Wednesdays, 7:30pm; Room 507, Mathematics
Topic: Modular Forms and Elliptic Curves
Text: A First Course in Modular Forms by Fred Diamond and Jerry Shurman
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ums [at] math.columbia.edu
| Date | Speaker | Title | Abstract |
| June 3 | Linus Hamann | Introduction | In this lecture, we will give a comprehensive review of the basic theorems in single variable complex analysis. Then we will proceed to define the two main objects of study of this lecture series at face value these are the analytic objects of modular forms and the geometric objects of elliptic curves.Finally, we conclude with a discussion of some of the deep connections between these two objects that will be discussed in the upcoming lectures and why one might want to study them. |
| June 10 | Linus Hamann | Complex tori | In this lecture, we will discuss complex tori and their description as a quotient of the complex plane by a lattice. We will discuss elliptic functions and complex analytic maps between complex tori, giving a full categorization of these analytic maps and explain how the isomorphism classes are isomorphic to the fundamental domain described last week. Lastly, we will define some useful constructions related to isogenies such as the dual isogeny and the weil pairing. |
| June 17 | Hardik Shah | Elliptic curves as complex tori | |
| June 24 | Sander Mack-Crane | Modular forms | First we expand the connection between the modular group and elliptic curves by defining certain subgroups of the modular group, known as "congruence subgroups", and stating their relation to "enhanced elliptic curves". Then we will carefully define modular forms for congruence subgroups. Finally, as a neat application, we will use modular forms to count the number of ways an integer can be expressed as a sum of four squares. |
| July 1 | Irit Huq-Kuruvilla | Modular curves as Riemann surfaces | |
| July 8 | Linus Hamann | Cusps and elliptic points | |
| July 15 | Nawaz Sultani |
The genus of a modular curve and automorphic forms |
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| July 22 | Irit Huq-Kuruvilla |
Meromorphic differentials and the Riemann-Roch theorem |
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| July 29 | Nawaz Sultani | The dimension formula of modular forms |