# Elliptic curves

Thursday, August 24, 2017 - 9:00am - 9:45am

Kiran Kedlaya (University of California, San Diego)

We give a survey of some of the ways that number theorists use Sage. Among the basic objects we will encounter are elliptic curves, modular forms, and L-functions (including zeta functions).

Monday, January 3, 2011 - 3:40pm - 4:40pm

Karl Rubin (University of California)

In joint work with Barry Mazur, we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions for an elliptic curve to have twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) with a given 2-Selmer rank. As a consequence, under appropriate hypotheses there are many twists with Mordell-Weil rank zero, and (assuming the Shafarevich-Tate conjecture) many others with Mordell-Weil rank one.

Monday, January 3, 2011 - 2:00pm - 3:00pm

Carl Pomerance (Dartmouth College)

In the past three decades there have been some exciting

applications of elliptic curves over finite fields

to integer factoring, primality testing, and

cryptography. These applications in turn have raised some interesting

problems often of an unconventional flavor. For example, how

often is the order of an elliptic curve group prime, or how

often does it have all small prime factors? In this talk we will

visit problems such as these, as well as other analytic-type

problems relating to ranks of elliptic curves over function

applications of elliptic curves over finite fields

to integer factoring, primality testing, and

cryptography. These applications in turn have raised some interesting

problems often of an unconventional flavor. For example, how

often is the order of an elliptic curve group prime, or how

often does it have all small prime factors? In this talk we will

visit problems such as these, as well as other analytic-type

problems relating to ranks of elliptic curves over function

Tuesday, January 4, 2011 - 3:30pm - 4:30pm

Manjul Bhargava (Princeton University)

No Abstract