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Pete L. Clark

Publications and source records attributed to Pete L. Clark.

At least 37 records · Page 2Linked to original sources
arXiv2015

Torsion Points on CM Elliptic Curves Over Real Number Fields

We study torsion subgroups of elliptic curves with complex multiplication (CM) defined over number fields which admit a real embedding. We give a complete classification of the groups which arise up to isomorphism as the torsion subgroup of a CM elliptic curve defined over a number field of odd degree: there are infinitely many. Restricting to the case of prime degree, we show that there are only finitely many isomorphism classes. More precisely, there are six "Olson groups" which arise as torsion subgroups of CM elliptic curves over number fields of every degree, and there are precisely 17 "non-Olson" CM elliptic curves defined over a prime degree number field.

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arXiv2015

Fattening up Warning's Second Theorem

We present a generalization of Warning's Second Theorem to polynomial systems over a finite local principal ring with suitably restricted input and output variables. This generalizes a recent result with Forrow and Schmitt (and gives a new proof of that result). Applications to additive group theory, graph theory and polynomial interpolation are pursued in detail.

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arXiv2015

Algebraic curves uniformized by congruence subgroups of triangle groups

We construct certain subgroups of hyperbolic triangle groups which we call "congruence" subgroups. These groups include the classical congruence subgroups of SL_2(ZZ), Hecke triangle groups, and 19 families of arithmetic triangle groups associated to Shimura curves. We determine the field of moduli of the curves associated to these groups and thereby realize the groups PSL_2(q) and PGL_2(q) regularly as Galois groups.

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arXiv2015

Anatomy of torsion in the CM case

Let $T_{\mathrm{CM}}(d)$ 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 $T_{\mathrm{CM}}(d)$ as an "arithmetic function". Whereas a recent result of the last two authors computes the upper order of $T_{\mathrm{CM}}(d)$, here we determine the lower order, the typical order and the average order of $T_{\mathrm{CM}}(d)$ as well as study the number of isomorphism classes of groups $G$ of order $T_{\mathrm{CM}}(d)$ which arise as the torsion subgroup of a CM elliptic curve over a degree $d$ number field. To establish these analytic results we need to extend some prior algebraic results. Especially, if $E_{/F}$ is a CM elliptic curve over a degree $d$ number field, we show that $d$ is divisible by a certain function of $\# E(F)[\mathrm{tors}]$, and we give a complete characterization of all degrees $d$ such that every torsion subgroup of a CM elliptic curve defined over a degree $d$ number field already occurs over $\mathbb{Q}$.

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arXiv2014

Warning's Second Theorem with Resricted 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 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 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.

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arXiv2014

There are genus one curves of every index over every infinite, finitely generated field

Every nontrivial abelian variety over a Hilbertian field in which the weak Mordell-Weil theorem holds admits infinitely many torsors with period any $n > 1$ which is not divisible by the characteristic. The corresponding statement with "period" replaced by "index" is plausible but much more challenging. We show that for every infinite, finitely generated field $K$, there is an elliptic curve $E_{/K}$ which admits infinitely many torsors with index any $n > 1$.

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arXiv2014

Abstract Geometry of Numbers: Linear Forms

This paper concerns the \textbf{abstract geometry of numbers}: namely the pursuit of certain aspects of geometry of numbers over a suitable class of normed domains. (The standard geometry of numbers is then viewed as geometry of numbers over Z endowed with its standard absolute value.) In this work we study normed domains of "linear type", in which an analogue of Minkowski's linear forms theorem holds. We show that S-integer rings in number fields and coordinate rings of (nice) affine algebraic curves over an arbitrary ground field are of linear type. The theory is applied to quadratic forms in two ways, yielding a Nullstellensatz and a Small Multiple Theorem.

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arXiv2013

Computation on Elliptic Curves with Complex Multiplication

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

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arXiv2013

Graph Derangements

We introduce the notion of a graph derangement, which naturally interpolates between perfect matchings and Hamiltonian cycles. We give a necessary and sufficient condition for the existence of graph derangements on a locally finite graph. This result was first proved by W.T. Tutte in 1953 by applying some deeper results on digraphs. We give a new, simple proof which amounts to a reduction to the (Menger-Egervary-Konig-)Hall(-Hall) Theorem on transversals of set systems. Finally, we consider the problem of classifying all cycle types of graph derangements on m x n checkerboard graphs. Our presentation does not assume any prior knowledge in graph theory or combinatorics: all definitions and proofs of needed theorems are given.

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arXiv2012

The Instructor's Guide to Real Induction

We introduce real induction, a proof technique analogous to mathematical induction but applicable to statements indexed by an interval on the real line. More generally we give an inductive principle applicable in any Dedekind complete linearly ordered set. Real and ordered induction is then applied to give streamlined, conceptual proofs of basic results in honors calculus, elementary real analysis and topology.

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arXiv2012

Euclidean Quadratic Forms and ADC Forms I

Motivated by classical results of Aubry, Davenport and Cassels, we define the notion of a Euclidean quadratic form over a normed integral domain and an ADC form over an integral domain. The aforementioned classical results generalize to: Euclidean forms are ADC forms. We then initiate the study and classification of these two classes of quadratic forms, especially over discrete valuation rings and Hasse domains.

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arXiv2012

Covering Numbers in Linear Algebra

We compute the minimal cardinality of a covering (resp. an irredundant covering) of a vector space over an arbitrary field by proper linear subspaces. Analogues for affine linear subspaces are also given.

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arXiv2012

A Note on Euclidean Order Types

Euclidean functions with values in an arbitrary well-ordered set were first considered in a 1949 work of Motzkin and studied in more detail in work of Fletcher, Samuel and Nagata in the 1970's and 1980's. Here these results are revisited, simplified, and extended. The two main themes are (i) consideration of Ord-valued functions on an Artinian poset and (ii) use of ordinal arithmetic, including the Hessenberg-Brookfield ordinal sum. In particular, to any Euclidean ring we associate an ordinal invariant, its Euclidean order type, and we initiate a study of this invariant. The main new result gives upper and lower bounds on the Euclidean order type of a finite product of Euclidean rings in terms of the Euclidean order types of the factor rings.

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arXiv2009

Period-index problems in WC-groups IV: a local transition theorem

Let K be a complete discretely valued field with perfect residue field k. Assuming upper bounds on the relation between period and index for WC-groups over k, we deduce corresponding upper bounds on the relation between period and index for WC-groups over K. Up to a constant depending only on the dimension of the torsor, we recover theorems of Lichtenbaum and Milne in a "duality free" context. Our techniques include the use of LLR models of torsors under abelian varieties with good reduction and a generalization of the period-index obstruction map to flat cohomology. In an appendix, we consider some related issues of a field-arithmetic nature.

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arXiv2009

Torsion points on elliptic curves with complex multiplication

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.

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arXiv2009

Curves over global fields violating the Hasse Principle

In response to a question of B. Poonen, we exhibit for each global field k an algebraic curve over k which violates the Hasse Principle. In fact we can find such examples among Atkin-Lehner twists of certain elliptic modular curves and -- in positive characteristic -- Drinfeld modular curves. Our main tool is a refinement of the "Twist Anti-Hasse Principle" (TAHP). We also use TAHP to construct further Hasse Principle violations, for instance among curves over any number field of any given genus g which is at least 2.

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