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Paul Pollack

Publications and source records attributed to Paul Pollack.

At least 55 records · Page 3Linked to original sources
arXiv2016

Torsion subgroups of CM elliptic curves over odd degree number fields

Let $\mathscr{G}_{\rm CM}(d)$ denote the collection of groups (up to isomorphism) that appear as the torsion subgroup of a CM elliptic curve over a degree $d$ number field. We completely determine $\mathscr{G}_{\rm CM}(d)$ for odd integers $d$ and deduce a number of statistical theorems about the behavior of torsion subgroups of CM elliptic curves. Here are three examples: (1) For each odd $d$, the set of natural numbers $d'$ with $\mathscr{G}_{\rm CM}(d') = \mathscr{G}_{\rm CM}(d)$ possesses a well-defined, positive asymptotic density. (2) Let $T_{\rm CM}(d) = \max_{G \in \mathscr{G}_{\rm CM}(d)} \#G$; under the Generalized Riemann Hypothesis, $$\left(\frac{12e^γ}π\right)^{2/3} \le \limsup_{\substack{d\to\infty\\d\text{ odd}}} \frac{T_{\rm CM}(d)}{(d\log\log{d})^{2/3}} \le \left(\frac{24e^γ}π\right)^{2/3}.$$ (3) For each $ε> 0$, we have $\#\mathscr{G}_{\rm CM}(d) \ll_ε d^ε$ for all odd $d$; on the other hand, for each $A> 0$, we have $\#\mathscr{G}_{\rm CM}(d) > (\log{d})^A$ for infinitely many odd $d$.

math.NT↗
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↗
arXiv2015

Digitally delicate primes

Tao has shown that in any fixed base, a positive proportion of prime numbers cannot have any digit changed and remain prime. In other words, most primes are "digitally delicate". We strengthen this result in a manner suggested by Tao: A positive proportion of primes become composite under any change of a single digit and any insertion a fixed number of arbitrary digits at the beginning or end.

math.NT↗
arXiv2015

Bounds for the first several prime character nonresidues

Let $\varepsilon > 0$. We prove that there are constants $m_0=m_0(\varepsilon)$ and $κ=κ(\varepsilon) > 0$ for which the following holds: For every integer $m > m_0$ and every nontrivial Dirichlet character modulo $m$, there are more than $m^κ$ primes $\ell \le m^{\frac{1}{4\sqrt{e}}+\varepsilon}$ with $χ(\ell)\notin \{0,1\}$. The proof uses the fundamental lemma of the sieve, Norton's refinement of the Burgess bounds, and a result of Tenenbaum on the distribution of smooth numbers satisfying a coprimality condition. For quadratic characters, we demonstrate a somewhat weaker lower bound on the number of primes $\ell \le m^{\frac14+ε}$ with $χ(\ell)=1$.

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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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arXiv2015

The length spectra of arithmetic hyperbolic 3-manifolds and their totally geodesic surfaces

In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of techniques from analytic number theory, we address the following problems: Is the commensurability class of an arithmetic hyperbolic 3-orbifold determined by the lengths of closed geodesics lying on totally geodesic surfaces?, Do there exist arithmetic hyperbolic 3-orbifolds whose "short" geodesics do not lie on any totally geodesic surfaces?, and Do there exist arithmetic hyperbolic 3-orbifolds whose "short" geodesics come from distinct totally geodesic surfaces?

math.GT↗
arXiv2015

Clusters of primes with square-free translates

Let $\mathcal{R}$ be a finite set of integers satisfying appropriate local conditions. We show the existence of long clusters of primes $p$ in bounded length intervals with $p-b$ squarefree for all $b \in \mathcal{R}$. Moreover, we can enforce that the primes $p$ in our cluster satisfy any one of the following conditions: (1) $p$ lies in a short interval $[N, N+N^{\frac{7}{12}+ε}]$, (2) $p$ belongs to a given inhomogeneous Beatty sequence, (3) with $c \in (\frac{8}{9},1)$ fixed, $p^c$ lies in a prescribed interval mod $1$ of length $p^{-1+c+ε}$.

