Pete L. Clark

There are genus one curves of every index over every number field

Abstract We show that there exist genus one curves of every index over the rational numbers, answering affirmatively a question of Lang and Tate. The proof is “elementary” in the sense that it does not assume the finiteness of any Shafarevich-Tate group. On the other hand, using Kolyvagins construction of a rational elliptic curve whose Mordell-Weil and Shafarevich-Tate groups are both trivial, we show that there are infinitely many genus one curves of every index over every number field.

TORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION (WITH AN APPENDIX BY ALEX RICE)

We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results refine bounds of Silverberg and Prasad-Yogananda by taking into account the class number of the CM order and the splitting of the prime in the CM field. In many cases we can show that our refined bounds are optimal or asymptotically optimal. We also derive asymptotic upper and lower bounds on the least degree of a CM-point on X_1(N). Upon comparison to bounds for the least degree for which there exist infinitely many rational points on X_1(N), we deduce that, for sufficiently large N, X_1(N) will have a rational CM point of degree smaller than the degrees of at least all but finitely many non-CM points.

Computation on elliptic curves with complex multiplication

We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number elds of degree 1-13. Addi- tionally we describe the algorithm used to compute these torsion subgroups and its implementation.

Warning's Second Theorem with restricted variables

We present a restricted variable generalization of Warning’s Second Theorem (a result giving a lower bound on the number of solutions of a low degree polynomial system over a finite field, assuming one solution exists). This is analogous to Schauz-Brink’s restricted variable generalization of Chevalley’s Theorem (a result giving conditions for a low degree polynomial system not to have exactly one solution). Just as Warning’s Second Theorem implies Chevalley’s Theorem, our result implies Schauz-Brink’s Theorem. We include several combinatorial applications, enough to show that we have a general tool for obtaining quantitative refinements of combinatorial existence theorems.Let q = pℓ be a power of a prime number p, and let Fq be “the” finite field of order q.For a1,...,an, N∈Z+, we denote by m(a1,...,an;N)∈Z+ a certain combinatorial quantity defined and computed in Section 2.1.

AN "ANTI-HASSE PRINCIPLE" FOR PRIME TWISTS

Given an algebraic curve C/ℚ having points everywhere locally and endowed with a suitable involution, we show that there exists a positive density family of prime quadratic twists of C violating the Hasse principle. The result applies in particular to wN-Atkin–Lehner twists of most modular curves X0(N) and to wp-Atkin–Lehner twists of certain Shimura curves XD+.

Anatomy of torsion in the CM case

Let

The Euclidean Criterion for Irreducibles

T_{\mathrm {CM}}(d)

Absolute Convergence in Ordered Fields

TCM(d) denote the maximum size of a torsion subgroup of a CM elliptic curve over a degree d number field. We initiate a systematic study of the asymptotic behavior of