Time:

Lecture 1: 3:30-5:30 pm , Feb 27 (Wed)

Lecture 2: 3:30-5:30 pm , Feb 28 (Thu)

Lecture 3: 3:30-5:30 pm , Mar 1 (Fri)

Lecture 4: 3:30-5:30 pm , Mar 2 (Sat)

place: 404 Math building

Title: The rational cuspidal group of J_0(N)

Abstract: For a positive integer N, consider J_0(N), the Jacobian variety of the modular curve X_0(N). It is an abelian variety over the field of rational numbers.

Sometimes, number theorists are interested in T(N), the group of rational torsion points of J_0(N), which is a finite abelian group by Mordell-Weil theorem.

When N is a prime, Ogg conjectured and Mazur proved that

T(N) is equal to the cuspidal group of J_0(N).

A generalization of this conjecture is as follows: Let C(N) be the subgroup of J_0(N)

generated by the images of degree 0 rational cuspidal divisors on X_0(N). Then, for any positive integer N, C(N)=T(N).

In the perspective of this conjecture, it is very important to understand the structure of the group C(N). It was known only for the following cases (before November 2018):

1. N is a prime power.

2. N is the product of two primes.

3. Some small numbers N by SAGE, a computer program.

In this lecture series, we will review all the previous results and discuss the structure of the group C(N) when N is the product of two prime powers, which is a new result by the speaker.