arXiv subjects

Félix Baril Boudreau

Publications and source records attributed to Félix Baril Boudreau.

6 recordsLinked to original sources

arXiv2025

Arithmetic rank bounds for abelian varieties over function fields

It follows from the Grothendieck-Ogg-Shafarevich formula that the rank of an abelian variety (with trivial trace) defined over the function field of a curve is bounded by a quantity which depends on the genus of the base curve and on bad reduction data. Using a function field version of classical $\ell$-descent techniques, we derive an arithmetic refinement of this bound, extending previous work of the second and third authors from elliptic curves to abelian varieties, and improving on their result in the case of elliptic curves. When the abelian variety is the Jacobian of a hyperelliptic curve, we produce a more explicit $2$-descent map. Then we apply this machinery to studying points on the Jacobians of certain genus $2$ curves over $k(t)$, where $k$ is some perfect base field of characteristic not $2$.

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arXiv2025

Quadratic Euler-Kronecker constants in positive characteristic

In 2006, Ihara defined and systematically studied a generalization of the Euler-Mascheroni constant for all global fields, named the Euler-Kronecker constants. This paper examines their distribution across geometric quadratic extensions of a rational global function field, via the values of logarithmic derivatives of Dirichlet $L$-functions at 1. Using a probabilistic model, we show that the values converge to a limiting distribution with a smooth, positive density function, as the genii of quadratic fields approach infinity. We then prove a discrepancy theorem for the convergence of the frequency of these values, and obtain information about the proportion of the small values. Finally, we prove omega results on the extreme values. Our theorems imply new distribution results on the stable Taguchi heights and logarithmic Weil heights of rank 2 Drinfeld modules with CM.

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arXiv2024

Idempotentization of Affine Schemes and Sheaves

In this article, we introduce the idempotentization process, which bears some philosophical and mathematical similarities with modern analytification and tropicalization. Idempotentization associates to any affine scheme an idempotent version of itself with respect to a fixed covering by distinguished affine open subschemes. Once this cover is fixed, we can functorially associate a Zariski sheaf of rings or modules to a sheaf of idempotent semiring or a sheaf of idempotent semimodules. We show that idempotentization is independent of the chosen cover and in the Noetherian case, the idempotentization of the structure sheaf recovers the global sections. Underlying our formalism is a combinatorial reflection of lattices of subobjects of ordered-theoretic objects seen as lattices coming from commutative algebra. This has topological consequences for the semiring of subtractive ideals of a commutative semiring $S$: On one hand, it is a topological retract of the semiring of congruence relations of $S$ for the coarse lower topology. On the other hand, it is a topological retract of the semiring of ideals of $S$ for the coarse upper topology.

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arXiv2024

Kummer theory for function fields

We develop Kummer theory for algebraic function fields in finitely many transcendental variables. We consider any finitely generated Kummer extension (possibly, over a cyclotomic extension) of an algebraic function field, and describe the structure of its Galois group. Our results show in a precise sense how the questions of computing the degrees of these extensions and of computing the group structures of their Galois groups reduce to the corresponding questions for the Kummer extensions of their constant fields.

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arXiv2023

Adelic perturbation of rational functions and applications

Let $\sum a_nx^n\in\bar{\mathbb{Q}}[[x]]$ be the power series representation of a rational function and let $f:\ \{0,1,\ldots\}\rightarrow \bar{\mathbb{Q}}$ be a so-called almost quasi-polynomial. Under a necessary stability condition, we prove that $\sum f(n)a_nx^n$ satisfies the Pólya-Carlson dichotomy: it is either a rational function or it cannot be extended analytically to a strictly larger domain than its disk of convergence. This latter property is much stronger than being transcendental. The first application and motivation of our result is the solution of a conjecture by Byszewski-Cornelissen. This gives a complete understanding of the analytic continuation behavior of the Artin-Mazur zeta function associated to a dynamical system on an abelian variety. Further applications include the solution of a conjecture by Bell-Miles-Ward and a significant case of an open problem by Royals-Ward.

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arXiv2022

$L$-Functions of Elliptic Curves Modulo Integers

In 1985, Schoof devised an algorithm to compute zeta functions of elliptic curves over finite fields by directly computing the numerators of these rational functions modulo sufficiently many primes (see \cite{schoof_1985}). If $E/K$ is an elliptic curve with nonconstant $j$-invariant defined over a function field $K$ of characteristic $p \geq 5$, we know that its $L$-function $L(T,E/K)$ is a polynomial in $\mathbb{Z}[T]$ (see \cite[p.11]{katz_2002}). Inspired by Schoof, we study the reduction of $L(T,E/K)$ modulo integers. We obtain three main results. Firstly, if $E/K$ has non-trivial $K$-rational $N$-torsion for some integer $N$ coprime with $p$, we extend a formula for $L(T,E/K) \bmod N$ due to Hall (see \cite[p.133, Theorem 4]{hall_2006}) to all quadratic twists $E_f/K$ with $f \in K^\times \smallsetminus K^{\times 2}$. Secondly, without any condition on the $2$-torsion subgroup of $E(K)$, we give a formula for the quotient modulo $2$ of $L$-functions of any two quadratic twists of $E/K$. Thirdly, we use these results to compute the global root numbers of an infinite family of quadratic twists of an elliptic curve and in most cases find the exact analytic rank of each of these twists. We also illustrate that in favourable situations our second main result allows one to compute much more efficiently $L(T,E_f/K) \bmod 2$ than an algorithm of Baig and Hall (see \cite{baig_hall_2012}). Finally, we use our formulas to compute directly some degree $2$ $L$-functions.

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