| dc.description.abstract | In this thesis, we are concerned with two problems on the distribution of primes, the distribution according to the size of torsion of a given algebraic group modulo primes and an Erdos-Pomerance conjecture for rank one Drinfeld modules.
First of all, we consider a commutative algebraic group $A$ which is defined over a global field $K$. Then, we fix a positive integer $n$. For a prime divisor $wp$ of $K$, let $F_{wp}$ denote the residue field. If $A$ has good reduction at $wp$, let $ ilde A$ be the reduction of $A$ modulo $wp$ and let $N_{wp,n}$ be the number of $n$-torsion points in $ ildeAleft(F_wp
ight)$, the set of $F_{wp}$-rational points in $ ilde A$. If $A$ has bad reduction at $wp$, let $N_{wp,n} = 0$. Let $
ormwp$ denote the norm of $wp$, equal to the cardinality of the residue field $F_wp$. We are interested in the average value of $N_{wp, n}$, where $wp$ runs through the prime divisors in $K$, namely the limit
$$
limlimits_{x
ightarrow infty } frac{1}{pi_{K}(x)}sumlimits_{
ormwp leq x}N_{wp,n},
$$
where $pi_{K}(x)$ is the number of primes $wp$ with $
ormwp leq x$. We denote this limit by $M(Bbb A_{/K}, n)$. We shall derive explicit formulas for the average value $M(Bbb A_{/K}, n)$ when $A$ is a commutative algebraic group of dimension one defined over $K$.
Secondly, we consider a global function field $k$ of positive characteristic containing a prime divisor $infty$ of degree one and whose field of constants is $Bbb F_q$. Let $A$ be the ring of elements of $k$ which are regular outside $infty$. Let $psi$ be a sgn-normalized rank one Drinfeld $A$-module defined over $mathcal{O}$, the integral closure of $A$ in the Hilbert class field of $A$. Given any $0
eq alpha in mathcal{O}$ and an ideal $frak{M}$ in $mathcal{O}$, let $f_{alpha}left(frak{M
ight) = left{f in A : psi_{f}left(alpha
ight)equiv 0 pmod{frak{M}}
ight}$ be the ideal in $A$. We denote by $omegaig(f_alphaleft(frak{M}
ight)ig)$ the number of distinct prime ideal divisors of $f_alphaleft(frak{M}
ight)$. If $q
eq 2$, we prove that the following quantity
$$
frac{omegaig(f_alphaleft(frak{M}
ight)ig)-frac{1}{2}left(logdegfrak{M}
ight)^2}{frac{1}{sqrt{3}}left(logdegfrak{M}
ight)^{3/2}}
$$
distributes normally. | en_US |