Rainer Schulze-Pillot

Siegel modular forms and theta series attached to quaternion algebras

The two main problems in the theory of the theta correspondence or lifting (between automorphic forms on some adelic orthogonal group and on some adelic symplectic or metaplectic group) are the characterization of kernel and image of this correspondence. Both problems tend to be particularly difficult if the two groups are approximately the same size.

Arithmetic and Equidistribution of Measures on the Sphere

Motivated by problems of mathematical physics (quantum chaos) questions of equidistribution of eigenfunctions of the Laplace operator on a Riemannian manifold have been studied by several authors. We consider here, in analogy with arithmetic hyperbolic surfaces, orthonormal bases of eigenfunctions of the Laplace operator on the two dimensional unit sphere which are also eigenfunctions of an algebra of Hecke operators which act on these spherical harmonics. We formulate an analogue of the equidistribution of mass conjecture for these eigenfunctions as well as of the conjecture that their moments tend to moments of the Gaussian as the eigenvalue increases. For such orthonormal bases we show that these conjectures are related to the analytic properties of degree eight arithmetic L-functions associated to triples of eigenfunctions. Moreover we establish the conjecture for the third moments and give a conditional (on standard analytic conjectures about these arithmetic L-functions) proof of the equidistribution of mass conjecture.

An algorithm for computing genera of ternary and quaternary quadratic forms

It is well known that due to the simple shape of the reduction conditions in these dimensions [Mil, Ca] it is in principle no problem to compute representatives of all classes of such quadratic forms whose discriminant is below a given bound. This has been done by hand in the ternary case for halfdiscriminant up to 1000 by Brandt and Intrau [BI], for quaternaries see [Ge, To]. It is, however, sometimes desirable to be able to quickly determine representatives of all classes in some fixed genus of quadratic forms of possibly high discriminant without having to generate along the way all forms of smaller discriminant.

On the global Gross-Prased conjecture for Yoshida liftings

We restrict a Siegel modular cusp form of degree 2 and square free level that is a Yoshida lifting (a lifting from the orthogonal group of a definite quaternion algebra) to the embedded product of two half planes and compute the Petersson product against the product of two elliptic cuspidal Hecke eigenforms. The square of this integral can be explicitly expressed in terms of the central critical value of an L-function attached to the situation. The result is related to a conjecture of Gross and Prasad about restrictions of automorphic representations of special orthogonal groups.

Representation of Quadratic Forms by Integral Quadratic Forms

The number of representations of a positive definite integral quadratic form of rank n by another positive definite integral quadratic form of rank m ≥ n has been studied by arithmetic, analytic, and ergodic methods. We survey and compare in this article the results obtained by these methods.

Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables

Let

Teilbarkeit und Primzahlen

-d

Faktorisierung von Polynomen

-d be a a negative discriminant and let T vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant