UCSB Math Department

Graduate Number Theory Seminar
Winter 2021

Organizers: David Nguyen, Mulun Yin.

Meets Fridays 3-4 PM PT.
Due to the COVID-19 we are still meeting virtually on Zoom.


This winter, our theme will be on problems in arithmetic statistics. In particular, we focus on the congruent number problem, following the book Introduction to Elliptic Curves and Modular Forms by Koblitz (Springer GTM, Volume 97) and, if time permits, surveying some recent researches on this problem.


Date Speaker Title
January 8 Everyone Organizational meeting
January 15 Mulun Yin CH I. From Congruent Numbers to Elliptic Curves
January 22 Mulun Yin CH II. The Hasse-Weil L-Function of an Elliptic Curve
The congruence zeta-function
The zeta function of the congruent elliptic curve \(E_n\)
Varrying the prime \(p \)
January 29 David Nguyen CH II. The Hasse-Weil L-Function of an Elliptic Curve (cont'd)
The prototype: the Riemann zeta-function
The Hasse-Weil L-function and its functional equation
The critical value
February 5 Garo Sarajian CH III. Modular forms
\(\mathrm{SL}_2(\mathbb{Z})\) and its congruence subgroups
Modular forms for \(\mathrm{SL}_2(\mathbb{Z})\)
February 12 Garo Sarajian CH III. Modular forms (cont'd)
Modular forms for \(\mathrm{SL}_2(\mathbb{Z})\) (cont'd)
Modular forms for congruence subgroups
February 19 Mulun Yin CH III. Modular forms (cont'd)
Transformation formula for the theta-function
The modular interpretation and Hecke operators
February 26 David Nguyen CH IV. Modular Forms of Half Integer Weight
Definitions and examples
Eisenstein series of half integer weight for \(\tilde{\Gamma}_0(4) \)
March 5 Mulun Yin CH IV. Modular Forms of Half Integer Weight (cont'd)
Hecke operators on forms of half integer weight
March 12 David Nguyen CH IV. Modular Forms of Half Integer Weight (cont'd)
The theorems of Shimura, Waldspurger, Tunnell, and the congruent number problem