Workshop on automorphic forms and L-functions

Institute of mathematics Simion Stoilow of the Romanian Academy

Bucharest, Romania, June 6-8, 2012

Home Participants Schedule of talks Abstracts Travel and accomodation
Andrade: Random Matrix Theory and L-Functions in Function Fields
The connection between Random Matrix Theory (RMT) and the theory of L-functions is mysterious and still we do not know why RMT is a good model for the distribution of zeros of L-functions. We will review some basic concepts of RMT and discuss moments of L-functions in families. We then discuss recent developments in the calculation of moments of L-functions in the function field setting using the recent techniques developed by Kurlberg and Rudnick to study the opposite limit in the Katz-Sarnak philosophy (this is joint work with Jon Keating).

Brunault: On Zagier's conjecture for base extensions of elliptic curves
Zagier's conjectures predicts strong relations between special values of L-functions and polylogarithms. Although the full conjecture remains out of reach at present, a weakened form of the conjecture can sometimes be established. In 1995, Goncharov and Levin proved that for any elliptic curve E over Q, the special value L(E,2) can be expressed in terms of elliptic dilogarithms. Their proof uses Beilinson's theorem on modular curves. In this talk, I will explain how to prove the analogue of Goncharov and Levin's result for the base change of E to an arbitrary abelian number field. The proof uses modular curves and modular forms in the adelic setting. If time permits, I will also explain what is expected in the case the extension is not abelian.

Clingher: K3 Surfaces of high Picard rank and Siegel modular forms
I will discuss a special family of complex algebraic K3 surfaces of Picard rank 17 or higher. These surfaces are naturally related to principally polarized abelian surfaces. I will outline the geometry of the correspondence as well as present an explicit classification of these special K3 surfaces in terms of Siegel modular forms.

Cojocaru: Frobenius fields for elliptic curves
Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\). For a prime \(p\) of good reduction for \(E\), let \(\pi_p\) be the \(p\)-Weil root of \(E\) and \(\mathbb{Q}(\pi_p)\) the associated imaginary quadratic field generated by \(\pi_p\). In 1976, Serge Lang and Hale Trotter formulated a conjectural asymptotic formula for the number of primes \(p \le x\) for which \(\mathbb{Q}(\pi_p)\) is isomorphic to a fixed imaginary quadratic field. I will discuss progress on this conjecture, in particular an average result confirming the predicted asymptotic formula. This is joint work with Henryk Iwaniec and Nathan Jones.

Diamantis: Cohen kernels and double Eisenstein series
We extend to general values of L-functions the interpretation of critical values given by Kohnen-Zagier using Cohen's kernel. In the same setting of general values of L-functions, double Eisenstein series are shown to take the role played by Rankin-Cohen brackets in Zagier's expression of critical values in terms of inner products. A characterization of the field containing an arbitrary value of an L-function is given. This is joint work with C. O'Sullivan.

Harcos: On the sup-norm of Maass cusp forms of large level
Let \(f\) be an \(L^2\)-normalized Hecke-Maass cuspidal newform of square-free level \(N\). Then \(\sup|f| \ll N^{-1/6+\epsilon}\) for any \(\epsilon>0\), with an implied constant depending continuously on the Laplacian eigenvalue of \(f\). This is joint work with Nicolas Templier.

Jones: An alternative view of primitivity of Dirichlet characters
Dirichlet characters and their associated L-functions were introduced by Dirichlet in his proof of the prime number theorem in arithmetic progressions. In this talk, I will discuss a modification of the classical treatment of imprimitive characters, which makes them behave primitively (e.g. the properties of the associated Gauss sum and the functional equation of the attached L-function take on a form usually associated to a primitive character). This is based on joint work with R. Daileda.

Kohnen: Conic theta functions
We study a class of polyhedral functions called conic theta functions, which are closely related to classical theta functions. This is very recent joint work with A. Folsom and S. Robins.

Raulf: Limits of Eisenstein series off the half-line
Luo and Sarnak proved a quantum unique ergodicity theorem for the non-holomorphic Eisenstein series E(z, 1/2+it) for SL_2(Z). We will discuss the dependence of their result on the spectral parameter. This is joint work with Y. Petridis and M. Risager.

Risager: Non-vanishing of Taylor coeffients of modular forms
We consider Taylor coefficients of modular forms at points in the upper half-plane. These coeffients are non-cuspidal analogues of the standard Fourier coefficients. We show, without too much effort, that "generically" these coefficients are all non-vanishing. Yet it is highly non-trivial to prove that at specific points the coefficients are non-vanishing. In the simplest case of the discriminant function the non-vanishing of Fourier coefficients at infinity is an old conjecture of Lehmer's. We show how the non-cuspidal analogue of this conjecture is true for certain CM-points. We also discuss applications of such non-vanishing results. This is joint work with Cormac O'Sullivan.

Strömberg: A dimension formula for vector-valued Hilbert modular forms
Dimension formulas for scalar-valued Hilbert modular forms were obtained by Freitag already in the 80's. These formulas, however, involve several terms which are complicated to evaluate. So far no one has implemented these formulas on a computer, making them, although beautiful, rather useless for practical purposes.
I will present theoretical and computational aspects of joint work with N.-P. Skoruppa. Our aim has first of all been to generalize the formulas of Freitag to vector-valued Hilbert modular forms of integral or half-integral weight. This generalization is necessary in order to be able to obtain liftings between Hilbert modular forms and Jacobi forms over (totally real) number fields. Since we want to actually use these dimension formulas we have also aimed making them as explicit as possible in order to implement them in, for example Sage.