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Tyler Genao

Publications and source records attributed to Tyler Genao.

12 recordsLinked to original sources
arXiv2025

The possible adelic indices for elliptic curves admitting a rational cyclic isogeny

In the 1970s, Serre proved that the adelic index of a non-CM elliptic curve over a number field is finite. More recently, Zywina conjectured the complete set of adelic indices for such curves over $\mathbb{Q}$. In this article, we prove that Zywina's conjecture is true for the family of non-CM elliptic curves over $\mathbb{Q}$ that admit a nontrivial rational cyclic isogeny. This strengthens a result of Lemos that resolved Serre's uniformity question for the same family of curves. Our proof proceeds by analyzing a collection of modular curves associated with each prime isogeny degree, using recent advances on $\ell$-adic images, isogeny-torsion graphs, and computations of models and rational points.

math.NT↗
arXiv2025

Uniform bounds on the level of cyclotomic division fields of elliptic curves

In this paper, we prove that for each number field $F$ there exists a uniform bound on the prime levels $p$ of elliptic curves $E/F$ for which $F(E[p])=F(\zeta_p)$. Under the Generalized Riemann Hypothesis, we also give uniform bounds on $p$ for which $F(E[p])/F$ is abelian, provided that $F$ has no rational complex multiplication. These are generalizations of results of Gonz\'{a}lez-Jim\'{e}nez and Lozano-Robledo to general number fields.

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arXiv2025

A uniform bound on the smallest surjective prime of an elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. A well-known theorem of Serre asserts that the $\ell$-adic Galois representation $\rho_{E,\ell^\infty}$ is surjective for all but finitely many prime numbers $\ell$. Considerable work has gone into bounding the largest possible nonsurjective prime; a uniform bound of $37$ has been proposed but is yet unproven. We consider an opposing direction, proving that the smallest prime $\ell$ such that $\rho_{E,\ell^\infty}$ is surjective is at most $7$. Moreover, we completely classify all elliptic curves $E/\mathbb{Q}$ for which the smallest surjective prime is exactly $7$.

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arXiv2024

Uniform polynomial bounds on torsion from rational geometric isogeny classes

In 1996, Merel showed there exists a function $B\colon \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for any elliptic curve $E/F$ defined over a number field of degree $d$, one has the torsion group bound $\# E(F)[\textrm{tors}]\leq B(d)$. Based on subsequent work, it is conjectured that one can choose $B$ to be polynomial in the degree $d$. In this paper, we show that such bounds exist for torsion from the family $\mathcal{I}_{\mathbb{Q}}$ of elliptic curves which are geometrically isogenous to at least one rational elliptic curve. More precisely, we show that for each $\epsilon>0$, there exists $c_\epsilon>0$ such that for any elliptic curve $E/F\in \mathcal{I}_{\mathbb{Q}}$, one has \[ E(F)[\textrm{tors}]\leq c_\epsilon\cdot [F:\mathbb{Q}]^{3+\epsilon}. \] This generalizes work of the second author for elliptic curves within a fixed rational geometric isogeny class. For the family of elliptic curves with rational $j$-invariant, we also obtain bounds which improve those of Clark and Pollack. In this case, our bounds on the exponent of $E(F)[\textrm{tors}]$ are optimal if one does not exclude elliptic curves with complex multiplication.

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arXiv2024

New isogenies of elliptic curves over number fields

Using Galois representations, we analyze fields of definition of cyclic isogenies on elliptic curves to prove the following uniformity result: for any number field $F$ which has no rational CM, under GRH there exists an effectively computable constant $B:=B(F)\in\mathbb{Z}^+$ such that for any finite extension $L/F$ whose degree $[L:F]$ is coprime to $B$, one has for all elliptic curves $E_{/F}$ that any $L$-rational isogeny on $E$ is $F$-rational. For any number field $F$, under GRH we also prove results for the mod-$\ell$ Galois representations of non-CM elliptic curves with an $F$-rational isogeny of uniformly large prime degree $\ell$.

