(See also my list of publications on arXiv)
Preprints
[+] Abstract
Borisov and Gunnells have proved that certain linear combinations of products of Eisenstein series are Eisenstein series themselves, in analogy with the Manin relations for modular symbols. We devise a new method to determine and prove such relations, by differentiating with respect to the parameters of the Eisenstein series.
[+] Abstract
We give two constructions of families of elliptic curves over cubic or
quartic fields with three, respectively four, `integral' elements in the
kernel of the tame symbol on the curves.
The fields are in general non-Abelian, and the elements linearly independent.
For the integrality of the elements we prove a new criterion that does not ignore any torsion.
We also verify Beilinson's conjecture numerically for just over 90 of
the curves.
[+] Abstract
We prove a conjecture of Boyd and Rodriguez Villegas relating the Mahler measure of the polynomial \((1+x)(1+y)+z\) and the value at \(s=3\) of the \(L\)-function of an elliptic curve of conductor \(15\). The proof makes use of the computation by Zudilin and the author of the regulator of certain \(K_4\) classes on modular curves.
See also the PARI/GP and SageMath accompanying scripts.
[+] Abstract
We compute explicitly the Goncharov regulator integral associated to \(K_4\) classes on modular curves in terms of \(L\)-values of modular forms. We use this expression to connect it with the Beilinson regulator integral.
[+] Abstract
We construct elements in the group \(K_4\) of modular curves using the polylogarithmic complexes of weight 3 defined by Goncharov and De Jeu. The construction is uniform in the level and uses new modular units obtained as cross-ratios of division values of the Weierstrass \(\wp\) function. These units provide explicit triangulations of the 3-term relations in \(K_2\) of modular curves, which in turn give rise to elements in \(K_4\). Based on numerical computations and on recent results of W. Wang, we conjecture that these elements are proportional to the Beilinson elements defined using the Eisenstein symbol.
See also the PARI/GP and Magma accompanying scripts.
Books
[+] Abstract
The Mahler measure is a fascinating notion and an exciting topic in contemporary mathematics, interconnecting with subjects as diverse as number theory, analysis, arithmetic geometry, special functions and random walks. This friendly and concise introduction to the Mahler measure is a valuable resource for both graduate courses and self-study. It provides the reader with the necessary background material, before presenting the recent achievements and the remaining challenges in the field. The first part introduces the univariate Mahler measure and addresses Lehmer's question, and then discusses techniques of reducing multivariate measures to hypergeometric functions. The second part touches on the novelties of the subject, especially the relation with elliptic curves, modular forms and special values of \(L\)-functions. Finally, the Appendix presents the modern definition of motivic cohomology and regulator maps, as well as Deligne–Beilinson cohomology. The text includes many exercises to test comprehension and challenge readers of all abilities.
Errata
Articles
[+] Abstract
We prove that certain sequences of Laurent polynomials, obtained from a fixed multivariate 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.
[+] Abstract
We show that any homomorphism between Jacobians of modular curves arises from a linear combination of Hecke modular correspondences. The proof uses the adelic language and is based on a study of the actions of \(\mathrm{GL}_2\) and Galois on the étale cohomology of the tower of modular curves. We also make this result explicit for Ribet's twisting operators on modular abelian varieties.
[+] Abstract
In this article we study the fields generated by the Fourier coefficients of modular forms at arbitrary cusps. We prove that these fields are contained in certain cyclotomic extensions of the field generated by the Fourier coefficients at infinity. We also show that this bound is tight in the case of newforms with trivial Nebentypus. The main tool is a result of Shimura on the interplay between the actions of \(\mathrm{GL}_2^+(\mathbf{Q})\) and \(\mathrm{Aut}(\mathbf{C})\) on modular forms.
See also our Note on Fourier expansions at cusps.
[+] Abstract
We develop a new method for relating Mahler measures of three-variable polynomials that define elliptic modular surfaces to \(L\)-values of modular forms. Using an idea of Deninger, we express the Mahler measure as a Deligne period of the surface and then apply the first author's extension of the Rogers-Zudilin method for Kuga-Sato varieties, to arrive at an \(L\)-value.
[+] Abstract
We prove an equivariant version of Beilinson's conjecture on non-critical \(L\)-values of strongly modular abelian varieties over number fields. The proof builds on Beilinson's theorem on modular curves as well as a modularity result for endomorphism algebras. As an application, we prove a weak version of Zagier's conjecture on \(L(E,2)\) and Deninger's conjecture on \(L(E,3)\) for non-CM strongly modular \(\mathbf{Q}\)-curves.
[+] Résumé
Nous calculons le régulateur des éléments de Beilinson-Deninger-Scholl en termes de valeurs spéciales de fonctions L de formes modulaires en utilisant la méthode de Rogers-Zudilin.
[+] Abstract
We present a formula for the regulator of two arbitrary Siegel units in terms of \(L\)-values of pairwise products of Eisenstein series of weight one. We give applications to Boyd's conjectures and Zagier's conjectures for elliptic curves of conductors 14, 21, 35, 48 and 54.
