. For p sufficiently big, the degree is [ k+1 2 ]. Moreover, there is a decomposition Zk (T ; p) = Pk (T ; p)Zek (T ; p), where Pk (T ; p) contains the so-called trivial factors. Fu-Wan give in loc.cit. an explicit expression for the trivial factors from which one can read that, for p big enough, ( k (1 − T )(1 − p 2 T ) if 4 divides k, Pk (T ; p) = (1 − T ) else. 1.4. Global L-functions. We now define a global L-function as the Euler product Y 1 Lk (s) = , Zek (p−s ; p) p prime
possibly modified at the bad primes p < k. This work arose as an attempt to understand experiments by Broadhurst and Roberts, who checked with amazing numerical precision that Lk (s) satisfies a functional equation. In what follows, we will focus on the odd symmetric powers, which are a bit easier to handle than the even ones. Let k = 2m + 1 be an odd integer ≥ 3. We define a “conductor” N as the product of the square-free parts of the odd integers smaller than or equal to k, that is N = 3s · 5s · · · ks . Broadhurst and Roberts consider the completed L-function m N sY s−j Λk (s) = Γ Lk (s). πm 2 j=1
Their computations provide strong evidence for the functional equation Λk (s) = ±Λk (k + 1 − s).
(3)
HODGE THEORY OF KLOOSTERMAN CONNECTIONS
3
1.5. Serre’s recipe. Let us recall Serre’s classical recipe [8] for the gamma factors at infinity in the conjectural functional equation for the L-function of a motive over Q. We first consider the following variants of the gamma function: ΓR (s) = π −s/2 Γ(s/2), 1 ΓC (s) = (2π)−s Γ(s) = ΓR (s)ΓR (s + 1). 2 If H is a real Hodge structure of weight w, we define w,w w,w Y p,w−p 2 2 2 2 ΓC (s − p)h ΓR (s − w/2)h+ ΓR (s − w/2 + 1)h− , ΓH (s) = p<w/2 w w
,
where hp,w−p is the dimension of H p,q and h±2 2 are the dimensions of the ±(−1)w/2 -eigenspaces for the action of complex conjugation on H w/2,w/2 . 1.6. Expectation. All the above suggests that, for each odd integer k ≥ 3, the “motive” 1 k 1 k M = Im Hc (Gm , Sym Kl2 ) −→ H (Gm , Sym Kl2 ) has Hodge types (2, k−1), (4, k−3), . . . , (k−1, 2) with all Hodge numbers equal to 1. Moreover, w/2,w/2 if k is congruent to 3 modulo 4, then complex conjugation should act as −Id on MC . 1.7. The work of Yun. By a beautiful result of Yun [9], M is indeed a usual motive. More precisely, Yun constructs a smooth projective variety X of dimension k − 1 over Q, an orthogonal Q` -vector space M` of dimension (k − 1)/2, and a Galois representation ρ` : Gal(Q/Q) −→ O(M` ) k+1 which appears as a subquotient of H´ek−1 t (XQ , Q` )( 2 ) and has the property that, for all prime numbers p 6= ` satisfying p > k or p = 2, the representation ρ` is unramified at p and
tr(Frobp | M` ) = −p−
k+1 2
(mk2 (p) + 1).
Remark 1.4. Yun also proves that this Galois representation is modular for k = 5, 7, which allows one to actually prove the functional equation (3) in those cases. 2. Hodge theory of Kloosterman connections The geometry behind Yun’s constructions is rather intricate. For the purpose of computing the Hodge numbers or relating the special values of the L-function to periods, it is more convenient to change gears and work in the framework of exponential motives, as introduced in my joint work with Peter Jossen [4]. 2.1. The Kloosterman connection. We now work over the field of complex numbers. Let x and t denote the coordinates on G2m and consider the diagram G2m π
}
Gm
f
!
A1 ,
4
´ JAVIER FRESAN
where f : G2m → A1 is the function x 7→ x + t/x and π : G2m → Gm is the projection to the t coordinate. After an obvious change of variables, we obtain the same diagram as in (1). The role of the Artin–Schreier sheaf is replaced by the exponential. Let E f = (OG2m , d − df ) denote the rank one connection whose local system of horizontal sections is spanned by ef . Definition 2.1. The Kloosterman connection is defined as Kl2 = π+ E f , where π+ stands for the direct image of D-modules. This is a priori a complex of D-modules, but one easily proves that it is concentrated in degree zero and has no singularity on Gm . Our goal is to study the cohomology of the symmetric powers of Kl2 : 1 HdR (Gm , Symk Kl2 ).
