## DEPARTMENTAL PHD THESIS EXAM: Peter Jaehyun Cho

Wednesday, February 8, 2012
2:10 p.m., in BA 6183, 40 St. George Street

PhD Candidate: Peter Jaehyun Cho

Thesis Title:  L-functions and Number Theory

Everyone welcome. The thesis can be viewed at http://www.math.toronto.edu/graduate/PhDthesis_PeterCho.pdf . Refreshments will be served in the Math Lounge before the exam.

Thesis abstract:

The main part of the thesis is the applications of the Strong Artin conjecture to number theory. We have two applications. One is generating number fields with extreme class numbers. The other is also generating extreme positive values and negative values of Euler-Kronecker constants.

For each number field $K$ of degree $n$, there is the Artin L-function $L(s,\rho)=\frac{\zeta_K(s)}{\zeta(s)}$ attached to $K$. When $L(s,\rho)$ is an entire function and has a broad zero-free region $[\alpha,1] \times [-(\log N)^2, (\log N)^2]$ where $N$ is the conductor of $L(s,\rho)$, we can estimate $\log L(1,\rho)$ and $\frac{L’}{L}(1,\rho)$ as a sum over small primes:

$$\log L(1,\rho) = \sum_{p\leq(\log N)^{k}}\lambda(p)p^{-1} + O_{l,k,\alpha}(1)$$

$$\frac{L’}{L}(1,\rho)=-\sum_{p\leq x^2} \frac{\lambda(p) \log{p}}{p} +O_{l,x,\alpha}(1).$$

where $0 < k < \frac{16}{1-\alpha}$ and $(\log N)^{\frac{16}{1-\alpha}} \leq x \leq N^{\frac{1}{4}}$. With these approximations, we can study extreme values of class numbers and Euler-Kronecker constants.

Let $\frak{K}(n,G,r_1,r_2)$ be the set of number fields of degree $n$ with signature $(r_1,r_2)$ whose normal closures are Galois $G$ extension over $\mathbb{Q}$. Let $f(x,t) \in \mathbb{Z}[t][x]$ be a parametric polynomial whose splitting field over $\mathbb{Q} (t)$ is a regular $G$ extension. By Cohen’s theorem, most specialization $t\in \mathbb{Z}$ corresponds to a number field $K_t$ in $\frak{K}(n,G,r_1,r_2)$ with a proper signature $(r_1,r_2)$ and hence we have a family of Artin L-functions $L(s,\rho)$. By counting zeros of L-functions over this family, we can obtain L-functions with the broad zero-free region above.

The second topic in the thesis is about simple zeros of Maass L-functions. We consider a Hecke Maass form $f$ for $SL(2,\mathbb{Z})$. In Chapter 6, we show that if the L-function $L(s,f)$ has a non-trivial simple zero, it has infinitely many simple zeros. This result is an extension of Conrey and Ghosh.