[+] Abstract
We investigate the ramification of modular parametrizations of elliptic curves at the cusps. We prove that if the modular form associated to the elliptic curve has minimal level among its twists by Dirichlet characters, then the modular parametrization is unramified at the cusps. The proof uses Bushnell's formula for the Godement-Jacquet local constant of a cuspidal automorphic representation of \(\mathrm{GL}(2)\). We also report on numerical computations indicating that in general, the ramification index at a cusp seems to be a divisor of 24.
[+] Abstract
We prove a weak version of Beilinson's conjecture for non-critical values of \(L\)-functions for the Rankin-Selberg product of two modular forms.
[+] Abstract
It is well-known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over \(\mathbf{Q}\) can be parametrized by modular units. This answers a question raised by Zudilin in a recent work on Mahler measures. Further, we give the list of all elliptic curves \(E\) of conductor up to 1000 parametrized by modular units supported in the rational torsion subgroup of \(E\). Finally, we raise several open questions.
[+] Abstract
Let \(E\) be an elliptic curve over \(\mathbf{Q}\), and let \(F\) be a finite abelian extension of \(\mathbf{Q}\). Using Beilinson’s theorem on a suitable modular curve, we prove a weak version of Zagier’s conjecture for \(L(E_F,2)\), where \(E_F\) is the base change of \(E\) to \(F\).
[+] Résumé
Dans cet article, nous utilisons le système d’Euler de Kato et la théorie de Perrin-Riou pour établir une formule reliant la valeur en 0 de la fonction \(L\) \(p\)-adique d’une courbe elliptique définie sur \(\mathbf{Q}\), et un régulateur p-adique sur la courbe modulaire \(X(N)\). En particulier, nous obtenons une relation explicite entre fonction \(L\) \(p\)-adique et régulateur p-adique pour la courbe elliptique \(X_0(20)\).
[+] Abstract
This article investigates explicit linear dependence relations in the \(K_2\)-group of modular curves. In particular, it is shown that the Beilinson-Kato elements in \(K_2\) of the modular curve \(Y(N)\) satisfy the Manin relations when \(N\) is not divisible by 3. Similar results are obtained for the modular curves \(X_1(N)\) and \(X_0(N)\) when \(N\) is prime. Finally we exhibit explicit generators of \(K_2\), assuming the Beilinson conjecture.
[+] Résumé
Nous montrons une version explicite du théorème de Beilinson pour la courbe modulaire \(X_1(N)\). Ce résultat est la première étape d'un travail reliant, d'une part, la valeur en 2 de la fonction \(L\) d'une forme primitive de poids 2, et d'autre part, la fonction dilogarithme associée à la courbe modulaire correspondante, dans l'esprit de la conjecture de Zagier pour les courbes elliptiques. Comme corollaire de notre théorème, dans le cas où \(N\) est premier, nous répondons à une question de Schappacher et Scholl concernant l'image de l'application régulateur de Beilinson.
[+] Résumé
Nous énonçons une version explicite du théorème de Beilinson pour la courbe modulaire \(X_1(N)\). Nous en déduisons, pour toute courbe elliptique \(E\) de conducteur \(N\) premier, une formule donnant \(L(E,2)\) en termes des valeurs tordues \(L(E,\chi,1)\), avec \(\chi\) caractère modulo \(N\). Nous illustrons ce résultat et ses conséquences dans le cas de la courbe elliptique \(E=X_1(11)\).
Other texts
Habilitation à diriger des recherches, soutenue le 19 septembre 2019 à l'École normale supérieure de Lyon.
[+] Résumé
Les travaux présentés dans ce mémoire concernent principalement la théorie des formes modulaires. Nous étudions en particulier leurs fonctions \(L\) et les liens avec la théorie des régulateurs, d'un point de vue à la fois théorique et effectif.
Nous commençons par démontrer des versions explicites de la conjecture de Beilinson pour les formes modulaires dans le cas de la première valeur non critique, en utilisant la méthode récente de Rogers et Zudilin. Nous donnons aussi un analogue pour la fonction \(L\) \(p\)-adique associée à une courbe elliptique. Par ailleurs, en collaboration avec Chida, nous traitons le cas du produit de Rankin de deux formes modulaires.
Nous appliquons ces formules pour montrer des cas particuliers des conjectures de Boyd sur les mesures de Mahler des polynômes en deux variables, et en collaboration avec Neururer, nous généralisons la méthode à certains polynômes en trois variables.
Nous étudions ensuite la conjecture de Beilinson pour les variétés abéliennes quotients de la jacobienne d'une courbe modulaire. Grâce au langage adélique, nous montrons que l'algèbre de leurs endomorphismes est engendrée par les correspondances de Hecke, ce qui permet d'établir une version équivariante de la conjecture.
Pour terminer, nous présentons deux résultats concernant le développement de Fourier des formes modulaires. Nous donnons une condition suffisante pour que la paramétrisation modulaire d'une courbe elliptique soit non ramifiée aux pointes de la courbe modulaire. Enfin, en collaboration avec Neururer, nous bornons le corps engendré par les coefficients de Fourier d'une forme modulaire en une pointe arbitraire.
