Siegel modular forms can be thought of as modular forms in more than one variable. Introduced in the 1930’s by Siegel in his analytic study of quadratic forms, they nowadays occur naturally in many unexpected places. We develop the basic theory from scratch, assuming only that the listener/reader has seen some rudiments of modular forms in one variable. We list some of the many applications and...

We determine conditions for the existence and non-existence of Ramanujan-type congruences for Jacobi forms. We extend these results to Siegel modular forms of degree 2 and as an application, we establish Ramanujan-type congruences for explicit examples of Siegel modular forms.

Hiroki Aoki On the structure theorem for modular forms ... Igusa’s result and beyond In this talk we treat the structure theorem of the graded ring of modular forms of several variables. Determining the structure is not easy in general. The first result was given by Professor Jun-Ichi Igusa in 1962. This was on the graded ring of Siegel modular forms of degree 2 of even weights. And then in 196...

There are a variety of characterizations of Saito-Kurokawa lifts from elliptic modular forms to Siegel modular forms of degree 2. In addition to giving a survey of known characterizations, we apply a recent result of Weissauer to provide a number of new and simpler characterizations of Saito-Kurokawa lifts.

‎We introduce a higher dimensional Atkin-Lehner theory for‎ ‎Siegel-Parahoric congruence subgroups of $GSp(2g)$‎. ‎Old‎ ‎Siegel forms are induced by geometric correspondences on Siegel‎ ‎moduli spaces which commute with almost all local Hecke algebras‎. ‎We also introduce an algorithm to get equations for moduli spaces of‎ ‎Siegel-Parahoric level structures‎, ‎once we have equations for prime l...