Number theory is at the junction of many mathematical fields. Among its important topics are elliptic curves, modular forms, Maass waveforms and Galois representations. Despite their diversity, these objects are different faces of a single one: automorphic representations.
Isolated automorphic forms however remain elusive to study, even in the case of GL2. A leading philosophy originating with Sarnak, is to consider automorphic forms in families and to seek results on average, called arithmetic statistics. Trace formulas are the central tools in this approach and led to various arithmetic statistics (Weyl law, equidistribution, subconvexity, etc.).
Very recent works successfully developed both the Arthur trace formula [3] and the relative trace formula [2, 5] for GSp4. In my research project, I aim at establishing arithmetic statistics on families of automorphic forms for GSp4, by studying the precise behavior of the transforms arising in the trace formulas to make their analysis possible, leading to applications to families of automorphic forms. This will be the first example of precise arithmetic statistics on nonholomorphic automorphic forms in families beyond GLn.

[2] Comtat A relative trace formula approach to the Kuznetsov formula on GSp4
, arXiv preprint (2022)
[5] Man, S. H. Symplectic automorphic forms and Kloosterman sums, PhD thesis, Bonn University (2021)