# Batalin-Vilkovisky formalism in the $p$-adic Dwork theory

## TL;DR

This paper develops a Batalin-Vilkovisky formalism within $p$-adic Dwork theory, constructing a $p$-adic dGBV algebra that links cohomology, Frobenius operators, and zeta functions of algebraic varieties over finite fields.

## Contribution

It introduces a novel $p$-adic dGBV algebra framework and provides a deformation theoretic interpretation of Dwork's zeta function theory.

## Key findings

- Constructed a $p$-adic dGBV algebra for smooth projective varieties.
- Expressed the $p$-adic Dwork Frobenius operator via homotopy Lie morphisms.
- Derived a formula for the Frobenius operator using Bell polynomials.

## Abstract

The goal of this article is to develop BV (Batalin-Vilkovisky) formalism in the $p$-adic Dwork theory. Based on this formalism, we explicitly construct a $p$-adic dGBV algebra (differential Gerstenhaber-Batalin-Vilkovisky algebra) for a smooth projective complete intersection variety $X$ over a finite field, whose cohomology gives the $p$-adic Dwork cohomology of $X$, and its cochain endomorphism (the $p$-adic Dwork Frobenius operator) which encodes the information of the zeta function $X$. As a consequence, we give a modern deformation theoretic interpretation of Dwork's theory of the zeta function of $X$ and derive a formula for the $p$-adic Dwork Frobenius operator in terms of homotopy Lie morphisms and the Bell polynomials.