The arithmetic of modular forms (Q2768083)

The 1999 IAS/Park City Mentoring Program for women took place at the Institute for Advanced Study, Princeton, NJ, the title of which was chosen to be ``The arithmetic of elliptic curves, modular forms, and Calabi-Yau varieties''. The purpose was to give a preview of the main subjects but also discussing the higher-dimensional analogue connected to geometry and physics. The lectures were delivered on the following subjects: (1) Wen-Ching W. Li: The arithmetic of modular forms (3 lectures), (2) Noriko Yui: Arithmetic of certain Calabi-Yau varieties and mirror symmetry (3 lectures), (3) Alice Silverberg: Conjectures about elliptic curves (2 lectures). NEWLINENEWLINENEWLINEHere the nicely written account of the first three lectures is given. The first lecture served as an overall introduction of the theme of the program (elliptic curves, modular forms, Calabi-Yau varieties, zeta-functions). The second lecture surveys the theory of classical modular forms (Eisenstein series, Hecke operators, newforms) and discusses certain proved and unproved conjectures in the area. The third lecture draws connections among modular forms (functional equations), elliptic curves (Sato-Tate conjecture), and representations of Galois groups over \(\mathbb{Q}\) with the common thread being the associated \(L\)-functions. Comparisons with progress made for automorphic forms of \(\text{GL}_2\) over function fields are also given there.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00034].