Product formula for p -adic epsilon
               factors

AbstractLet $X$ be a smooth proper curve over a finite field of characteristic $p$. We prove a product formula for $p$-adic epsilon factors of arithmetic $\mathscr{D}$-modules on $X$. In particular we deduce the analogous formula for overconvergent $F$-isocrystals, which was conjectured previously. The $p$-adic product formula is a counterpart in rigid cohomology of the Deligne–Laumon formula for epsilon factors in $\ell$-adic étale cohomology (for $\ell \neq p$). One of the main tools in the proof of this $p$-adic formula is a theorem of regular stationary phase for arithmetic $\mathscr{D}$-modules that we prove by microlocal techniques.

Download Full-text

On middle cohomology of special Artin–Schreier varieties and finite Heisenberg groups

Forum Mathematicum ◽

10.1515/forum-2017-0085 ◽

2019 ◽

Vol 31 (1) ◽

pp. 83-110

Author(s):

Takahiro Tsushima

Keyword(s):

Finite Field ◽

Cohomology Group ◽

Heisenberg Groups ◽

Base Field ◽

Etale Cohomology ◽

Étale Cohomology ◽

Coprime Integers ◽

Affine Scheme

Abstract We introduce a certain Artin–Schreier scheme over a finite field associated to a pair of coprime integers {(m,n)} with {n\geq 3} divisible by the characteristic of the base field, and study the middle étale cohomology group of it. If m is even, the variety admits actions of some finite Heisenberg groups. We study the middle cohomology as representations of the Heisenberg groups. If m is odd, we compute the Frobenius eigenvalues of it concretely. This affine scheme comes from the reduction of a certain affinoid in a Lubin–Tate space.

Download Full-text

Galois representations on the cohomology of hyper-Kähler varieties

Mathematische Zeitschrift ◽

10.1007/s00209-021-02923-3 ◽

2022 ◽

Author(s):

Salvatore Floccari

Keyword(s):

Finite Field ◽

Galois Representations ◽

Finite Extension ◽

Tate Conjecture ◽

Etale Cohomology ◽

Technical Assumption ◽

Étale Cohomology

AbstractWe show that the André motive of a hyper-Kähler variety X over a field $$K \subset {\mathbb {C}}$$ K ⊂ C with $$b_2(X)>6$$ b 2 ( X ) > 6 is governed by its component in degree 2. More precisely, we prove that if $$X_1$$ X 1 and $$X_2$$ X 2 are deformation equivalent hyper-Kähler varieties with $$b_2(X_i)>6$$ b 2 ( X i ) > 6 and if there exists a Hodge isometry $$f:H^2(X_1,{\mathbb {Q}})\rightarrow H^2(X_2,{\mathbb {Q}})$$ f : H 2 ( X 1 , Q ) → H 2 ( X 2 , Q ) , then the André motives of $$X_1$$ X 1 and $$X_2$$ X 2 are isomorphic after a finite extension of K, up to an additional technical assumption in presence of non-trivial odd cohomology. As a consequence, the Galois representations on the étale cohomology of $$X_1$$ X 1 and $$X_2$$ X 2 are isomorphic as well. We prove a similar result for varieties over a finite field which can be lifted to hyper-Kähler varieties for which the Mumford–Tate conjecture is true.

Download Full-text

Étale Steenrod operations and the Artin–Tate pairing

Compositio Mathematica ◽

10.1112/s0010437x20007216 ◽

2020 ◽

Vol 156 (7) ◽

pp. 1476-1515

Author(s):

Tony Feng

Keyword(s):

Finite Field ◽

Algebraic Topology ◽

Affirmative Answer ◽

Characteristic Classes ◽

Brauer Group ◽

Tate Pairing ◽

Steenrod Operations ◽

Etale Cohomology ◽

Étale Cohomology ◽

Topology Of Manifolds

We prove a 1966 conjecture of Tate concerning the Artin–Tate pairing on the Brauer group of a surface over a finite field, which is the analog of the Cassels–Tate pairing. Tate asked if this pairing is always alternating and we find an affirmative answer, which is somewhat surprising in view of the work of Poonen–Stoll on the Cassels–Tate pairing. Our method is based on studying a connection between the Artin–Tate pairing and (generalizations of) Steenrod operations in étale cohomology. Inspired by an analogy to the algebraic topology of manifolds, we develop tools allowing us to calculate the relevant étale Steenrod operations in terms of characteristic classes.

Download Full-text