## When is the product of Siegel eigenforms an eigenform? (Jim Brown, Occidental College)

### February 26 @ 12:15 pm - 1:10 pm

Modular forms are ubiquitous in modern number theory. For instance, showing that elliptic curves are secretly modular forms was the key to the proof of Fermat’s Last Theorem. In addition to number theory, modular forms show up in diverse areas such as coding theory and particle physics. Roughly speaking, a modular form is a complex-valued function defined on the complex upper half-plane that satisfies a large number of symmetries. A modular form has two invariants: weight and level. If one fixes a weight and level, the collection of modular forms of that weight and level form a finite-dimensional complex vector space. One has a collection of operators on these spaces referred to as Hecke operators. A natural question is if one takes two eigenforms of these operators and multiplies them, when is the product still an eigenform? It was shown in independent work by Duke and Ghate that there is a finite list of pairs of eigenforms whose product is again an eigenform. In this talk we will report on the case when one replaces modular forms with the more general case of Siegel modular forms. This is work that was partially conducted during an REU in summer 2018. No prior familiarity with modular forms is assumed.