Research

Last update: March 2024
Status: Work in progress
Abstract:  We extend the recent study of Diophantine properties for special values of Dedekind zeta functions at non-integral points carried out by Généreaux and Lalín. More precisely, we use results of Stankus and Soundararajan to show that the special values of the Dedekind zeta functions associated to quadratic number fields do not satisfy the Bogomolov property in the right half of the critical strip. Moreover, we study the Diophantine properties associated to the special values of the completed Dedekind zeta functions.

Last update: February 2024
Status: Work in progress
Abstract:  We provide an efficient presentation of an algorithm which computes the Mahler measure of a Laurent polynomial P (in any number of variables) by computing the differential equation corresponding to the linear recurrence with polynomial coefficients satisfied by the constant terms of the powers of the Laurent polynomial P P* - k, where P* is the conjugate reciprocal of P, and k is the constant term of P P*.

Last update: February 2024
Status: Work in progress
Abstract:  We give a cohomological and rigorous interpretation of the notion of successive exactness of a Laurent polynomial, which was introduced by Lalín developing ideas of Maillot. We show moreover how this notion is particularly useful to predict the existence of some relations between the Mahler measure of a Laurent polynomial and special values of certain L-functions at different integers.

Last update: March 2024
Status: Work in progress
Abstract:  We study the standard conjectures proposed by Gillet and Soulé for arithmetically ample line bundles on arithmetic varieties. In particular, we prove that the strong form of these conjectures holds true for projective spaces, generalizing work of Künnemann-Maillot and Takeda. We provide moreover an almost complete classification of the arithmetic line bundles which satisfy these standard conjectures on the projective space. This classification is based on a classification of those (1,1)-forms that satisfy analogues of the standard conjectures on suitable spaces of differential forms, which can be of independent interest.

Last update: October 2023
Status: Submitted
Links: arXiv, HAL, ResearchGate
Abstract:  We provide two different proofs of an irreducibility criterion for the preimages of a transverse subvariety of a product of elliptic curves under a diagonal endomorphism of sufficiently large degree. For curves, we present an arithmetic proof of the aforementioned irreducibility result, which enlightens connections to methods used in the context of the Torsion Anomalous Conjecture. On the other hand, we generalize the result for higher dimensional varieties using a more geometric approach. Finally, we give some applications of these results. More precisely, we establish the irreducibility of some explicit families of polynomials, we provide new estimates for the normalized heights of certain intersections and images, and we give new lower bounds for the essential minima of preimages.

Last update: October 2023
Status: Submitted
Links: arXiv, HAL, ResearchGate, MPIM
Abstract: Using the special value at u=1 of Artin-Ihara L-functions, we associate to every ℤ-cover of a finite graph a polynomial which we call the Ihara polynomial. We show that the number of spanning trees for the finite intermediate graphs of such a cover can be expressed in terms of the Pierce-Lehmer sequence associated to a factor of the Ihara polynomial. This allows us to express the asymptotic growth of the number of spanning trees in terms of the Mahler measure of this polynomial. Specializing to the situation where the base graph is a bouquet or the dumbbell graph gives us back previous results in the literature for circulant and I-graphs (including the generalized Petersen graphs). We also express the p-adic valuation of the number of spanning trees of the finite intermediate graphs in terms of the p-adic Mahler measure of the Ihara polynomial. When applied to a particular ℤ-cover, our result gives us back Lengyel's calculation of the p-adic valuations of Fibonacci numbers.

Last update: November 2022
Status: Accepted
Type: Journal article
Journal:   Annales de l'Institut Fourier
Links: DOI, arXiv, HAL, ResearchGate
Abstract: We prove that certain sequences of Laurent polynomials, obtained from a fixed Laurent polynomial P by monomial substitutions, give rise to sequences of Mahler measures which converge to the Mahler measure of P. This generalizes previous work of Boyd and Lawton, who considered univariate monomial substitutions. We provide moreover an explicit upper bound for the error term in this convergence, generalizing work of Dimitrov and Habegger, and a full asymptotic expansion for a family of 2-variable polynomials, whose Mahler measures were studied independently by the third author.

Last update:   June 2022
Status: Published
Type: Book chapter
Book: Arithmetic, Geometry, Cryptography, and Coding Theory 2021
Links: arXiv, HAL, ResearchGate, DOI
Abstract: Using an analogue of Serre's open image theorem for elliptic curves with complex multiplication, one can associate to each CM elliptic curve E defined over a number field F a natural number which describes how big the image of the Galois representation associated to E is. We show how one can compute this number, using a closed formula that we obtain from the classical theory of complex multiplication.

Last update: November 2023
Status: Published
Type: Journal article
Journal: Revista Matematica Iberoamericana, Vol. 40, No. 1
Links: arXiv, HAL, ResearchGate, DOI
Abstract: We propose an investigation on the Northcott, Bogomolov and Lehmer properties for special values of L-functions. We first introduce an axiomatic approach to these three properties. We then focus on the Northcott property for special values of L-functions. We prove that such a property holds for the special value at zero of Dedekind zeta functions of number fields. In the case of L-functions of pure motives, we prove a Northcott property for special values located at the left of the critical strip, assuming the validity of the functional equation.

Last update: March 2022
Status: Published
Type: Journal article
Journal: Pacific Journal of Mathematics,  Vol. 317, No. 1, 2022
Links: arXiv, HAL, ResearchGate, DOI
Abstract: For every elliptic curve E with complex multiplication (CM) defined over a number field F which contains the CM field K, we prove that the family of p^∞-division fields of E, where p ranges over the rational primes, becomes linearly disjoint over F after removing an explicit finite subfamily of fields. We then give a necessary condition for this finite subfamily to be entangled over F, which is always met when F = K. In this case, and under the further assumption that the elliptic curve E is obtained as a base-change from ℚ, we describe in detail the entanglement in the family of division fields of E. 

Last Update: May 2020
Status: Under revision
Links: arXiv, HAL, ResearchGate
Abstract: Given an elliptic curve E defined over ℚ, which has potential complex multiplication by the ring of integers of an imaginary quadratic field K, we construct a polynomial P ∈ ℤ[x,y] which is a planar model of E, and such that the Mahler measure of P is related to the special value of the L-function L(E,s) at s = 2.

Type: PhD thesis
Defense date: October 9th, 2020
Advisers: Ian Kiming, Fabien Pazuki
Jury: José Ignacio Burgos Gil, Morten Risager, Wadim Zudilin
University: University of Copenhagen
Links: KU, TEL
Abstract: The first four chapters of this thesis contain a survey of some parts of the theories of heights, motives, L-functions and Mahler measures, and the seventh chapter contains a survey of the theory of complex multiplication. The fifth chapter exposes a summary of the joint work in progress with François Brunault. Moreover, the sixth chapter and part of the seventh concern joint work in progress with Francesco Campagna, and the last section of the third chapter contains a summary of the article On the Northcott property for special values of L-functions, written jointly with Fabien Pazuki. Finally, the last two chapters consist of expositions of the two articles Entanglement in the family of division fields of elliptic curves with complex multiplication (joint with Francesco Campagna) and Mahler's measure and elliptic curves with potential complex multiplication.

Type: MSc. Thesis
Defense date: July 24th, 2017
Adviser: Peter Bruin
Jury: Martin Bright, Bas Edixhoven
Universities: University of Milan, Leiden University
Links: ALGANT, Leiden University
Abstract: The goal of this MSc. project was to extend to compact modular curves the adelic description of affine modular curves that is given in Proposition 2.7 of James Milne's paper "Canonical models of Shimura curves".