Adriano Marmora The motivation: the product formula for p

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