arXiv subjects

Bella Tobin

Publications and source records attributed to Bella Tobin.

9 recordsLinked to original sources

arXiv2025

Local data of elliptic curves under quadratic twist

Let $K$ be the field of fractions of a complete discrete valuation ring with a perfect residue field. In this article, we investigate how the Tamagawa number of $E/K$ changes under quadratic twist. To accomplish this, we introduce the notion of a strongly-minimal model for an elliptic curve $E/K$, which is a minimal Weierstrass model satisfying certain conditions that lead one to easily infer the local data of $E/K$. Our main results provide explicit conditions on the Weierstrass coefficients of a strongly-minimal model of $E/K$ to determine the local data of a quadratic twist $E^{d}/K$. We note that when the residue field has characteristic $2$, we only consider the special case $K=\mathbb{Q}_{2}$. In this setting, we also determine the minimal discriminant valuation and conductor exponent of $E$ and $E^d$ from further conditions on the coefficients of a strongly-minimal model for $E$.

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arXiv2024

Dynamical Irreducibility of Certain Families of Polynomials over Finite Fields

We determine necessary and sufficient conditions for unicritical polynomials to be dynamically irreducible over finite fields. This result extends the results of Boston-Jones and Hamblen-Jones-Madhu regarding the dynamical irreducibility of particular families of unicritical polynomials. We also investigate dynamical irreducibility conditions for cubic and shifted linearized polynomials.

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arXiv2024

Supercongruences arising from Ramanujan-Sato Series

Recently, the authors with Lea Beneish established a recipe for constructing Ramanujan-Sato series for $1/\pi$, and used this to construct 11 explicit examples of Ramanujan-Sato series arising from modular forms for arithmetic triangle groups of non-compact type. Here, we use work of Chisholm, Deines, Long, Nebe and the third author to prove a general $p$-adic supercongruence theorem through an explicit connection to CM hypergeometric elliptic curves that provides $p$-adic analogues of these Ramanujan-Sato series. We further use this theorem to construct explicit examples related to each of our explicit Ramanujan-Sato series examples.

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arXiv2024

On a Slice of the Cubic 2-adic Mandelbrot Set

Consider the one-parameter family of cubic polynomials defined by $f_t(z) =-\frac 32 t(-2z^3+3z^2)+1, t \in \mathbb{C}_2$. This family corresponds to a slice of the parameter space of cubic polynomials in $\mathbb{C}_2[z]$. We investigate which parameters in this family belong to the cubic $2$-adic Mandelbrot set, a $p$-adic analog of the classical Mandelbrot set. When $t=1$, $f_t(z)$ is post-critically finite with a strictly preperiodic critical orbit. We establish that this is a non-isolated boundary point on the cubic $2$-adic Mandelbrot set and show asymptotic self-similarity of the Mandelbrot set near this point. Subsequently, we investigate the Julia set for polynomial on the boundary and demonstrate a similarity between the Mandelbrot set at this point and the Julia set, similar to what is seen in the classical complex case.

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arXiv2022

Generalized Ramanujan-Sato Series Arising from Modular Forms

Motivated by work of Chan, Chan, and Liu, we obtain a new general theorem which produces Ramanujan-Sato series for $1/\pi$. We then use it to construct explicit examples related to non-compact arithmetic triangle groups, as classified by Takeuchi. Some of our examples are new, and some reproduce existing examples.

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arXiv2021

Stochastic Equidistribution and Generalized Adelic Measures

We study the dynamics of stochastic families of rational maps on the projective line. As such families can be infinite and may not typically be defined over a single number field, we introduce the concept of generalized adelic measures, generalizing previous notions introduced by Favre and Rivera-Letelier and Mavraki and Ye. Generalized adelic measures are defined over the measure space of places of an algebraic closure of the rationals, using a framework established by Allcock and Vaaler. This turns our heights from sums over places into integrals. We prove an equidistribution result for generalized adelic measures, and use this result to prove an equidistribution result for random backwards orbits in stochastic arithmetic dynamics.

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arXiv2021

Some Applications of Dynamical Belyi Polynomials

We give necessary and sufficient conditions for post-critically finite polynomials to have persistent bad reduction at a given prime. We also answer in the negative a pair of questions posed by Silverman about conservative polynomials. Our proofs rely on conservative dynamical Belyi polynomials as exemplars of PCF (resp. conservative) maps.

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arXiv2020

Cubic post-critically finite polynomials defined over $\mathbb{Q}$

We describe and implement an algorithm to find all post-critically finite (PCF) cubic polynomials defined over $\mathbb{Q}$, up to conjugacy over $\text{PGL}_2(\bar{\mathbb{Q}})$. We describe normal forms that classify equivalence classes of cubic polynomials while respecting the field of definition. Applying known bounds on the coefficients of post-critically bounded polynomials to these normal forms simultaneously at all places of $\mathbb{Q}$, we create a finite search space of cubic polynomials over $\mathbb{Q}$ that may be PCF. Using a computer search of these possibly PCF cubic polynomials, we find fifteen which are in fact PCF.

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arXiv2019

Dessins d'Enfants for Single-Cycle Belyi Maps

Riemann's Existence Theorem gives the following bijections: (1) Isomorphism classes of Belyi maps of degree $d$. (2) Equivalence classes of generating systems of degree $d$. (3) Isomorphism classes of dessins d'enfants with $d$ edges. In previous work, the first author and collaborators exploited the correspondence between Belyi maps and their generating systems to provide explicit equations for two infinite families of dynamical Belyi maps. We complete this picture by describing the dessins d'enfants for these two families.

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