Remark 2.2 (A family of exponential motives). We think of Kl2 as the de Rham realisation of a family of exponential motives parametrised by Gm . Indeed, for a fixed value of t, H 1 (Gm , x + t/x)
(4)
is a rank two exponential motive in the sense of [4]. A basis of the de Rham realisation is given by the differential forms dx/x and dx. Besides, the rapid decay homology is spanned by a loop encircling 0 counterclockwise and by a second unbounded cycle which can be chosen to be (0, ∞) if t is a positive real number. Among the exponential periods of (4), we find the modified Bessel function of the first kind I ∞ X tn −x− xt dx e = 2πi x (n!)2 n=1
which, as a function of t, is a solution of the second order differential equation d2 u 1 du + − u = 0. dt2 t dt The object Kl2 is nothing but this differential equation seen as a connection on Gm . It has a regular singularity at t = 0 and an irregular singularity at infinity. 2.2. The irregular Hodge filtration. Let X be a smooth quasi-projective variety, together with a regular function f : X → A1 . After Deligne, Sabbah, Yu, Kontsevich, and Esnault– Sabbah–Yu, the de Rham cohomology n HdR (X, f ) = H n (X, (Ω•X , d − df ))
is equipped with an irregular Hodge filtration. A possible way to define it is as follows. We choose a smooth compactification X of X such that D = X \ X is a simple normal crossing divisor and such that the function f extends to a rational function: X f
1
A
/X
f
/ P1
HODGE THEORY OF KLOOSTERMAN CONNECTIONS
5
Let P = f ∗ (∞) be the pole divisor of f . For each α ∈ Q ∩ [0, 1), we define the sheaf p p+1 Ωpf (α) = {ω ∈ ΩX (log D) ⊗ OX ([αP ]) | df ∧ ω ∈ ΩX (log D) ⊗ OX ([αP ])},
where [·] denotes the integral part. Together with the differential d − df , the Ωpf (α) form a complex of coherent sheaves on X called the Kontsevich complex. A basic result is that it computes the de Rham cohomology n HdR (X, f ) = H n (X, (Ω•f (α), d − df )).
As in usual Hodge theory, we can then consider the bˆete filtration of this complex. By a theorem of Esnault–Sabbah–Yu [3], the natural map n • n H n (X, (Ω•≥p f (α), d − df )) −→ H (X, (Ωf (α), d − df )) = HdR (X, f )
is injective, which makes the following definition reasonable: n (X, f ) is given by Definition 2.3. The irregular Hodge filtration on HdR n n n (X, Ω•≥p (X, f ) = Im HdR F p−α HdR (X, f ) . (α)) −→ H dR f
The significant values of α are the rational numbers with denominator the multiplicities of the irreducible components of Pred , as one can already see in the following example: Example 2.4. If X = A1 and f = xn , the differentials dx, xdx, . . . , xn−2 dx form a basis of the first de Rham cohomology and one has 1
2
1 (A1 , xn ) = F 0 = F n ⊃ F n ⊃ · · · ⊃ F HdR
n−1 n
⊃ {0}
j
with F 1− n spanned by dx, . . . , xj−1 dx. This corresponds to Deligne’s intuition [2] that the special value of the gamma function Γ(j/n) has Hodge type (1 − j/n, j/n). 2.3. Main results. Returning to the symmetric powers of the Kloosterman connection, we 1 consider the function fk : Gk+1 m → A defined by fk =
k X i=1
k X 1 xi + t . xi i=1
In order to equip the cohomology of Symk Kl2 with the irregular Hodge filtration, we view it as a direct factor 1 1 HdR (Gm , Symk Kl2 ) ⊂ HdR (Gm , Kl2⊗k ) and observe that, by the K¨ unneth formula and the degeneration of the Leray spectral sequence associated with the projection π, the rightmost term is isomorphic to k+1 HdR (Gk+1 m , fk ).
