Siegel theta series for indefinite quadratic forms

The modular transformation behavior of theta series for indefinite quadratic forms is well understood in the case of elliptic modular forms due to Vignéras, who deduced that solving a differential equation of second order serves as a criterion for modularity. In this paper, we will give a generalization of this result to Siegel theta series.

Hecke operators on Hilbert–Siegel theta series

We consider the action of Hecke-type operators on Hilbert–Siegel theta series attached to lattices of even rank. We show that the average Hilbert–Siegel theta series are eigenforms for these operators, and we explicitly compute the eigenvalues.

Quantum modularity of partial theta series with periodic coefficients

We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich–Zagier series F t ⁢ ( q ) \mathscr{F}_{t}(q) which matches (at a root of unity) the colored Jones polynomial for the family of torus knots T ⁢ ( 3 , 2 t ) T(3,2^{t}) , t ≥ 2 t\geq 2 , is a weight 3 2 \frac{3}{2} quantum modular form. This generalizes Zagier’s result on the quantum modularity for the “strange” series F ⁢ ( q ) F(q) .

EXTREME VALUES OF GEODESIC PERIODS ON ARITHMETIC HYPERBOLIC SURFACES

Given a closed geodesic on a compact arithmetic hyperbolic surface, we show the existence of a sequence of Laplacian eigenfunctions whose integrals along the geodesic exhibit nontrivial growth. Via Waldspurger’s formula we deduce a lower bound for central values of Rankin-Selberg L-functions of Maass forms times theta series associated to real quadratic fields.

On the Rankin–Selberg method for vector-valued Siegel modular forms

In this work, we use the Rankin–Selberg method to obtain results on the analytic properties of the standard [Formula: see text]-function attached to vector-valued Siegel modular forms. In particular we provide a detailed description of its possible poles and obtain a non-vanishing result of the twisted [Formula: see text]-function beyond the usual range of absolute convergence. Our results include also the case of metaplectic Siegel modular forms. We remark that these results were known in this generality only in the case of scalar weight Siegel modular forms. As an interesting by-product of our work we establish the cuspidality of some theta series.