Dan Fretwell

University of Bristol

Definite orthogonal modular forms: Computations, Excursions and Discoveries

Heilbronn Number Theory Seminar

2nd March 2022, 4:00 pm – 5:00 pm
Fry Building, 2.04

The theory of (positive definite) integral quadratic forms and lattices has a long and rich history. For many years it has been known how to study these objects via their theta series, modular forms whose Fourier coefficients encode arithmetic data. A less well known fact is that isometry classes of lattices (in a genus) can themselves be viewed as automorphic forms, for the corresponding (definite) orthogonal group. These forms also contain a wealth of arithmetic information.

In general, algorithms for computing spaces of automorphic forms for higher rank groups are few and far between. However, the case of definite orthogonal groups is concrete enough to be amenable to computation, and provides a significant testing ground for general conjectures in the Langlands Program (e.g. explicit Functoriality).

Recently, E. Assaf and J. Voight have developed a new magma package for computing spaces of orthogonal modular forms. We will take a stroll through a zoo of explicit examples computed using this package, outlining links with conjectures of Arthur on endoscopy and discoveries of new Eisenstein congruences.

(Joint work with E. Assaf, C. Ingalls, A. Logan, S. Secord and J. Voight)

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