(5)
Thanks to the analogue of the Grothendieck–Ogg–Shafarevich formula, the dimension of the cohomology is equal to k+1 1 k k dim HdR (Gm , Sym Kl2 ) = Irr∞ (Sym Kl2 ) = . 2
´ JAVIER FRESAN
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Note that the result agrees with the degree of the characteristic polynomial of Frobenius for a big enough prime number p (Remark 1.3). Moreover, the classes of ⊗k dx dt k−1 j t , j = 0, . . . , x t 2 form a basis of the first cohomology. Our first result is that this basis is adapted to the irregular Hodge filtration induced from (5). Theorem 2.5. Assume k is odd. Then * + ⊗k dx dt k + 1 − p 1 F p HdR (Gm , Symk Kl2 ) = tj . |0≤j≤ x t 2 In particular, the jumps of the irregular Hodge filtration occur at the even integers p = 2r, for r = 1, . . . , (k + 1)/2 all with multiplicity 1. The theorem gives Hodge types (2, k − 1), (4, k − 3), . . . , (k − 1, 2), (k + 1, 0). The last one corresponds to (dx/x)⊗k dt/t which does not belong to the image of the compactly supported cohomology, so that the result fully agrees with the expectaction of 1.6. 2.4. A Kloosterman motive. The second main result is an elementary construction of a motive over Q whose de Rham realisation is 1 1 (Gm , Symk Kl2 ) . (Gm , Symk Kl2 ) −→ HdR Im HdR,c It is first constructed as an object of the category of exponential motives M exp (Q), then one shows that it actually lies in the full subcategory of usual motives M (Q). In a nutshell, one proceeds as follows: (a) As above, one realises H 1 (Gm , Symk Kl2 ) as the factor of the exponential motive H 1 (Gm , Kl2⊗k ) = H k+1 (Gk+1 m , fk ) cut off by the alternating projector. (b) Consider the function ϕk = y1 + · · · + yk + y1−1 + · · · + yk−1 . Passing to the degree two ´etale cover Gm → Gm that sends t to s = t2 and making the change of variables yi = sxi , one can recover (a) from the exponential motive H k+1 (Gk+1 m , sϕk ) and the action of the group Z/2Z coming from the cover. (c) One replaces (b) with H k+1 (A1 × Gkm , sϕk ) and proves that the latter is a classical motive, essentially the middle cohomology of the zero locus of ϕk . A concrete consequence of the construction of the Kloosterman motive is the following lower bound for the p-adic valuations of the coefficients of the characteristic polynomial of Frobenius, which sharpens a result of Haessig [6]: Corollary 2.6 (Newton above Hodge). Let k be integer and p > k a prime number. Pan odd m Write the local L-function of 1.3 as Zk (T ; p) = cm T . Then ordp cm ≥ m(m − 1).
HODGE THEORY OF KLOOSTERMAN CONNECTIONS
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1 Figure 1. The irregular Hodge polygon of HdR,c (Gm , Symk Kl2 )
1 (Gm , Symk Kl2 ) has vertices (m, m(m − 1)), as depicted Since the Hodge polygon of HdR,c in Figure 1, this corollary can be thought of as a “Newton above Hodge” statement.
References [1] P. Deligne. Applications de la formule des traces aux sommes trigonom´etriques. In Cohomologie ´etale (S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 4 21 ). Lectures Notes in Math. 569. Springer-Verlag, 1977. [2] P. Deligne, B. Malgrange, and J-P. Ramis. Singularit´es irr´eguli`eres. Correspondance et documents. Documents Math´ematiques 5. Soci´et´e Math´ematique de France, Paris, 2007. [3] H. Esnault, C. Sabbah, and J-D. Yu. E1 degeneration of the irregular Hodge filtration. With an appendix by Morihiko Saito. J. reine und angew. Math. 729 (2017), 171–227. ´ n and P. Jossen. Exponential motives. Preprint. [4] J. Fresa [5] L. Fu and D. Wan. L-functions for symmetric products of Kloosterman sums. J. reine und angew. Math. 589 (2005), 79–103. [6] C. D. Haessig. L-functions of symmetric powers of Kloosterman sums (unit root L-functions and p-adic estimates). Math. Ann. 369 (2017), no. 1-2, 17–47. [7] P. Robba. Symmetric powers of the p-adic Bessel equation. J. reine und angew. Math. 366 (1986), 194–220. [8] J.-P. Serre. Facteurs locaux des fonctions zˆeta des variet´es alg´ebriques (d´efinitions et conjectures). S´eminaire Delange–Pisot–Poitou. 11e ann´ee: 1969/70. Th´eorie des nombres. Fasc. 1: Expos´e 1 a ` 15; Fasc. 2: Expos´e 16 a ` 24, 15 pp., Secr´etariat Math., Paris, 1970. [9] Z. Yun. Galois representations attached to moments of Kloosterman sums and conjectures of Evans. Appendix B by Christelle Vincent. Compos. Math. 151 (2015), no. 1, 68–120.
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´ JAVIER FRESAN
´ CMLS, Ecole Polytechnique, F-91128 Palaiseau, France E-mail address: [email protected]