[+] Abstract
The work presented here is concerned with the theory of modular forms, their associated \(L\)-functions and the links with the theory of regulators, from a theoretical and effective point of view.
We begin by proving explicit versions of the Beilinson conjecture for modular forms in the case of the first non-critical \(L\)-value, by using the recent method of Rogers and Zudilin. We also give an analogue for the \(p\)-adic \(L\)-function associated to an elliptic curve. Moreover, in joint work with Chida, we treat the case of the Rankin product of two modular forms.
We apply these formulas to show particular cases of the Boyd conjectures on Mahler measures of polynomials in two variables, and in joint work with Neururer, we generalise this method to some polynomials in three variables.
We then study the Beilinson conjecture for abelian varieties which are quotients of the Jacobian of a modular curve. We show with the adelic language that their endomorphism algebras are generated by the Hecke correspondences, which enables us to establish an equivariant version of this conjecture.
We end this text by presenting two results concerning Fourier expansions of modular forms. We give a sufficient condition for the modular parametrisation of an elliptic curve being unramified at the cusps of the modular curve. Finally, in joint work with Neururer, we bound the field generated by the Fourier coefficients of a modular form at an arbitrary cusp.
Thèse de l'Université Paris 7 Denis-Diderot, sous la direction de Loïc Merel, soutenue le 9 décembre 2005.
[+] Résumé
Nous étudions dans cette thèse la valeur
spéciale des fonctions \(L\) des courbes elliptiques, et plus généralement des formes modulaires de poids 2, au premier point entier non critique, à savoir \(s=2\). Nous démontrons une version explicite d'un théorème de Beilinson relatif à cette valeur spéciale : pour toute forme parabolique primitive \(f\) de poids 2, niveau \(N \geqslant 1\) et caractère \(\psi\), et pour tout caractère de Dirichlet \(\chi\) modulo \(N\) (pair, primitif et distinct du conjugué de \(\psi\)), nous exprimons \(L(f,2) L(f,\chi,1)\) comme régulateur d'un symbole de Milnor explicite associé à des unités modulaires de \(X_1(N)\). Lorsque \(N=p\) est premier, nous en déduisons que les symboles de Milnor associés aux unités modulaires de \(X_1(p)\) engendrent l'espace d'arrivée du régulateur de Beilinson. En utilisant l'appendice par Merel, nous donnons une formule explicite et universelle pour \(L(E,2)\), où \(E\) est une courbe elliptique de conducteur \(p\) premier, en termes des valeurs tordues \(L(E,\chi,1)\), avec \(\chi\) caractère de conducteur \(p\). Nous suggérons également une reformulation de la conjecture de Zagier pour \(L(E,2)\) au niveau de la jacobienne \(J_1(N)\) de \(X_1(N)\), où \(N\) est le conducteur de \(E\). En ce sens, nous proposons un analogue du dilogarithme elliptique pour la jacobienne \(J\) d'une courbe algébrique : c'est une fonction \(R_J\) des points complexes de \(J\) vers le dual de l'espace des \(1\)-formes différentielles holomorphes sur \(J\). Nous montrons que \(L(f,2) L(f,\chi,1)\) est combinaison linéaire explicite de valeurs de \(R_{J_1(N)}\), appliquée à \(f\), en des points \(\mathbf{Q}\)-rationnels du sous-groupe cuspidal de \(J_1(N)\).
Talks
Unpublished notes
[+] Abstract
We prove a Sturm bound for modular forms on general congruence subgroups (improving known results in the case the width of the cusp is greater than 1). We also give some examples.
[+] Abstract
We show that the Mahler measure of a defining equation of the modular curve \(X_1(13)\) is equal to the derivative at \(s=0\) of the \(L\)-function of a cusp form of weight 2 and level 13 with integral Fourier coefficients. The proof combines Deninger's method, an explicit version of Beilinson's theorem together with an idea of Merel to express the regulator integral as a linear combination of periods. Finally, we present further examples related to the modular curves of level 16, 18 and 25.
[+] Abstract
This was originally an appendix to our paper Fourier expansions at cusps. The purpose of this note is to give a proof of a theorem of Shimura on the action of \(\mathrm{Aut}(\mathbf{C})\) on modular forms for \(\Gamma(N)\) from the perspective of algebraic modular forms. As the theorem is well-known, we do not intend to publish this note but want to keep it available as a preprint.
[+] Résumé
Nous détaillons dans ce texte une preuve due à Isaacs du résultat suivant : si \(a\) et \(b\) sont des nombres algébriques de degrés premiers entre eux, alors le degré de \(a+b\) est égal au produit des degrés de \(a\) et de \(b\). La démonstration utilise la théorie des représentations des groupes finis.
[+] Résumé
Nous étudions dans ce texte la loi de groupe sur une courbe elliptique particulière donnée comme intersection de deux quadriques. Nous interprétons géométriquement cette loi et, en explicitant des résultats de Lange-Ruppert et Kohel, nous en déterminons toutes les expressions algébriques possibles en degré 2.
