Vector valued theta functions associated with binary quadratic forms

We study the space of vector valued theta functions for the Weil representation of a positive definite even lattice of rank two with fundamental discriminant. We work out the relation of this space to the corresponding scalar valued theta functions of weight one and determine an orthogonal basis with respect to the Petersson inner product. Moreover, we give an explicit formula for the Petersson norms of the elements of this basis.

On ratios of Petersson norms for Yoshida lifts

We prove an algebraicity property for a certain ratio of Petersson norms associated to a Siegel cusp form of degree 2 (and arbitrary level) whose adelization generates a weak endoscopic lift. As a preparation for this, we explicate various features of the correspondence between scalar valued Siegel cusp forms of degree

THE JACQUET–LANGLANDS CORRESPONDENCE AND THE ARITHMETIC RIEMANN–ROCH THEOREM FOR POINTED CURVES

We show how the Jacquet–Langlands correspondence and the arithmetic Riemann–Roch theorem for pointed curves, relate the arithmetic self-intersection numbers of the sheaves of modular forms — with their Petersson norms — on modular and Shimura curves: these are equal modulo ∑l∈S ℚ log l, where S is a controlled set of primes. These quantities were previously considered by Bost and Kühn (modular curve case) and Kudla–Rapoport–Yang and Maillot–Roessler (Shimura curve case). By the work of Maillot and Roessler, our result settles a question raised by Soulé.

Transcendental Nature of Special Values of L-Functions

In this paper, we study the non-vanishing and transcendence of special values of a varying class of L-functions and their derivatives. This allows us to investigate the transcendence of Petersson norms of certain weight one modular forms.