There are two types of mathematicians, the problem solvers and the theory builders. I’m more like a problem solver. Number theory is such a wonderful world for me and there are so many amazing subjects I enjoy and work on. During my PhD I mainly got the following results on congruence primes for automorphic forms, additive number theory, and distribution of the trace of Frobenius of elliptic curves.

This video shows why 691 is an amazing number. On the congruence primes of modular forms and automorphic forms, my advisor Prof. Jim Brown guides me on two projects, one with respect to the Saito-Kurokawa liftings over Hida families, and the other with respect to Siegel Hilbert automorphic forms.

The Saito-Kurokawa lifting is a sequence of Hecke equivariant isomorphisms from the space of elliptic cuspforms, to the space of Jacobi cuspforms, and to the space of Siegel cuspforms. The Saito-Kurokawa liftings are very interesting. Even though the liftings are between cuspforms, if we look at the eigenvalues of the cuspforms, they behave like Eisenstein series. These CAP forms have some interesting arithmetic propositions, where CAP means cuspidal associated to parabolics. For example, they do not satisfy the Ramanujan’s conjecture. Moreover, the Saito-Kurokawa liftings can be used via congruences to non-CAP cuspidal eigenforms to provide evidence for the Bloch-Kato conjecture and the Birch and Swinnerton-Dyer conjecture.

We are also working on the congruence properties of the Fourier coefficients of the Hilbert Siegel Eisenstein series. We are trying to provide a sufficient condition for a prime to be a congruence prime for a Hilbert Siegel eigenform for a large class of totally real fields F via the divisibility of a special value of the standard L-function associated to the eigenform. Here is what Brown and Klosin did in the unitary group case.

In additive number theory, the Goldbach’s conjecture is a famous old puzzle, which states that every even integer greater than 2 is a sum of two primes. Even though it has been tested to be true for even integers up to 4*10^18, a proof is still missing. So far the best result is Chen’s theorem, which states that every sufficiently large even integer can be represented as a sum of a prime and an almost prime with at most 2 prime divisors. Using the same idea from his proof, I was able to show that every odd integer can be written as a sum of a prime and an almost prime with at most 3 prime divisors. Together with Ross’s observation, Chen’s theorem and my theorem imply that every sufficiently large integer can be written as the sum of a prime and a square-free integer with at most 3 prime divisors. This improves Estermann’s theorem that every sufficiently large integer can be written as the sum of a prime and a square-free integer.

Given an elliptic curve over rational numbers, Hasse’s theorem asserts that the normalized trace of Frobenius bp = ap / (2 √ p) = (p + 1 – E(Fp))/ (2 √ p) is in (-1, 1). For elliptic curves with complex multiplication (CM), a theorem by Deuring and Hecke gives the distribution of bp; while for elliptics curve without CM, the distribution of bp is described by the Sato-Take conjecture, now a theorem by Clozel, Harris, Shepherd-Barron and Taylor in 2006. Using their theorems, we can get an asymptotic formula for the number of primes p up to x such that the trace of Frobenius ap is in a large sub-interval of (-2 √ p, 2 √ p) with positive density. Recently I worked as a graduate student mentor for a group at the Clemson REU in 2017 summer, and we generalized the results on extremal primes by Giberson, James and Pollack. More explicitly, for elliptic curves with CM, we gave the asymptotic formula for the number of primes p up to x such that ap is in a small sub-interval of (-2 √ p, 2 √ p) of density zero. For elliptic curves without CM, we gave the average asymptotic formula for the number of primes up to x satisfying the same conditions over a family of elliptic curves.