arXiv subjects

Dominique Bernardi

Publications and source records attributed to Dominique Bernardi.

5 recordsLinked to original sources

arXiv2020

Idéaux premiers totalement décomposés et sommes de Newton

Let $K$ be a number field and $f\in K[X]$ an irreducible monic polynomial with coefficients in $O_K$, the ring of integers of $K$. We aim to enounce an effective criterion, in terms of the Galois group of $f$ over $K$ and a linear recurrence sequence associated to $f$, allowing sometimes to characterize the prime ideals of $O_K$ modulo which $f$ completely splits. If $α$ is a root of $f$, this criterion therefore gives a characterization of the prime ideals of $O_K$ which split completely in $K(α)$. It does apply if the degree of $f$ is at least $4$ and the Galois group of $f$ is the symmetric group or the alternating group.

math.NT↗

arXiv2020

Symboles modulaires et produit de Petersson

We revisit some papers by Eichler and Shimura in order to give an algebraic formulation (based on Farey symbols) for the intersection product on the space of modular symbols, as described by Pollack and Stevens. We define the period homomorphism of an Eisenstein series (Eisenstein-Dedekind-Stevens symbol) and extend the definition of the intersection product to these objects. We construct a computationally convenient basis for the space of Eisenstein series for $Γ$ 0 (N) with rational periods. Given a Farey symbol for a subgroup $Γ$ of the modular group and a subgroup $Γ$ ' of finite index of $Γ$, we give an algorithmic construction for a Farey symbol for $Γ$ '.

math.NT↗

arXiv2019

Polygones fondamentaux d'une courbe modulaire

A few pages in Siegel describe how, starting with a fundamental polygon for a compact Riemann surface, one can construct a symplectic basis of its homology. This note retells that construction, specializing to the case where the surface is associated to a congruence subgroup $Γ$ of $SL_2(Z)$. One then obtains by classical procedures a generating system for $Γ$ with a minimal number of hyperbolic elements and a presentation of the $Z[Γ]$-module $Z[P^1(Q)]_0$.

math.NT↗

arXiv2018

La constante de Manin et le degré modulaire d'une courbe elliptique

We revisit the calculation of the strong Weil curve in an isogeny class of elliptic curves over Q, of the Manin constant and modular degree of an elliptic curve, using modular symbols as defined in [Pollack-Stevens], now implemented in Pari/GP. There is no innovation claim.

math.NT↗

arXiv2009

On empty lattice simplices in dimension 4

We give an almost complete classification of empty lattice simplices in dimension 4 using the conjectural results of Mori-Morrison-Morrison, later proved by Sankaran and Bober. In particular, all of these simplices correspond to cyclic quotient singularities, and all but finitely many of them have width bounded by 2.

math.AG↗