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Pete L. Clark

Publications and source records attributed to Pete L. Clark.

At least 19 recordsLinked to original sources
arXiv2024

Functional degrees and arithmetic applications III: Beyond Prime Exponent

Continuing our work on group-theoretic generalizations of the prime Ax-Katz Theorem, we give a lower bound on the $p$-adic divisibility of the cardinality of the set of simultaneous zeros $Z(f_1,f_2,\ldots,f_r)$ of $r$ maps $f_j:A\rightarrow B_j$ between arbitrary finite commutative groups $A$ and $B_j$ in terms of the invariant factors of $A, B_1,B_2,\dotsc,B_r$ and the \emph{functional degrees} of the maps $f_1,f_2,\dotsc,f_r$.

math.NT↗
arXiv2023

Functional degrees and arithmetic applications II: The Group-Theoretic Prime Ax-Katz Theorem

We give a version of Ax-Katz's $p$-adic congruences and Moreno-Moreno's $p$-weight refinement that holds over any finite commutative ring of prime characteristic. We deduce this from a purely group-theoretic result that gives a lower bound on the $p$-adic divisibility of the number of simultaneous zeros of a system of maps $f_j: A\to B_j$ from a fixed ``source'' finite commutative group $A$ of exponent $p$ to varying ``target'' finite commutative $p$-groups $B_j$. Our proof combines Wilson's proof of Ax-Katz over $\mathbb{F}_p$ with the functional calculus of Aichinger-Moosbauer.

math.GR↗
arXiv2023

Densities of integer sets represented by quadratic forms

Let $f(t_1,\ldots,t_n)$ be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function $D_f(X)$, the number of integers of absolute value up to $X$ represented by $f$. When $f$ is isotropic or $n$ is at least $3$, we show that there is a $δ(f) \in \mathbb{Q} \cap (0,1)$ such that $D_f(X) \sim δ(f) X$ and call $δ(f)$ the density of $f$. We consider the inverse problem of which densities arise. Our main technical tool is a Near Hasse Principle: a quadratic form may fail to represent infinitely many integers that it locally represents, but this set of exceptions has density $0$ within the set of locally represented integers.

math.NT↗
arXiv2022

CM Elliptic Curves: Volcanoes, Reality and Applications, Part II

Let $M \mid N$ be positive integers, and let $Δ$ be the discriminant of an order in an imaginary quadratic field $K$. When $Δ_K < -4$, the first author determined the fiber of the morphism $X_0(M,N) \rightarrow X(1)$ over the closed point $J_Δ$ corresponding to $Δ$ and showed that all fibers of the map $X_1(M,N) \rightarrow X_0(M,N)$ over $J_Δ$ were connected. Here we complement this prior work by addressing the most difficult cases $Δ_K \in \{-3,-4\}$. These works provide all the information needed to compute, for each positive integer $d$, all subgroups of $E(F)[\operatorname{tors}]$, where $F$ is a number field of degree $d$ and $E_{/F}$ is an elliptic curve with complex multiplication.

math.NT↗
arXiv2022

CM Elliptic Curves: Volcanoes, Reality and Applications

For positive integers $M \mid N$ and an order of discriminant $Δ$ in an imaginary quadratic field $K$ with discriminant $Δ_K < -4$, we determine the fiber of the morphism $X_0(M,N) \rightarrow X(1)$ over the closed point $J_Δ$ corresponding to $Δ$. We also show that the fiber of the natural map $X_1(M,N) \rightarrow X_0(M,N)$ over $J_Δ$ is connected. Putting this together we deduce the number of points in the fiber of $X_1(M,N) \rightarrow X(1)$ over $J_Δ$ and their residual degrees. In the continuation of this work with F. Saia, these results will be extended to $Δ_K \in \{-4,3\}$. These works provide all the information needed to compute, for each positive integer $d$, all subgroups of $E(F)[\operatorname{tors}]$, where $F$ is a number field of degree $d$ and $E_{/F}$ is an elliptic curve with complex multiplication (CM).

math.NT↗
arXiv2022

Restricted Variable Chevalley-Warning Theorems

We pursue various restricted variable generalizations of the Chevalley-Warning theorem for low degree polynomial systems over a finite field. Our first such result involves variables restricted to Cartesian products of the Vandermonde subsets of $\F_q$ defined by Gács-Weiner and Sziklai-Takáts. We then define an invariant $\uomega(X)$ of a nonempty subset of $\F_q^n$. Our second result involves $X$-restricted variables when the degrees of the polynomials are small compared to $\uomega(X)$. We end by exploring various classes of subsets for which $\uomega(X)$ can be bounded from below.

math.NT↗
arXiv2019

Torsion points and Galois representations on CM elliptic curves

We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic isogenies of a $K$-CM elliptic curve $E$ defined over $K(j(E))$.

math.NT↗
arXiv2019

Torsion points and isogenies on CM elliptic curves

Let $\mathcal{O}$ be an order in the imaginary quadratic field $K$. For positive integers $M \mid N$, we determine the least degree of an $\mathcal{O}$-CM point on the modular curve $X(M,N)_{/K(ζ_M)}$ and also on the modular curve $X(M,N)_{/\mathbb{Q}(ζ_M)}$: that is, we treat both the case in which the complex multiplication is rationally defined and the case in which we do not assume that the complex multiplication is rationally defined. To prove these results we establish several new theorems on rational cyclic isogenies of CM elliptic curves. In particular, we extend a result of Kwon that determines the set of positive integers $N$ for which there is an $\mathcal{O}$-CM elliptic curve $E$ admitting a cyclic, $\mathbb{Q}(j(E))$-rational $N$-isogeny.

