Research

Publications and preprints

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.

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.

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.

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.

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.

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. 

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.

Work in progress

I. Mahler measures of successively exact polynomials (joint with François Brunault)

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.

II. Standard conjectures for arithmetic Grassmannians and toric varieties
(joint with Paolo Dolce and Roberto Gualdi)

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.

III. Diophantine properties for special values of Dedekind zeta functions
 (joint with Jerson Caro and Fabien Pazuki)

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.

IV. Computing Mahler measures via differential equations (joint with Berend Ringeling)

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*.

Theses

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.

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".

Collaborators

François Brunault
Francesco Campagna
Paolo Dolce
Roberto Gualdi