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Elliptic curves, moduli spaces and modular forms I SemesterWiSe 2021 / 22 Lecturer Guido Kings Type of course (Veranstaltungsart) Vorlesung German title Elliptische Kurven, Modulräume und Modulformen I Contents In this lecture we will consider algebraic families of elliptic curves and construct their moduli space. This leads to an algebraic description of modular forms and is the basis to study topics like $p$-adic modular forms, Galois modules attached to modular forms or Iwasawa theory of modular forms. All this is the basis for Wiles proof of the Shimura-Taniyama conjecture and of essential importance in arithmetic geometry. The lecture starts with developing the theory of elliptic curves over a scheme without assuming any previous knowledge of elliptic curves. We will assume only very few results from algebraic geometry and will provide an overview of the results needed with precise references. Many finer points in algebraic geometry will be proven or discussed in the lecture. After these basics on elliptic curves we proceed to the construction of the moduli space of elliptic curves and will consider some of its basic properties. Then $p$-adic modular forms are considered and we construct Galois representations associated to modular forms. It is planed to continue the lecture in the summer term to cover more of the basics occurring in the proof of Wiles celebrated theorem. Literature will be announced at the beginning of the course Time/Date Tue and Fri 14 - 16 Location Tue 2-4 p.m. (M103 and online); Fri 2-4 p.m. (online) Course homepage https://www.uni-regensburg.de/mathematik/mathematik-kings/startseite/index.html (Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern) Registration
BV, MV, MArGeo, MGAGeo, LA-GyGeo ECTS 9 |