This is an expository paper on zeta functions of abelian varieties over finite fields. We would like to go through how zeta function is defined, and discuss the Weil conjectures. The main purpose of this paper is to fill in more details to the proofs provided in Milne. Subject to length constrain, we will not include a detailed proof for Riemann hypothesis in this paper. We will mainly be follo...

In this article, following an insight of Kontsevich, we extend the Weil conjecture, as well strong form Tate from realm algebraic geometry to broad noncommutative setting dg categories. Moreover, establish a functional equation for Hasse-Weil zeta functions, compute l-adic and p-adic absolute values eigenvalues cyclotomic Frobenius, provide complete description category numerical motives in ter...

Let OK be any domain with field of fractions K . Let F(x, y) ∈ OK [x, y] be a homogeneous polynomial of degree n, coprime to y, and assumed to have unit content (i.e., the coefficients of F generate the unit ideal in OK ). Assume that gcd(n, char(K )) = 1. Let h ∈ OK and assume that the polynomial hzn − F(x, y) is irreducible in K[x, y, z]. We denote by X F,h/K the nonsingular complete model of...

We give a construction of Connes-Moscovici’s cyclic cohomology for any Hopf algebra equipped with a character. Furthermore, we introduce a non-commutative Weil complex, which connects the work of Gelfand and Smirnov with cyclic cohomology. We show how the Weil complex arises naturally when looking at Hopf algebra actions and invariant higher traces, to give a non-commutative version of the usua...

Let Sg be a closed surface of genus g and Mg be the moduli space of Sg endowed with the Weil-Petersson metric. In this paper we investigate the Weil-Petersson curvatures of Mg for large genus g. First, we study the asymptotic behavior of the extremal Weil-Petersson holomorphic sectional curvatures at certain thick surfaces in Mg as g → ∞. Then we prove two curvature properties on the whole spac...

We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for WeilPetersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equivalent condition for Weil-Petersson geodesics. As an application, we show the...