About p-adic Local Fourier Transform Adriano Marmora Institut de Recherche Mathématique Avancée Institut de Recherche Mathématique Avancée UMR 7501, Université de Strasbourg [email protected] The motivation: the product formula for p-adic epsilon factor P.Berthelot defined a “good” category of p-adic coefficients on a scheme over a field of characteristic p. It is the category of overconvergent F -isocrystals and the associated cohomology theory is called rigid cohomology. In this poster we review the local constants – irregularity and epsilon factor – attached to an overconvergent F -isocrystal in any point of the scheme. We state a global conjecture: a formula connecting the product of the epsilon factors over the points of a proper and smooth curve with the rigid cohomology. We are inspired by the case of `-adic coefficients (` 6= p), where the analogous formula was proved by Laumon. Notations Local constants • Let k be a finite field of characteristic p > 0, C/ Frac(W(k)) a totally ramified finite extension and C an algebraic closure of C. Irregularity. It is a natural integer irrs(M ) = irr(js∗M ), defined by Christol and Mebkhout as the height of the polygon of p-adic slopes [CM02]. • Let X be a proper and smooth curve over k, geometrically connected. Epsilon Factors. Fix a primitive p-root of unit ζ ∈ C and let ω be a nonzero meromorphic form in Ω1Ks/k(s). Deligne and Langlands associate to a Weil-Deligne representation a non-zero element of C, • For all closed point s ∈ |X|, denote OX,s the local ring at s, ms / OX,s its maximal ideal, k(s) = OX,s/ms, its residue field, b O X,s its ms-completion, C(s) = C ⊗W(k) W(k(s)), | · |p a p-adic absolute value, b ) The Robba’s rings associated to Ks = Frac(O X,s Rs ∼ = g(t) = R+s ∼ = g(t) = X n an t n∈Z X 7−→ WD(js∗M ) 7−→ ε(js∗M, js∗ω) ∈C ∗ which generalizes the constant appearing in the local function equation of Tate’s thesis for the dimension one case [Del73]. Remark. If s ∈ |U |, then irrs(M ) = 0 and ε(js∗M, js∗ω) is completely explicit. g(t) an ∈ C(s), ∃ η < 1 s.t. converges for η < |t|p < 1 antn ∈ n∈Z js∗M Rs ∀ n The product formula Conjecture. Let U ⊂ X be a non-empty open, M a F -isocristal overconvergent along X\U , and ω a non-zero meromorphic differential form. Then ∈ Z, |an|p ≤ 1 Some properties: – R+s is an henselian discrete valuation ring (not complete) 2 Y i (−1)i+1 det(−F, Hrig,c(U, M )) =q (1−g)rank(M ) i=0 – its residue field identifies to Ks, by t̄ 7→ ts =an unif. of Ks. – Frac(R+s ) ⊂ Rs is dense. b1 ∼ Ω Rs/C(s) = Rsdt continuous differential forms Let σ : Rs → Rs be a Frobenius, i.e. an endomorphism lifting x 7→ xp and fixing an uniformizer. ε(jx∗M, jx∗ω) Y x∈|X| where q = #k = pf , g = genus(X), and F is the f -th power of the Frobenius which acts linearly on the rigid cohomology with compact supi (U, M ). port Hrig,c Theorem. The product formula is satisfied for: 1. Overconvergent F -isocrystals of rank one, 2. Unit-root overconvegent F -isocrystals with finite global monodromy. p-adic differential equations Let U ⊂ X be a non-empty open. Roughly speaking, an F -isocrystal on U overconvergent along X\U is a linear system of p-adic differential equations with singularities contained in X\U (cf. [Ber96] for a precise definition). Remark. The product formula generalizes the Grothendieck-OggShafarevich formula for p-adic coefficients (Christol-Mebkhout) 2 X (−1) i i dimC Hrig,c (U, M ) = χc(U )rk(M ) − i=0 • a Frobenius ϕ : js∗M → js∗M , which is a σ-linear horizontal homomorphism, whose image spans js∗M . deg(s) irrs(M ) s6∈|U | Local description Let M be an overconvergent F -isocrystal, for every s ∈ |X| closed, the pullback of M by js : Spec Ks → X gives a (ϕ, ∇)-module over Rs: a free Rs-module js∗M of finite rank, endowed with b1 • a connection ∇ : js∗M → js∗M ⊗Rs Ω Rs/C(s), X The product formula for `-adic epsilon factors The analogous statement for `-adic sheaves (` 6= p) was conjectured by Deligne and proved by Laumon [Lau87]. Replace: • overconvergent F -isocrystals over U by `-adic constructible sheaves, smooth over U . i i • rigid cohomology Hrig,c by étale cohomology Het,c js∗M horizontal sections ) s ∈ |U | It is isomorphic to Rrank(M with Ker ∇ = ϕ-module over C(s) s the trivial connection s ∈ X\U more interesting object by taking horizontal sections over the “universal” covering Remark. Thanks to the p-adic mon- of R , we get a Weil-Deligne s odromy theorem, the dimension of representation WD(j ∗M ) of s ∗ WD(js M ) is equal to rank(M ). the Weil group W (Kssep/Ks). `-adic Perverse complexes of sheaves p-adic? Holonomic Arithmetic D-modules Global Fourier Transform (Deligne) Global Fourier Transform (Huyghe) Local Fourier Transform (Laumon) Local Fourier Transform (Crew) `-adic stationary phase (Laumon) A p-adic stationary phase? Table: Main tools for the proof of the general case References I [Ber96] Berthelot, P.: Cohomologie rigide et cohomologie rigide à supports propres. Première partie. Prépublication IRMR 96-03 (1996). [CM02] Christol, G.; Mebkhout, Z.: Équations différentielles p-adiques et coefficients p-adiques sur les courbes, Astérisque 279 (2002), 125-183. [Del73] Deligne, P.: Les constantes locales des équations fonctionnelles des fonctions L, in: Modular functions of one variable II, LNM 349 (1973), 501-597. [Lau87] Laumon, G.: Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Publ. Math. de I.H.E.S. 65 (1987), 131-210. [Mar08] Marmora, A.: Facteurs epsilon p-adiques, Compositio Mathematica 144 (2008), 439-483. [Noo04] Noot-Huyghe, C.: Transformation de Fourier des D-modules arithémetiques I, in: Geometric aspects of Dwork theory, Vol. II, WdG, (2004), 857-907. Journées de Géométrie Arithmétique de Rennes, 6th-10th July 2009, Institut de Recherche Mathématique de Rennes, Université de Rennes I, Rennes, France