Zeta functions of trinomial curves and maximal curves

Zeta functions for function fields

We introduce new non-abelian zeta functions for curves defined over finite fields. There are two types, i.e., pure non-abelian zetas defined using semi-stable bundles, and group zetas defined for pairs consisting of (reductive group, maximal parabolic subgroup). Basic properties such as rationality and functional equation are obtained. Moreover, conjectures on their zeros and uniformity are given. We end this paper with an explanation on why these zetas are non-abelian in nature, using our up-coming works on 'parabolic reduction, stability and the mass'. The constructions and results were anno…

On some curve singularity invariants and reductions

This paper deals with the study of the behaviour of the value semigroup of a curve singularity define over a global field reduced modulo a maximal ideal. We also define a global zeta function of the curve by means of motivic integration over a suitable ring of adéles, whose reduction modulo a maximal ideal will coincide with some already known zeta functions of the singularity.

On the coefficients of the Stöhr Zeta Function

Let $O$ be a one-dimensional Cohen-Macaulay local ring having a finite field as a coefficient field. The aim of this work is to extend the explicit computations of the Stöhr Zeta Function of $O$ for one and two branches to an arbitrary number of them, obtaining in this general case an upper bound for the coefficients of the zeta function, instead of an equality. The calculations are based on the use of the value semigroup of a curve singularity and a suitable classification of the maximal points of the semigroup.

Lower bound for the poles of Igusa's p-adic zeta functions

Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P. Let f be a polynomial over R in n>1 variables and let χbe a character of R^{\times}. Let M_i(u) be the number of solutions of f=u in (R/P^i)^n for i \in \mathbb{Z}_{\geq 0} and u \in R/P^i. These numbers are related with Igusa's p-adic zeta function Z_{f,χ}(s) of f. We explain the connection between the M_i(u) and the smallest real part of a pole of Z_{f,χ}(s). We also prove that M_i(u) is divisible by q^{\ulcorner(n/2)(i-1)\urcorner}, where the corners indicate that we …

Igusa zeta functions and the non-archimedean SYZ fibration

We explain the proof, obtained in collaboration with Chenyang Xu, of a 1999 conjecture of Veys about poles of maximal order of Igusa zeta functions. The proof technique is based on the Minimal Model Program in birational geometry, but the proof was heavily inspired by ideas coming from non-archimedean geometry and mirror symmetry; we will outline these relations at the end of the paper. This text is intended to be a low-tech introduction to these topics; we only assume that the reader has a basic knowledge of algebraic geometry.

Motivic Euler products and motivic height zeta functions

A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes, in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a topology on the Grothendieck ring of varieties coming from the th…

Elliptic curves with large rank over function fields

We produce explicit elliptic curves over \Bbb F_p(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related elliptic surfaces) and then use zeta functions to determine the rank. In contrast to earlier examples of Shafarevitch and Tate, our curves are not isotrivial. Asymptotically these curves have maximal rank for their conductor. Motivated by this fact, we make a conjecture about the growth of ranks of elliptic curves over number fields.

平面曲线的多变量强单值群猜想

The Multivariable Strong Monodromy Conjecture for Plane Curves

该研究针对全纯芽组建立迭代残差障碍，证明局部多变量拓扑ζ函数的极大阶极超平面及约化平面曲线芽组的实际极超平面均含于Bernstein–Sato零轨迹，从而证明平面曲线的拓扑多变量强单值群猜想。

On the motivic oscillation index and bound of exponential sums modulo $p^m$ via analytic isomorphisms

Let $f$ be a polynomial in $n$ variables over some number field and $Z$ a subscheme of affine $n$-space. The notion of motivic oscillation index of $f$ at $Z$ was initiated by Cluckers (2008) and Cluckers-Mustaţǎ-Nguyen (2019). In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta functions of $f$ has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with b…

Improved upper bounds for the number of points on curves over finite fields

We give new arguments that improve the known upper bounds on the maximal number N_q(g) of rational points of a curve of genus g over a finite field F_q for a number of pairs (q,g). Given a pair (q,g) and an integer N, we determine the possible zeta functions of genus-g curves over F_q with N points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus-g curve over F_q with N points must have a low-degree map to another curve over F_q, and often this is enough to give us a contradiction. In particular, we able to provide eight previously unk…