math.NT↗
arXiv2014

The average of the first invariant factor for reductions of CM elliptic curves mod $p$

Let $E/\mathbb{Q}$ be a fixed elliptic curve. For each prime $p$ of good reduction, write $E(\mathbb{F}_p) \cong \mathbb{Z}/d_p \mathbb{Z} \oplus \mathbb{Z}/e_p \mathbb{Z}$, where $d_p \mid e_p$. Kowalski proposed investigating the average value of $d_p$ as $p$ runs over the rational primes. For CM curves, he showed that $x\log\log{x}/\log{x} \ll \sum_{p \le x} d_p \ll x\sqrt{\log{x}}$. It was shown recently by Felix and Murty that in fact $\sum_{p \le x} d_p$ exceeds any constant multiple of $x\log\log{x}/\log{x}$, once $x$ is sufficiently large. In the opposite direction, Kim has shown that the expression $x\sqrt{\log{x}}$ in the upper bound can be replaced by $x\log\log{x}$. In this paper, we obtain the correct order of magnitude for the sum: $\sum_{p \le x} d_p \asymp x$ for all large $x$.

math.NT↗
arXiv2014

Arithmetic functions at consecutive shifted primes

For each of the functions $f \in \{ϕ, σ, ω, τ\}$ and every natural number $k$, we show that there are infinitely many solutions to the inequalities $f(p_n-1) < f(p_{n+1}-1) < \dots < f(p_{n+k}-1)$, and similarly for $f(p_n-1) > f(p_{n+1}-1) > \dots > f(p_{n+k}-1)$. We also answer some questions of Sierpiński on the digit sums of consecutive primes. The arguments make essential use of Maynard and Tao's method for producing many primes in intervals of bounded length.

math.NT↗
arXiv2014

Bounded gaps between primes with a given primitive root, II

Let $m$ be a natural number, and let $\mathcal{Q}$ be a set containing at least $\exp(C m)$ primes. We show that one can find infinitely many strings of $m$ consecutive primes each of which has some $q\in\mathcal{Q}$ as a primitive root, all lying in an interval of length $O_{\mathcal{Q}}(\exp(C'm))$. This is a bounded gaps variant of a theorem of Gupta and Ram Murty. We also prove a result on an elliptic analogue of Artin's conjecture. Let $E/\mathbb{Q}$ be an elliptic curve with an irrational $2$-torsion point. Assume GRH. Then for every $m$, there are infinitely many strings of $m$ consecutive primes $p$ for which $E(\mathbb{F}_p)$ is cyclic, all lying an interval of length $O_E(\exp(C'' m))$. If $E$ has CM, then the GRH assumption can be removed. Here $C$, $C'$, and $C''$ are absolute constants.

math.NT↗
arXiv2014

Besicovitch, Bisection, and the normality of $0.(1)(4)(9)(16)(25)\dots$

We revisit Besicovitch's 1935 paper in which he introduced several techniques that have become essential elements of modern combinatorial methods of normality proofs. Despite his paper's influence, the results he inspired are not strong enough to reprove his original result. We provide a new proof of the normality of the constant $0.(1)(4)(9)(16)(25)\dots$ formed by concatenating the squares, updating Besicovitch's methods.

math.NT↗
arXiv2014

Bounded gaps between primes with a given primitive root

Fix an integer $g \neq -1$ that is not a perfect square. In 1927, Artin conjectured that there are infinitely many primes for which $g$ is a primitive root. Forty years later, Hooley showed that Artin's conjecture follows from the Generalized Riemann Hypothesis (GRH). We inject Hooley's analysis into the Maynard--Tao work on bounded gaps between primes. This leads to the following GRH-conditional result: Fix an integer $m \geq 2$. If $q_1 < q_2 < q_3 < \dots$ is the sequence of primes possessing $g$ as a primitive root, then $\liminf_{n\to\infty} (q_{n+(m-1)}-q_n) \leq C_m$, where $C_m$ is a finite constant that depends on $m$ but not on $g$. We also show that the primes $q_n, q_{n+1}, \dots, q_{n+m-1}$ in this result may be taken to be consecutive.