math.NT↗
arXiv2017

A generalization of the theorems of Chevalley-Warning and Ax-Katz via polynomial substitutions

We give conditions under which the number of solutions of a system of polynomial equations over a finite field F_q of characteristic p is divisible by p. Our setup involves the substitution t_i |-> f_i(t_i) for auxiliary polynomials f_1,...,f_n in F_q[t]. We recover as special cases results of Chevalley-Warning and Morlaye-Joly. Then we investigate higher p-adic divisibilities, proving a result that recovers the Ax-Katz Theorem. We also consider p-weight degrees, recovering work of Moreno-Moreno, Moreno-Castro and Castro-Castro-Velez.

math.NT↗
arXiv2017

Typically bounding torsion

We formulate the notion of \emph{typical boundedness} of torsion on a family of abelian varieties defined over number fields. This means that the torsion subgroups of elements in the family can be made uniformly bounded by removing from the family all abelian varieties defined over number fields of degree lying in a set of arbitrarily small density. We show that for each fixed $g$, torsion is typically bounded on the family of all $g$-dimensional CM abelian varieties. We show that torsion is \emph{not} typically bounded on the family of all elliptic curves, and we establish results -- some unconditional and some conditional -- on typical boundedness of torsion of elliptic curves for which the degree of the $j$-invariant is fixed.

math.NT↗
arXiv2017

On zeros of a polynomial in a finite grid

A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain "Condition (D)" on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further Generalized Alon-Füredi Theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon-Füredi. We then discuss the relationship between Alon-Füredi and results of DeMillo-Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon-Füredi Theorem and its generalization in terms of Reed--Muller type affine variety codes is shown which gives us the minimum Hamming distance of these codes. Then we apply the Alon-Füredi Theorem to quickly recover (and sometimes strengthen) old and new results in finite geometry, including the Jamison/Brouwer-Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.

math.CO↗
arXiv2017

Pursuing polynomial bounds on torsion

We show that for all epsilon > 0, there is a constant C(epsilon) > 0 such that for all elliptic curves E defined over a number field F with j(E) in Q we have #E(F)[tors] \leq C(epsilon)[F:Q]^{5/2+epsilon}. We pursue further bounds on the size of the torsion subgroup of an elliptic curve over a number field E/F that are polynomial in [F:Q] under restrictions on j(E). We give an unconditional result for j(E) lying in a fixed quadratic field that is not imaginary of class number one as well as two further results, one conditional on GRH and one conditional on the strong boundedness of isogenies of prime degree for non-CM elliptic curves.

math.NT↗
arXiv2016

The truth about torsion in the CM case, II

Let $T_{\rm CM}(d)$ be the largest size of the torsion subgroup of an elliptic curve with complex multiplication (CM) defined over a degree $d$ number field. Work of Breuer and Clark--Pollack showed $\limsup_{d \to \infty} \frac{T_{\rm CM}(d)}{d \log \log d} \in (0,\infty)$. Here we show that the above limit supremum is precisely $\frac{e^γ π}{\sqrt{3}}$. We also study -- in part, out of necessity -- the upper order of the size of the torsion subgroup of various restricted classes of CM elliptic curves over number fields.

math.NT↗
arXiv2016

Hasse Principle Violations for Atkin-Lehner Twists of Shimura Curves

Let $D > 546$ be the discriminant of an indefinite rational quaternion algebra. We show that there are infinitely many imaginary quadratic fields $l/\mathbb Q$ such that the twist of the Shimura curve $X^D$ by the main Atkin-Lehner involution $w_D$ and $l/\mathbb Q$ violates the Hasse Principle over $\mathbb Q$.

math.NT↗
arXiv2016

The Euclidean criterion for irreducibles

We recast Euclid's proof of the infinitude of prime numbers as a Euclidean Criterion for a domain to have infinitely many atoms. We make connections with Furstenberg's "topological" proof of the infinitude of prime numbers and show that our criterion applies even in certain domains in which not all nonzero nonunits factor into products of irreducibles.

math.AC↗
arXiv2016

A note on rings of finite rank

The rank of a ring $R$ is the supremum of minimal cardinalities of generating sets of $I$ as $I$ ranges over ideals of $R$. Matson showed that every positive integer occurs as the rank of some ring $R$. Motivated by the result of Cohen and Gilmer that a ring of finite rank has Krull dimension $0$ or $1$, we give four different constructions of rings of rank $n$ (for all positive integers n). Two constructions use one-dimensional domains, and the former of these directly generalizes Matson's construction. Our third construction uses Artinian rings (dimension zero), and our last construction uses polynomial rings over local Artinian rings (dimension one, irreducible, not a domain).

math.AC↗
arXiv2015

The number of atoms in an atomic domain

We study the number of atoms and maximal ideals in an atomic domain with finitely many atoms and no prime elements. We show in particular that for all $m,n \in \mathbb{Z}^+$ with $n \geq 3$ and $4 \leq m \leq \frac{n}{3}$ there is an atomic domain with precisely $n$ atoms, precisely $m$ maximal ideals and no prime elements. The proofs use both commutative algebra and additive number theory.

math.AC↗