arXiv subjects

Ashvin Swaminathan

Publications and source records attributed to Ashvin Swaminathan.

24 records · Page 2Linked to original sources

arXiv2017

Permutations that Destroy Arithmetic Progressions in Elementary $p$-Groups

Given an abelian group $G$, it is natural to ask whether there exists a permutation $π$ of $G$ that "destroys" all nontrivial 3-term arithmetic progressions (APs), in the sense that $π(b) - π(a) \neq π(c) - π(b)$ for every ordered triple $(a,b,c) \in G^3$ satisfying $b-a = c-b \neq 0$. This question was resolved for infinite groups $G$ by Hegarty, who showed that there exists an AP-destroying permutation of $G$ if and only if $G/Ω_2(G)$ has the same cardinality as $G$, where $Ω_2(G)$ denotes the subgroup of all elements in $G$ whose order divides $2$. In the case when $G$ is finite, however, only partial results have been obtained thus far. Hegarty has conjectured that an AP-destroying permutation of $G$ exists if $G = \mathbb{Z}/n\mathbb{Z}$ for all $n \neq 2,3,5,7$, and together with Martinsson, he has proven the conjecture for all $n > 1.4 \times 10^{14}$. In this paper, we show that if $p$ is a prime and $k$ is a positive integer, then there is an AP-destroying permutation of the elementary $p$-group $(\mathbb{Z}/p\mathbb{Z})^k$ if and only if $p$ is odd and $(p,k) \not\in \{(3,1),(5,1), (7,1)\}$.

math.NT↗

arXiv2016

Elliptic Curve Variants of the Least Quadratic Nonresidue Problem and Linnik's Theorem

Let $E_1$ and $E_2$ be $\overline{\mathbb{Q}}$-nonisogenous, semistable elliptic curves over $\mathbb{Q}$, having respective conductors $N_{E_1}$ and $N_{E_2}$ and both without complex multiplication. For each prime $p$, denote by $a_{E_i}(p) := p+1-\#E_i(\mathbb{F}_p)$ the trace of Frobenius. Under the assumption of the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power $L$-functions $L(s, \mathrm{Sym}^i E_1\otimes\mathrm{Sym}^j E_2)$ where $i,j\in\{0,1,2\}$, we prove an explicit result that can be stated succinctly as follows: there exists a prime $p\nmid N_{E_1}N_{E_2}$ such that $a_{E_1}(p)a_{E_2}(p)<0$ and \[ p < \big( (32+o(1))\cdot \log N_{E_1} N_{E_2}\big)^2. \] This improves and makes explicit a result of Bucur and Kedlaya. Now, if $I\subset[-1,1]$ is a subinterval with Sato-Tate measure $μ$ and if the symmetric power $L$-functions $L(s, \mathrm{Sym}^k E_1)$ are functorial and satisfy GRH for all $k \le 8/μ$, we employ similar techniques to prove an explicit result that can be stated succinctly as follows: there exists a prime $p\nmid N_{E_1}$ such that $a_{E_1}(p)/(2\sqrt{p})\in I$ and \[ p < \left((21+o(1)) \cdot μ^{-2}\log (N_{E_1}/μ)\right)^2. \]

math.NT↗

arXiv2015

On Logarithmically Benford Sequences

Let $\mathcal{I} \subset \mathbb{N}$ be an infinite subset, and let $\{a_i\}_{i \in \mathcal{I}}$ be a sequence of nonzero real numbers indexed by $\mathcal{I}$ such that there exist positive constants $m, C_1$ for which $|a_i| \leq C_1 \cdot i^m$ for all $i \in \mathcal{I}$. Furthermore, let $c_i \in [-1,1]$ be defined by $c_i = \frac{a_i}{C_1 \cdot i^m}$ for each $i \in \mathcal{I}$, and suppose the $c_i$'s are equidistributed in $[-1,1]$ with respect to a continuous, symmetric probability measure $μ$. In this paper, we show that if $\mathcal{I} \subset \mathbb{N}$ is not too sparse, then the sequence $\{a_i\}_{i \in \mathcal{I}}$ fails to obey Benford's Law with respect to arithmetic density in any sufficiently large base, and in fact in any base when $μ([0,t])$ is a strictly convex function of $t \in (0,1)$. Nonetheless, we also provide conditions on the density of $\mathcal{I} \subset \mathbb{N}$ under which the sequence $\{a_i\}_{i \in \mathcal{I}}$ satisfies Benford's Law with respect to logarithmic density in every base. As an application, we apply our general result to study Benford's Law-type behavior in the leading digits of Frobenius traces of newforms of positive, even weight. Our methods of proof build on the work of Jameson, Thorner, and Ye, who studied the particular case of newforms without complex multiplication.

