Modulformen

Abstract

The workshop Modulformen, organised by Siegfried Böcherer (Mannheim), Tomoyoshi Ibukiyama (Osaka) and Winfried Kohnen (Heidelberg) was held October 28–November 3, 2007. This meeting was attended by 24 participants with different backgrounds reflecting some of the many aspects of the theory of modular forms. One of the callenging features of this theory is that techniques using Whittaker models do not apply here. On the other hand, the theory is very rich in explicit structures and has immediate connections to arithmetic and geometry.

Some of the main topics (presented in one-hour talks) are

• Maass forms and their role in arithmetic and geometry (Bruinier, Bringman)
• Application to Algebraic Geometry (Dummigan, Yoshida, Gritsenko)
• The fine structure of modular forms for $GSp(4)$ (Poor, Wakatski, Roberts, Schmidt)
• Properties of explicit liftings (Panchishkin, Katsurada, Heim, Schulze-Pillot)

The third and forth topic are not only interesting in their own right but they are also a testing ground for general conjectures. We explictly mention the theory of newforms by Roberts and Schmidt for $GSp(4)$.

Explicit liftings can be viewed as examples for Langlands functoriality. The emphasis is on the explicit description of these liftings in terms of modular forms (not only representations). A prototype is the Duke–Imamoglu–Ikeda-lifting, which appeared in several talks.

Other talks dealt with theta series, Poincare series, generation of spaces of modular forms, converse theorems and spherical functions on p-adic spaces. Beyond the talks, there was much opportunity for scientific interaction among the participants.