Arithmetic Geometry and Number Theory RTG Seminar: “The cuspidal subgroup, bigly”

Seminar | September 18 | 3:10-5 p.m. | 891 Evans Hall

 Kenneth A. Ribet, University of California, Berkeley

 Department of Mathematics

Let $X$ be the modular curve $X_0(N)$, where $N$ is a positive integer. The $cusps$ on $X$ are the points of $X$ “at infinity.'' According to the theorem of Manin–Drinfeld, the group of degree-0 cuspidal divisors on $X$ maps to a finite subgroup of the Jacobian of $X$. This group is the cuspidal subgroup $C$ of the Jacobian.

There is a general feeling that $C$ ought to be big in various respects. For example, the generalized Ogg conjecture (first made by William Stein on the basis of explicit calculations) states that each rational torsion point on the Jacobian is contained in $C$.

One encounters quite a bit of literature in the direction of proving that $C$ is “big” in various senses, but this literature leverages explicit calculations of $C$. Ogg calculated $C$ when $N$ is either prime or the product of two distinct primes; his method works more generally when $N$ is square free.

My talk describes a joint project with Bruce Jordan and Anthony Scholl: I will explain a “pure thought” approach that proves (away from some bad primes) that $C$ is “big” in the sense that its annihilator (in the ring of Hecke operators acting on the Jacobian) is small. Our result is useful in studying the kernel on the Jacobian of “Eisenstein primes” in the ring of Hecke operators.

Seminar Format: The seminar consists of two 50-minute talks, a pre-talk (3:10-4:00) and an advanced talk (4:10-5:00), with a 10-minute break (4:00-4:10) between them. The advanced talk is a regular formal presentation about recent research results to general audiences in arithmetic geometry and number theory; the pre-talk (3:10-4:00) is to introduce some prerequisites or background for the advanced talk to audiences consisting of graduate students.