Exposés

Résumé :
  We show how to use the recent work of D. McKinnon and M. Roth on generalizations of Diophantine approximation to algebraic varieties to formulate a local version of the Batyrev-Manin principle on the distribution of rational points. We present several toric varieties for which the result is known.

Résumé :
  Sieves are an important tool in analytic number theory. In a typical sieve problem, one is given a list of \(p\)-adic conditions for all primes \(p\), and the challenge is to count the number of integers which satisfy all these \(p\)-adic conditions. In this talk we present some versions of sieves for varieties whose rational points are equidistributed, and give applications to counting rational points in thin sets. This is joint work with Tim Browning.

Résumé :
  In this talk we study results towards almost prime solutions for systems of Diophantine equations in many variables. This is joint work with Efthymios Sofos.

Résumé :
  For any Fuchsian subgroup \(\Gamma\subset\mathrm{PSL}_{2}(\mathbf{R})\) of the first kind, Selberg introduced the Selberg zeta function in analogy to the Riemann zeta function using the lengths of simple closed geodesics on \(\Gamma\backslash\mathbf{H}\) instead of prime numbers. In this talk, we report on a formula that determines the special value at \(s=1\) of the derivative of the Selberg zeta function for \(\Gamma=\mathrm{PSL}_2(\mathbf{Z})\). This formula is obtained as an application of a generalized Riemann-Roch isometry for the trivial sheaf on \(\overline{\Gamma\backslash\mathbf{H}}\), equipped with the Poincaré metric. This is joint work with Gerard Freixas.

Résumé :
  Chavdarov and Zywina showed that after passing to a suitable field extension, every abelian surface \(A\) with real multiplication over some number field has geometrically simple reduction modulo \(\mathfrak{p}\) for a density one set of primes \(\mathfrak{p}\). One may ask whether its complement, the density zero set of primes \(\mathfrak{p}\) such that the reduction of \(A\) modulo \(\mathfrak{p}\) is not geometrically simple, is infinite. Such question is analogous to the study of exceptional mod \(\mathfrak{p}\) isogeny between two elliptic curves in the recent work of Charles. In this talk, I will discuss how to apply Charles's method to the setting of certain abelian surfaces with real multiplication. This is joint work with Ananth Shankar.

Résumé :
  The Bogomolov conjecture for an abelian variety is a problem concerning non-density of points with small canonical height on closed subvarieties of an abelian variety over number fields or function fields. Over number fields, this conjecture was proved in 1998 by Ullmo and Zhang; over function fields, it is still open in full generality. In the proof of Ullmo and Zhang, they used the equidistribution of small points with respect to canonical measures over an archimedean place. In this talk, we explain that also over function fields, an analogous method using nonarchimedean analytic spaces and nonarchimedean canonical measures works well when the abelian variety is suitably degenerate at some place. The technical key in this argument is the investigation of the "strict supports" of canonical measures.

Résumé :
  The goal of this talk is to introduce a Gross--Zagier type formula, which relates the arithmetic self-intersection number of the Hodge bundle of a quaternionic Shimura curve over a totally real field to the logarithmic derivative of the Dedekind zeta function of the base field at 2. The proof is inspired by the previous works of Yuan, S. Zhang and W. Zhang on the Gross--Zagier formula and the averaged Colmez conjecture.