math.NT↗

arXiv2015

Linnik's Theorem for Sato-Tate Laws on Elliptic Curves with Complex Multiplication

Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication (CM), and for each prime $p$ of good reduction, let $a_E(p) = p + 1 - \#E(\mathbb{F}_p)$ denote the trace of Frobenius. By the Hasse bound, $a_E(p) = 2\sqrt{p} \cos θ_p$ for a unique $θ_p \in [0, π]$. In this paper, we prove that the least prime $p$ such that $θ_p \in [α, β] \subset [0, π]$ satisfies \[ p \ll \left(\frac{N_E}{β- α}\right)^A, \] where $N_E$ is the conductor of $E$ and the implied constant and exponent $A > 2$ are absolute and effectively computable. Our result is an analogue for CM elliptic curves of Linnik's Theorem for arithmetic progressions, which states that the least prime $p \equiv a \pmod q$ for $(a,q)=1$ satisfies $p \ll q^L$ for an absolute constant $L > 0$.

math.NT↗

arXiv2015

On Arboreal Galois Representations of Rational Functions

The action of the absolute Galois group $\text{Gal}(K^{\text{ksep}}/K)$ of a global field $K$ on a tree $T(ϕ, α)$ of iterated preimages of $α\in \mathbb{P}^1(K)$ under $ϕ\in K(x)$ with $\text{deg}(ϕ) \geq 2$ induces a homomorphism $ρ: \text{Gal}(K^{\text{ksep}}/K) \to \text{Aut}(T(ϕ, α))$, which is called an arboreal Galois representation. In this paper, we address a number of questions posed by Jones and Manes about the size of the group $G(ϕ,α) := \text{im} ρ= \underset{\leftarrow n}\lim\text{Gal}(K(ϕ^{-n}(α))/K)$. Specifically, we consider two cases for the pair $(ϕ, α)$: (1) $ϕ$ is such that the sequence $\{a_n\}$ defined by $a_0 = α$ and $a_n = ϕ(a_{n-1})$ is periodic, and (2) $ϕ$ commutes with a nontrivial Mobius transformation that fixes $α$. In the first case, we resolve a question posed by Jones about the size of $G(ϕ, α)$, and taking $K = \mathbb{Q}$, we describe the Galois groups of iterates of polynomials $ϕ\in \mathbb{Z}[x]$ that have the form $ϕ(x) = x^2 + kx$ or $ϕ(x) = x^2 - (k+1)x + k$. When $K = \mathbb{Q}$ and $ϕ\in \mathbb{Z}[x]$, arboreal Galois representations are a useful tool for studying the arithmetic dynamics of $ϕ$. In the case of $ϕ(x) = x^2 + kx$ for $k \in \mathbb{Z}$, we employ a result of Jones regarding the size of the group $G(ψ, 0)$, where $ψ(x) = x^2 - kx + k$, to obtain a zero-density result for primes dividing terms of the sequence $\{a_n\}$ defined by $a_0 \in \mathbb{Z}$ and $a_n = ϕ(a_{n-1})$. In the second case, we resolve a conjecture of Jones about the size of a certain subgroup $C(ϕ, α) \subset \text{Aut}(T(ϕ, α))$ that contains $G(ϕ, α)$, and we present progress toward the proof of a conjecture of Jones and Manes concerning the size of $G(ϕ, α)$ as a subgroup of $C(ϕ, α)$.

math.NT↗

arXiv2015

Analysis on Surreal Numbers

The class $\mathbf{No}$ of surreal numbers, which John Conway discovered while studying combinatorial games, possesses a rich numerical structure and shares many arithmetic and algebraic properties with the real numbers. Some work has also been done to develop analysis on $\mathbf{No}$. In this paper, we extend this work with a treatment of functions, limits, derivatives, power series, and integrals. We propose surreal definitions of the arctangent and logarithm functions using truncations of Maclaurin series. Using a new representation of surreals, we present a formula for the limit of a sequence, and we use this formula to provide a complete characterization of convergent sequences and to evaluate certain series and infinite Riemann sums via extrapolation. A similar formula allows us to evaluates limits (and hence derivatives) of functions. By defining a new topology on $\mathbf{No}$, we obtain the Intermediate Value Theorem even though $\mathbf{No}$ is not Cauchy complete, and we prove that the Fundamental Theorem of Calculus would hold for surreals if a consistent definition of integration exists. Extending our study to defining other analytic functions, evaluating power series in generality, finding a consistent definition of integration, proving Stokes' Theorem to generalize surreal integration, and studying differential equations remains open.

math.CA↗