Adriano Marmora The motivation: the product formula for p

Institut de Recherche
Mathématique Avancée
About p-adic Local Fourier Transform
Adriano Marmora
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
•Let kbe a finite field of characteristic p > 0,C/ Frac(W(k)) a totally rami-
fied finite extension and Can algebraic closure of C.
•Let Xbe a proper and smooth curve over k, geometrically connected.
•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
OX,s its ms-completion,
C(s) = C⊗W(k)W(k(s)),|·|pap-adic absolute value,
The Robba’s rings associated to Ks= Frac( b OX,s)
Rs∼
=
g(t) = X
n∈Z
antn
an∈C(s),∃η < 1s.t. g(t)
converges for η < |t|p<1
R+
s∼
=
g(t) = X
n∈Z
antn∈ Rs
∀n∈Z,|an|p≤1
Some properties:
–R+
sis an henselian discrete valuation ring (not complete)
–its residue field identifies to Ks, by ¯ t7→ ts=an unif. of Ks.
–Frac(R+
s)⊂ Rsis dense.
b
Ω1
Rs/C(s)∼
=Rsdt continuous differential forms
Let σ:Rs→ Rsbe a Frobenius, i.e. an endomorphism lifting x7→ xpand
fixing an uniformizer.
p-adic differential equations
Let U⊂Xbe a non-empty open. Roughly speaking, an F-isocrystal on U
overconvergent along X\Uis a linear system of p-adic differential equations
with singularities contained in X\U(cf. [Ber96] for a precise definition).
Local description
Let Mbe an overconvergent F-isocrystal, for every s∈ |X|closed, the
pullback of Mby js: Spec Ks→Xgives a (ϕ, ∇)-module over Rs: a
free Rs-module j∗
sMof finite rank, endowed with
•a connection ∇:j∗
sM→j∗ sM⊗Rsb
Ω1
Rs/C(s),
•a Frobenius ϕ:j∗
sM→j∗ sM, which is a σ-linear horizontal homomor-
phism, whose image spans j∗
sM.
j∗
sMhorizontal sections
s∈ |U|It is isomorphic to Rrank(M)
swith
the trivial connection
Ker ∇=ϕ-module over C(s)
s∈X\Umore interesting object by taking horizontal sections
Remark. Thanks to the p-adic mon-
odromy theorem, the dimension of
WD(j∗
sM)is equal to rank(M).
over the “universal” covering
of Rs, we get a Weil-Deligne
representation WD(j∗
sM)of
the Weil group W(Ksep
s/Ks).
Local constants
Irregularity. It is a natural integer irrs(M) = irr(j∗
sM), defined by Christol
and Mebkhout as the height of the polygon of p-adic slopes [CM02].
Epsilon Factors. Fix a primitive p-root of unit ζ∈Cand let ωbe a non-
zero meromorphic form in Ω1
Ks/k(s). Deligne and Langlands associate to a
Weil-Deligne representation a non-zero element of C,
j∗
sM7−→ WD(j∗ sM)7−→ ε(j∗ sM, j∗ sω)∈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)=0and ε(j∗
sM, j∗ sω)is completely explicit.
The product formula
Conjecture. Let U⊂Xbe a non-empty open, MaF-isocristal overconver-
gent along X\U, and ωa non-zero meromorphic differential form. Then
2
Y
i=0
det(−F, Hi
rig,c(U, M))(−1)i+1 =q(1−g)rank(M)Y
x∈|X|
ε(j∗
xM, j∗ xω)
where q= #k=pf,g=genus(X), and Fis the f-th power of the
Frobenius which acts linearly on the rigid cohomology with compact sup-
port Hi
rig,c(U, M).
Theorem. The product formula is satisfied for:
1. Overconvergent F-isocrystals of rank one,
2. Unit-root overconvegent F-isocrystals with finite global monodromy.
Remark. The product formula generalizes the Grothendieck-Ogg-
Shafarevich formula for p-adic coefficients (Christol-Mebkhout)
2
X
i=0
(−1)idimCHi
rig,c(U, M) = χc(U)rk(M)−X
s6∈|U|
deg(s) irrs(M)
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.
•rigid cohomology Hi
rig,c by étale cohomology Hi et,c
-adic p-adic?
Perverse complexes of sheaves Holonomic Arithmetic D-modules
Global Fourier Transform (Deligne) Global Fourier Transform (Huyghe)
Local Fourier Transform (Laumon) Local Fourier Transform (Crew)
-adic stationary phase (Laumon) Ap-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