arXiv subjects

Vincent Maillot

Publications and source records attributed to Vincent Maillot.

6 recordsLinked to original sources

arXiv2026

The conjecture of Colmez and reciprocity laws for modular forms

In an article published in 1993, P. Colmez formulated a remarkable conjecture, which asserts that the Faltings height of a CM abelian variety can be computed as a linear combination of logarithmic derivatives of Artin $L$-functions. Noting that the Faltings height is an average of transcendental quantities summed over the embeddings of a number field of definition of the abelian variety, we propose a refinement of this conjecture, which identifies each of these transcendental quantities. We also show how our conjecture would imply the existence of fine reciprocity laws for Siegel modular forms with rational coefficients evaluated at CM points, and we prove our conjecture for elliptic curves, using old results of Siegel and Hasse on elliptic units.

math.NT↗

arXiv2018

Conjectures on the logarithmic derivatives of Artin L-functions II

We formulate a general conjecture relating Chern classes of subbundles of Gauss-Manin bundles in Arakelov geometry to logarithmic derivatives of Artin L-functions of number fields. This conjecture may be viewed as a far-reaching generalisation of the (Lerch-)Chowla-Selberg formula computing logarithms of periods of elliptic curves in terms of special values of the $Γ$-function. We prove several special cases of this conjecture in the situation where the involved Artin characters are Dirichlet characters. This article contains the computations promised in the article {\it Conjectures sur les dérivées logarithmiques des fonctions L d'Artin aux entiers négatifs}, where our conjecture was announced. We also give a quick introduction to the Grothendieck-Riemann-Roch theorem and to the geometric fixed point formula, which form the geometric backbone of our conjecture.

math.AG↗

arXiv2014

On a canonical class of Green currents for the unit sections of abelian schemes

We show that on any abelian scheme over a complex quasi-projective smooth variety, there is a Green current for the zero-section, which is axiomatically determined up to $\partial$ and $\bar\partial$-exact differential forms. This current generalizes the Siegel functions defined on elliptic curves. We prove generalizations of classical properties of Siegel functions, like distribution relations, limit formulae and reciprocity laws.

math.AG↗

arXiv2008

Formes automorphes et theoremes de Riemann-Roch arithmetiques

Nous construisons trois familles de formes automorphes au moyen du theoreme de Riemann-Roch arithmetique et de la formule de Lefschetz arithmetique. Deux de ces familles ont deja ete construites par Yoshikawa et notre construction met en lumiere leur origine arithmetique. ----- We construct three families of automorphic forms following the arithmetic Riemann-Roch theorem and the arithmetic Lefschetz formula. Two of these families were already constructed by Yoshikawa and our construction illuminates their arithmetic origin.

math.AG↗

arXiv2007

On the determinant bundles of abelian schemes

Let $π:\CA\ra S$ be an abelian scheme over a scheme $S$ which is quasi-projective over an affine noetherian scheme and let $\CL$ be a symmetric, rigidified, relatively ample line bundle on $\CA$. We show that there is an isomorphism \det(π_*\CL)^{øtimes 24}\simeq\big(π_*ω_{\CA}^{\vee}\big)^{øtimes 12d} of line bundles on $S$, where $d$ is the rank of the (locally free) sheaf $π_*\CL$. We also show that the numbers 24 and $12d$ are sharp in the following sense: if $N>1$ is a common divisor of 12 and 24, then there are data as above such that \det(π_*\CL)^{øtimes (24/N)}\not\simeq\big(π_*ω_{\CA}^{\vee}\big)^{øtimes (12d/N)}.

math.AG↗

arXiv2002

Conjectures sur les dérivées logarithmiques des fonctions L d'Artin aux entiers négatifs

We formulate several variants of a conjecture relating the arithmetic degree of certain hermitian fibre bundles with the values of the logarithmic derivative of Artin's L-functions at negative integers. This generalizes conjectures by Colmez and Gross-Deligne and complements Beilinson's conjectures for the Artin motives. We announce several results in the direction of these statements.

math.AG↗