| Hörsaal D2
Vortrag: Modular and mock modular generating series
Abstract: Given an integer-indexed family of quantities, for example cardinalities or dimensions, it is natural to organize them as a power series, called their generating series. Properties of a generating series reflect properties of its coefficients. The case of Hilbert polynomials and Hilbert series is classical: In the former case all but finitely many coefficients vanish—in some settings a property far from obvious—and in the latter case recursion relations among graded dimensions of a module are implied if the Hilbert series is a rational function.
Beyond the case of rational functions, a generating series might equal the q-expansion associated with a modular form or a mock modular form, in which case encoded relations among its coefficients are significantly more subtle. For instance, the Modularity Conjecture for elliptic curves over the rationals, which is behind the proof of Fermat's Last Theorem, can be phrased in terms of a generating series associated with traces on cohomology groups. We will present the definition of modular forms and mock modular forms and survey some of their Fourier coefficients' properties.