The arithmetic of cyclic subgroups.

摘要

In this thesis, we consider several problems relating to cyclic subgroups of the group (Z/nZ) x. Each element of (Z/n Z)x has a unique representative in one of the two intervals [special characters omitted] and [special characters omitted]. A subgroup H of (Z/ nZ)x is balanced if every coset of H intersects these two intervals equally. Such subgroups have repercussion for ranks of elliptic curves over function fields. We investigate several statistical questions about balanced subgroups, including the following: For a fixed integer g ≠ 0,1 , how frequently is the cyclic subgroup ⟨g mod n⟩ balanced? We prove a conjecture of Pomerance and Ulmer that this distribution is essentially determined by two special families of balanced subgroups. One of the deeper analytic tools we require is a prime number theorem of Grantham which counts primes in an arithmetic progression splitting completely in a given number field.;The behavior of the cyclic subgroup ⟨2 mod p⟩ as p ranges over odd primes is closely connected to the arithmetic of the Mersenne numbers. Let [special characters omitted], the reciprocal sum of the primes dividing the nth Mersenne number 2n -- 1 . Erdo&huml;s's showed that f(n) < log log log n + C for some constant C. Apart from the exact value of C, it is easy to prove that this inequality is tight. (Although it would be more interesting to understand the maximal order of [special characters omitted], the function f( n) is more tractable, albeit still difficult.) We give a conditional answer to Erdo&huml;s's question on the exact value of C. Moreover, we show that Erdo&huml;s's theorem is true when the Mersenne number 2n -- 1 is replaced with the nth Fibonacci number. Finally, we consider values of the order of the subgroup ⟨2 mod p⟩ .