Heilbronn Focused Research Workshop Opens Up New Direction in the Study of Converse Theorems

24 Apr 2017, by chrystalcherniwchan in News

In January 2017, Andy Booker and Min Lee hosted a Heilbronn Focused Research Workshop on the “Sarnak Rigidity Conjecture”. This led to some highly interesting new ideas concerning an important and long-standing problem in number theory which relates to “converse theorems” in the theory of modular forms. Participants included two current Heilbronn Fellows, one former Heilbronn Fellow, and several international visitors. Their results have now been written up in a paper that is attracting considerable attention. One of the Heilbronn Fellows involved, Tom Oliver, said

Modular forms satisfy simple relations associated to symmetries of the hyperbolic plane. Our paper is concerned with the question “under which conditions is a sequence of complex numbers the set of Fourier coefficients of a modular form?”. Previous results suggest that there is a tension between the assumption of multiplicative relationships involving these numbers and that of twisted functional equations for the associated L-function. We offer a conjecture which replaces the non-linear product structure with a linearization, and verify several simple cases“.