Problems in parameterization, zero-estimates, point-counting, and o-minimality

Thesis

In this thesis we consider problems of effectivity for zero-bounds, point-counting, computation, and class numbers.

Using the tameness (in the sense of o-minimality and Khovanskii) of the modular function (j-function) and Weierstrass elliptic functions on certain complex arcs we prove new zero-bounds for some polynomials involving these functions. Using the method of mild parameterizations, we prove a new result on the number of algebraic points on a curve involving the j-function, and give a new proof of the six exponentials theorem.

We develop new algorithms to invert the j-function without using the elliptic curve representation of a j-invariant, to determine the torsion subgroup of rational elliptic curves in the manner of Doud, and to test an elliptic curve for complex multiplication. All these algorithms either improve on previous methods, or are of equivalent optimal complexity. We develop a new algorithm to find numbers satisfying a set of modular conditions, which is significantly faster than the fastest previous method, with which we find some new Cunningham chains of primes.

Finally we consider Gauss' conjecture that there are no negative discriminants with one class of binary quadratic forms in each genus which have at least 32 genera. We apply our algorithm to find numbers satisfying modular conditions to prove a new general lower bound for such discriminants, adapt the methods of Watkins in his determination of discriminants with class number at most 100 to prove a large lower bound for discriminants indivisible by 2, 3 or 5, and adapt Baker's method for class numbers one and two to prove that such a discriminant cannot have a particularly large prime factor.

Authors: Armitage, J

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: zeros of polynomials, rational functions, and other analytic functions; computational number theory; modular and automorphic functions; number theory; elliptic functions and integrals; class numbers, class groups, discriminants
• Subjects: Mathematics