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Title:  Automorphic Lfunctions and their applications to Number Theory 
Authors:  Cho, Jaehyun 
Advisor:  Kim, Henry H. 
Department:  Mathematics 
Keywords:  Automorphic Lfunction Strong Artin Conjecture class number EulerKronecker constant simple zero Maass Lfunction 
Issue Date:  21Aug2012 
Abstract:  The main part of the thesis is 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 generating extreme positive and negative values of EulerKronecker constants.
For a given number field $K$ of degree $n$, let $\widehat{K}$ be the normal closure of $K$ with $Gal(\widehat{K}/\Bbb Q)=G.$ Let $Gal(\widehat{K}/K)=H$ for some subgroup $H$ of $G$. Then,
$$
L(s,\rho,\widehat{K}/\Bbb Q)=\frac{\zeta_K(s)}{\zeta(s)}
$$
where $Ind_H^G1_H = 1_G + \rho$.
When $L(s,\rho)$ is an entire function and has a zerofree 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:
$$
\log L(1,\rho) = \sum_{p\leq(\log N)^{k}}\lambda(p)p^{1} + O_{l,k,\alpha}(1)$$
$$
\frac{L'}{L}(1,\rho)=\sum_{p\leq x} \frac{\lambda(p) \log{p}}{p} +O_{l,x,\alpha}(1).
$$
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 EulerKronecker 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 $\Bbb Q$. Let $f(x,t) \in \Bbb Z[t][x]$ be a parametric polynomial whose splitting field over $\Bbb Q (t)$ is a regular $G$ extension. By Cohen's theorem, most specialization $t\in \Bbb Z$ corresponds to a number field $K_t$ in $\frak{K}$ $(n,G,r_1,r_2)$ with signature $(r_1,r_2)$ and hence we have a family of Artin Lfunctions $L(s,\rho,t)$. By counting zeros of Lfunctions over this family, we can obtain Lfunctions with the zerofree region above.
In Chapter 1, we collect the known cases for the Strong Artin conjecture and prove it for the cases of $G=A_4$ and $S_4$. We explain how to obtain the approximations of $\log (1,\rho)$ and $\frac{L'}{L}(1,\rho)$ as a sum over small primes in detail. We review the theorem of KowalskiMichel on counting zeros of automorphic Lfunctions in a family.
In Chapter 2, we exhibit many parametric polynomials giving rise to regular extensions. They contain the cases when $G=C_n,$ $3\leq n \leq 6$, $D_n$, $3\leq n \leq 5$, $A_4, A_5, S_4, S_5$ and $S_n$, $n \geq 2$.
In Chapter 3, we construct number fields with extreme class numbers using the parametric polynomials in Chapter 2.
In Chapter 4, We construct number fields with extreme EulerKronecker constants also using the parametric polynomials in Chapter 2.
In Chapter 5, we state the refinement of Weil's theorem on rational points of algebraic curves and prove it.
The second topic in the thesis is about simple zeros of Maass Lfunctions. We consider a Hecke Maass form $f$ for $SL(2,\Bbb Z)$. In Chapter 6, we show that if the Lfunction $L(s,f)$ has a nontrivial simple zero, it has infinitely many simple zeros. This result is an extension of the result of Conrey and Ghosh. 
URI:  http://hdl.handle.net/1807/32684 
Appears in Collections:  Doctoral

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