Searcharxiv⌕ Search

arXiv subjects

Alexey Zykin

Publications and source records attributed to Alexey Zykin.

9 recordsLinked to original sources
arXiv2017

Dense families of modular curves, prime numbers and uniform symmetric tensor rank of multiplication in certain finite fields

We obtain new uniform bounds for the symmetric tensor rank of multiplication in finite extensions of any finite field Fp or Fp2 where p denotes a prime number greater or equal than 5. In this aim, we use the symmetric Chudnovsky-type generalized algorithm applied on sufficiently dense families of modular curves defined over Fp2 attaining the Drinfeld-Vladuts bound and on the descent of these families to the definition field Fp. These families are obtained thanks to prime number density theorems of type Hoheisel, in particular a result due to Dudek (2016).

math.NT↗
arXiv2017

On M-functions associated with modular forms

Let $f$ be a primitive cusp form of weight $k$ and level $N,$ let $χ$ be a Dirichlet character of conductor coprime with $N,$ and let $\mathfrak{L}(f\otimes χ, s)$ denote either $\log L(f\otimes χ, s)$ or $(L'/L)(f\otimes χ, s).$ In this article we study the distribution of the values of $\mathfrak{L}$ when either $χ$ or $f$ vary. First, for a quasi-character $ψ\colon \mathbb{C} \to \mathbb{C}^\times$ we find the limit for the average $\mathrm{Avg}\_χψ(L(f\otimesχ, s)),$ when $f$ is fixed and $χ$ varies through the set of characters with prime conductor that tends to infinity. Second, we prove an equidistribution result for the values of $\mathfrak{L}(f\otimes χ,s)$ by establishing analytic properties of the above limit function. Third, we study the limit of the harmonic average $\mathrm{Avg}^h\_f ψ(L(f, s)),$ when $f$ runs through the set of primitive cusp forms of given weight $k$ and level $N\to \infty.$ Most of the results are obtained conditionally on the Generalized Riemann Hypothesis for $L(f\otimesχ, s).$

math.NT↗
arXiv2014

On the Number of Rational Points of Jacobians over Finite Fields

In this article we prove lower and upper bounds for class numbers of algebraic curves defined over finite fields. These bounds turn out to be better than most of the previously known bounds obtained using combinatorics. The methods used in the proof are essentially those from the explicit asymptotic theory of global fields. We thus provide a concrete application of effective results from the asymptotic theory of global fields and their zeta-functions.

math.NT↗
arXiv2013

Asymptotic properties of zeta functions over finite fields

In this paper we study asymptotic properties of families of zeta and $L$-functions over finite fields. We do it in the context of three main problems: the basic inequality, the Brauer--Siegel type results and the results on distribution of zeroes. We generalize to this abstract setting the results of Tsfasman, Vlăduţ and Lachaud, who studied similar problems for curves and (in some cases) for varieties over finite fields. In the classical case of zeta functions of curves we extend a result of Ihara on the limit behaviour of the Euler--Kronecker constant. Our results also apply to $L$-functions of elliptic surfaces over finite fields, where we approach the Brauer--Siegel type conjectures recently made by Kunyavskii, Tsfasman and Hindry.

math.NT↗
arXiv2009

Asymptotic properties of Dedekind zeta functions in families of number fields

The main goal of this paper is to prove a formula that expresses the limit behaviour of Dedekind zeta functions for $\Re s > 1/2$ in families of number fields, assuming that the Generalized Riemann Hypothesis holds. This result can be viewed as a generalization of the Brauer--Siegel theorem. As an application we obtain a limit formula for Euler--Kronecker constants in families of number fields.

math.NT↗
arXiv2008

Jacobians among Abelian threefolds: a formula of Klein and a question of Serre

Let k be a field and f be a Siegel modular form of weight h \geq 0 and genus g>1 over k. Using f, we define an invariant of the k-isomorphism class of a principally polarized abelian variety (A,a)/k of dimension g. Moreover when (A,a) is the Jacobian of a smooth plane curve, we show how to associate to f a classical plane invariant. As straightforward consequences of these constructions, when g=3 and k is a subfield of the complex field, we obtain (i) a new proof of a formula of Klein linking the modular form χ_{18} to the square of the discriminant of plane quartics ; (ii) a proof that one can decide when (A,a) is a Jacobian over k by looking whether the value of χ_{18} at (A,a) is a square in k. This answers a question of J.-P. Serre. Finally, we study the possible generalizations of this approach for g>3.

math.NT↗
arXiv2004

The Brauer-Siegel and Tsfasman-Vladut Theorems for Almost Normal Extensions of Number Fields

The classical Brauer-Siegel theorem states that if $k$ runs through the sequence of normal extensions of $\mathbb{Q}$ such that $n_k/\log|D_k|\to 0,$ then $\log h_k R_k/\log \sqrt{|D_k|}\to 1.$ First, in this paper we obtain the generalization of the Brauer-Siegel and Tsfasman-Vlăduţ theorems to the case of almost normal number fields. Second, using the approach of Hajir and Maire, we construct several new examples concerning the Brauer-Siegel ratio in asymptotically good towers of number fields. These examples give smaller values of the Brauer-Siegel ratio than those given by Tsfasman and Vlăduţ

math.NT↗