arXiv subjects

Pete L. Clark

Publications and source records attributed to Pete L. Clark.

48 records · Page 3Linked to original sources

arXiv2008

Period, index and potential sha

In this paper we advance the theory of O'Neil's period-index obstruction map and derive consequences for the arithmetic of genus one curves over global fields. Our first result implies that for every pair of positive integers (P,I) with P dividing I and I dividing P^2, there exists a number field K and a genus one curve C over K with period P and index I. Second, let E be any elliptic curve over a global field K, and let P > 1 be any integer indivisible by the characteristic of K. We construct infinitely many genus one curves C over K with period P, index P^2, and Jacobian E. We deduce strong consequences on the structure of Sharevich-Tate groups under field extension.

math.NT↗

arXiv2008

Elliptic Dedekind Domains Revisited

We give an affirmative answer to a 1976 question of M. Rosen: every abelian group is isomorphic to the class group of an elliptic Dedekind domain R. We can choose R to be the integral closure of a PID in a separable quadratic field extension. In particular, this yields new and -- we feel -- simpler proofs of theorems of L. Claborn and C.R. Leedham-Green.

math.AC↗

arXiv2006

On the indices of curves over local fields

Fix a non-negative integer g and a positive integer I dividing 2g-2. For any Henselian, discretely valued field K whose residue field is perfect and admits a degree I cyclic extension, we construct a curve C over K of genus g and index I. We can in fact give a complete description of the finite extensions L/K such that C has an L-rational point. Applications are discussed to the corresponding problem over number fields. S. Sharif, in his 2006 Berkeley thesis, has independently obtained similar (but not identical) results. Our proof, however, is different: via deformation theory, we reduce to the problem of finding suitable actions of cyclic groups on finite graphs.

math.NT↗

arXiv2006

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 w_N-Atkin-Lehner twists of most modular curves X_0(N) and to w_p-Atkin-Lehner twists of certain Shimura curves X^{D+}.

math.NT↗

arXiv2006

Abelian points on algebraic curves

We study the question of whether algebraic curves of a given genus g defined over a field K must have points rational over the maximal abelian extension K^{ab} of K. We give: (i) an explicit family of diagonal plane cubic curves with Q^{ab}-points, (ii) for every number field K, a genus one curve C_{/Q} with no K^{ab}-points, and (iii) for every g \geq 4 an algebraic curve C_{/Q} of genus g with no Q^{ab}-points. In an appendix, we discuss varieties over Q((t)), obtaining in particular a curve of genus 3 without (Q((t)))^{ab}-points.

math.NT↗

arXiv2005

On the Hasse principle for Shimura curves

Let C be an algebraic curve defined over a number field K, of positive genus and without K-rational points. We conjecture that there exists some extension field L over which C violates the Hasse principle, i.e., has points everywhere locally but not globally. We show that our conjecture holds for all but finitely many Shimura curves of the form X^D_0(N)_{/Q} or X^D_1(N)_{/Q}, where D > 1 and N are coprime squarefree positive integers. The proof uses a variation on a theorem of Frey, a gonality bound of Abramovich, and an analysis of local points of small degree.

math.NT↗

arXiv2005

Galois Groups Via Atkin-Lehner Twists

Using Serre's proposed complement to Shih's Theorem, we obtain PSL_2(F_p) as a Galois group over Q for at least 614 new primes p. Under the assumption that rational elliptic curves with odd analytic rank have positive rank, we obtain Galois realizations for 3/8 of the primes not covered by previous results; it would also suffice to assume a certain (plausible, and perhaps tractable) conjecture concerning class numbers of quadratic fields. The key issue is to understand rational points on Atkin-Lehner twists of X_0(N). In an appendix, we explore the existence of local points on these curves.

math.NT↗

arXiv2004

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

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 Kolyvagin's construction of a rational elliptic curve whose Mordell-Weil and Shafarevich-Tate groups are both trivial, we show that there are infinitely many curves of every index over every number field.

math.NT↗

arXiv2004

Bounds for torsion on abelian varieties with integral moduli

We give a function F(d,n,p) such that if K/Q_p is a degree n field extension and A/K is a d-dimensional abelian variety with potentially good reduction, then #A(K)[tors] is at most F(d,n,p). Separate attention is given to the prime-to-p torsion and to the case of purely additive reduction. These latter bounds are applied to classify rational torsion on CM elliptic curves over number fields of degree at most 3, on elliptic curves over Q with integral j (recovering a theorem of Frey), and on abelian surfaces over Q with integral moduli. In the last case, our efforts leave us with 11 numbers which may, or may not, arise as the order of the full torsion subgroup. The largest such number is 72.

math.NT↗

arXiv2004

On elementary equivalence, isomorphism and isogeny of arithmetic function fields

Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and of quadric surfaces. These results are applied to deduce new instances of "elementary equivalence implies isomorphism": for all genus zero curves over a number field, and for certain genus one curves over a number field, including some which are not elliptic curves.

math.LO↗

arXiv2004

The period-index problem in WC-groups II: abelian varieties

We study the relationship between the period and the index of a principal homogeneous space over an abelian variety, obtaining results which generalize work of Cassels and Lichtenbaum on curves of genus one. In addition, we show that the p-torsion in the Shafarevich-Tate group of a fixed abelian variety over a number field k grows arbitrarily large when considered over field extensions l/k of bounded degree. Essential use is made of an abelian variety version of O'Neil's period-index obstruction.

math.NT↗

arXiv2004

The period-index problem in WC-groups I: elliptic curves

Let E/K be an elliptic curve defined over a number field, and let p be a prime number such that E(K) has full p-torsion. We show that the order of the p-part of the Shafarevich-Tate group of E/L is unbounded as L varies over degree p extensions of K. The proof uses O'Neil's period-index obstruction. We deduce the result from the fact that, under the same hypotheses, there exist infinitely many elements of the Weil-Chatelet group of E/K of period p and index p^2.

math.NT↗