math.NT↗
arXiv2014

Bounded gaps between primes in number fields and function fields

The Hardy--Littlewood prime $k$-tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem. In this work, we extend the Maynard-Tao method to both number fields and the function field $\mathbb{F}_q(t)$.

math.NT↗
arXiv2014

Averages of the number of points on elliptic curves

If $E$ is an elliptic curve defined over $\mathbb Q$ and $p$ is a prime of good reduction for $E$, let $E(\mathbb F_p)$ denote the set of points on the reduced curve modulo $p$. Define an arithmetic function $M_E(N)$ by setting $M_E(N):= \#\{p: \#E(\mathbb F_p)= N\}$. Recently, David and the third author studied the average of $M_E(N)$ over certain "boxes" of elliptic curves $E$. Assuming a plausible conjecture about primes in short intervals, they showed the following: for odd $N$, the average of $M_E(N)$ over a box with sufficiently large sides is $\sim \frac{K^{\ast}(N)}{\log{N}}$ for an explicitly-given function $K^{\ast}(N)$. The function $K^{\ast}(N)$ is somewhat peculiar: defined as a product over the primes dividing $N$, it resembles a multiplicative function at first glance. But further inspection reveals that it is not, and so one cannot directly investigate its properties by the usual tools of multiplicative number theory. In this paper, we overcome these difficulties and prove a number of statistical results about $K^{\ast}(N)$. For example, we determine the mean value of $K^{\ast}(N)$ over all $N$, odd $N$ and prime $N$, and we show that $K^{\ast}(N)$ has a distribution function. We also explain how our results relate to existing theorems and conjectures on the multiplicative properties of $\# E(\mathbb F_p)$, such as Koblitz's conjecture.

math.NT↗
arXiv2013

Some normal numbers generated by arithmetic functions

Let $g \geq 2$. A real number is said to be g-normal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let ϕdenote Euler's totient function, let σbe the sum-of-divisors function, and let λbe Carmichael's lambda-function. We show that if f is any function formed by composing ϕ, σ, or λ, then the number \[ 0. f(1) f(2) f(3) \dots \] obtained by concatenating the base g digits of successive f-values is g-normal. We also prove the same result if the inputs 1, 2, 3, \dots are replaced with the primes 2, 3, 5, \dots. The proof is an adaptation of a method introduced by Copeland and Erdos in 1946 to prove the 10-normality of 0.235711131719\ldots.

math.NT↗
arXiv2013

Variations on a theorem of Davenport concerning abundant numbers

Let σ(n) = \sum_{d \mid n}d be the usual sum-of-divisors function. In 1933, Davenport showed that that n/σ(n) possesses a continuous distribution function. In other words, the limit D(u):= \lim_{x\to\infty} \frac{1}{x}\sum_{n \leq x,~n/σ(n) \leq u} 1 exists for all u \in [0,1] and varies continuously with u. We study the behavior of the sums \sum_{n \leq x,~n/σ(n) \leq u} f(n) for certain complex-valued multiplicative functions f. Our results cover many of the more frequently encountered functions, including φ(n), τ(n), and μ(n). They also apply to the representation function for sums of two squares, yielding the following analogue of Davenport's result: For all u \in [0,1], the limit \[ \tilde{D}(u):= \lim_{R\to\infty} \frac{1}{πR}\#\{(x,y) \in \Z^2: 0<x^2+y^2 \leq R \text{ and } \frac{x^2+y^2}{σ(x^2+y^2)} \leq u\} \] exists, and \tilde{D}(u) is both continuous and strictly increasing on [0,1].

math.NT↗
arXiv2012

Uncertainty principles connected with the Möbius inversion formula

We say that two arithmetic functions f and g form a Mobius pair if f(n) = \sum_{d \mid n} g(d) for all natural numbers n. In that case, g can be expressed in terms of f by the familiar Mobius inversion formula of elementary number theory. In a previous paper, the first-named author showed that if the members f and g of a Mobius pair are both finitely supported, then both functions vanish identically. Here we prove two significantly stronger versions of this uncertainty principle. A corollary is that in a nonzero Mobius pair, either \sum_{n \in supp(f)} 1/n or \sum_{n \in supp(g)} 1/n diverges.

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