We obtain explicit formulas for the number of nonisomorphic elliptic curves with a given group structure (considered as an abstract abelian group) and the number of distinct group structures of all elliptic curves over a finite field. We use these formulas to derive some asymptotic estimates and tight upper and lower bounds for various counting functions related to classification of elliptic curves according to their group structure. Finally, we present results of some numerical tests that exhibit several interesting phenomena in the distribution of group structures.

We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection that correspond to curves over prime fields or to curves with a prescribed torsion. Some of our results are rigorous and are based on recent advances in analytic number theory; some are conditional under certain widely believed conjectures; and others are purely heuristic in nature.

We address the problem of computing bounds for the selfintersection number (the minimum number of generic self intersection points) of members of a free homotopy class of curves in the doubly punctured plane as a function of their combinatorial length $L$ ; this is the number of letters required for a minimal description of the class in terms of a set of standard generators of the fundamental group and their inverses. We prove that the self-intersection number is bounded above by $L^2/4 + L/2 − 1$, and that when $L$ is even, this bound is sharp; in that case, there are exactly four distinct classes attaining that bound. For odd L we conjecture a smaller upper bound, $(L^2 − 1)/4$, and establish it in certain cases in which we show that it is sharp. Furthermore, for the doubly punctured plane, these self-intersection numbers are bounded below, by $L/2 − 1$ if $L$ is even, and by $(L − 1)/2$ if $L$ is odd. These bounds are sharp.

We consider invariants of a finite group related to the number of random (independent, uniformly distributed) conjugacy classes that are required to generate it. These invariants are intuitively related to problems of Galois theory.We find group-theoretic expressions for them and investigate their values both theoretically and numerically.

This paper numerically computes the topological and smooth invariants of Eschenburg–Kruggel spaces with small fourth cohomology group, following Kruggel’s determination of the Kreck–Stolz invariants of Eschenburg spaces satisfying condition C. It is shown that each topological Eschenburg–Kruggel space with small fourth cohomology group has each of its 28 oriented smooth structures represented by an Eschenburg–Kruggel space. Our investigations also suggest that there is an action of $\mathbb{Z}_{12}$ on the set of homotopy classes of Eschenburg–Kruggel spaces, the nature of which remains to be understood.

The calculations are done in C++ with the GNU GMP arbitrary precision library and Jon Wilkening’s C++ wrapper.

In this article, we present analogues of supercongruences of Ramanujan type observed by L. Van Hamme and W. Zudilin. Our congruences are inspired by Ramanujan-type series that involve quadratic algebraic numbers.

Given a homotopy connecting two polynomial systems, we provide a rigorous algorithm for tracking a regular homotopy path connecting an approximate zero of the start system to an approximate zero of the target system. Our method uses recent results on the complexity of homotopy continuation rooted in the alpha theory of Smale. Experimental results obtained with an implementation in the numerical algebraic geometry package Macaulay2 demonstrate the practicality of the algorithm. In particular, we confirm the theoretical results for random linear homotopies and illustrate the plausibility of a conjecture by Shub and Smale on a good initial pair.

The Sato–Tate conjecture has been recently settled in great generality. One natural question now concerns the rate of convergence of the distribution of the Fourier coefficients of modular newforms to the Sato–Tate distribution. In this paper, we address this issue, imposing congruence conditions on the primes and on the Fourier coefficients as well. Assuming a proper error term in the convergence to a conjectural limiting distribution, supported by experimental data, we prove the Lang–Trotter conjecture, and in the direction of Lehmer’s conjecture, we prove that $\tau (p) = 0$ has at most finitely many solutions. In fact, we propose a conjecture, much more general than Lehmer’s, about the vanishing of Fourier coefficients of any modular newform.