**Title:**Introduction to Hilbert modular forms and its determination by square-free Fourier coefficients.

**Speaker:**Rishabh Agnihotri

**Affiliation:**HRI, Prayagraj (Allahabad)

**Date and Time:**September 10, 2021 at 03:30PM

**Venue:**Seminar Hall, Kerala School of Mathematics

**Abstract: **We introduce two notions of Hilbert modular forms namely classical and adelic. After that we see the relation between them. We also talk about the determination of adelic hilbert modular forms. More concretely we discuss the following result.

Let $\mathbf{f}$ be as above with $C_{\mathbf{f}}(\mathfrak{m})$ denote its Fourier coefficients. Then there exists a square-free ideal $\mathfrak{m}$ with $N(\mathfrak{m})\ll k_0^{3n+\epsilon}N(\mathfrak{n})^{\frac{6n^2+1}{2}+\epsilon}$ such that $C_{\mathbf{f}}(\mathfrak{m})\neq 0$. The implied constant depends only on $\epsilon, F$.