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arXiv2022

Growth of torsion groups of elliptic curves upon base change from number fields

Given a number field $F_0$ that contains no Hilbert class field of any imaginary quadratic field, we show that under GRH there exists an effectively computable constant $B:=B(F_0)\in\mathbb{Z}^+$ for which the following holds: for any finite extension $L/F_0$ whose degree $[L:F_0]$ is coprime to $B$, one has for all elliptic curves $E_{/F_0}$ that the $L$-rational torsion subgroup $E(L)[\textrm{tors}]=E(F_0)[\textrm{tors}]$. This generalizes a previous result of Gonz\'{a}lez-Jim\'{e}nez and Najman over $F_0=\mathbb{Q}$. Towards showing this, we also prove a result on relative uniform divisibility of the index of a mod-$\ell$ Galois representation of an elliptic curve over $F_0$. Additionally, we show that the main result's conclusion fails when we allow $F_0$ to have rationally defined CM, due to the existence of $F_0$-rational isogenies of arbitrarily large prime degrees satisfying certain congruency conditions.

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arXiv2022

Polynomial bounds on torsion from a fixed geometric isogeny class of elliptic curves

We show there exist polynomial bounds on torsion of elliptic curves which come from a fixed geometric isogeny class. More precisely, for an elliptic curve $E_0$ defined over a number field $F_0$, for each $\epsilon>0$ there exist constants $c_\epsilon:=c_\epsilon(E_0,F_0),C_\epsilon:=C_\epsilon(E_0,F_0)>0$ such that for any elliptic curve $E_{/F}$ geometrically isogenous to $E_0$, if $E(F)$ has a point of order $N$ then \[ N\leq c_\epsilon\cdot [F:\mathbb{Q}]^{1/2+\epsilon}, \] and one also has \[ \# E(F)[\textrm{tors}] \leq C_\epsilon\cdot [F:\mathbb{Q}]^{1+\epsilon}. \]

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arXiv2021

Typically bounding torsion on elliptic curves isogenous to rational $j$-invariant

We prove that the family $\mathcal{I}_{F_0}$ of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with $F_0$-rational $j$-invariant is typically bounded in torsion. Under an additional uniformity assumption, we also prove that the family $\mathcal{I}_{d_0}$ of elliptic curves over number fields that are geometrically isogenous to an elliptic curve with degree $d_0$ $j$-invariant is typically bounded in torsion.

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arXiv2021

Computational study of non-unitary partitions

Following Cayley, MacMahon, and Sylvester, define a non-unitary partition to be an integer partition with no part equal to one, and let $\nu(n)$ denote the number of non-unitary partitions of size $n$. In a 2021 paper, the sixth author proved a formula to compute $p(n)$ by enumerating only non-unitary partitions of size $n$, and recorded a number of conjectures regarding the growth of $\nu(n)$ as $n\to \infty$. Here we refine and prove some of these conjectures. For example, we prove $p(n) \sim \nu(n)\sqrt{n/\zeta(2)}$ as $n\to \infty$, and give Ramanujan-like congruences between $p(n)$ and $\nu(n)$ such as $p(5n)\equiv \nu(5n)\ (\operatorname{mod} 5)$.

math.CO↗
arXiv2021

Typically bounding torsion on elliptic curves with rational $j$-invariant

A family $\mathcal{F}$ of elliptic curves defined over number fields is said to be typically bounded in torsion if the torsion subgroups $E(F)[$tors$]$ of those elliptic curves $E_{/F}\in \mathcal{F}$ can be made uniformly bounded after removing from $\mathcal{F}$ those whose number field degrees lie in a subset of $\mathbb{Z}^+$ with arbitrarily small upper density. For every number field $F$, we prove unconditionally that the family $\mathcal{E}_F$ of elliptic curves defined over number fields and with $F$-rational $j$-invariant is typically bounded in torsion. For any integer $d\in\mathbb{Z}^+$, we also strengthen a result on typically bounding torsion for the family $\mathcal{E}_d$ of elliptic curves defined over number fields and with degree $d$ $j$-invariant.

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arXiv2015

The density of primes dividing a particular non-linear recurrence sequence

Define the sequence $\{b_n\}$ by $b_0=1,b_1=1, b_2=2,b_3=1$, and $$b_n=\begin{cases} \frac{b_{n-1}b_{n-3}-b_{n-2}^2}{b_{n-4}}&\textrm{if}~ n\not\equiv 0\pmod 3, \frac{b_{n-1}b_{n-3}-3b_{n-2}^2}{b_{n-4}}&\textrm{if}~ n\equiv 0\pmod 3. We relate this sequence $\{b_n\}$ to the coordinates of points on the elliptic curve $E:y^2+y=x^3-3x+4$. We use Galois representations attached to $E$ to prove that the density of primes dividing a term in this sequence is equal to $\frac{179}{336}$. Furthermore, we describe an infinite family of elliptic curves whose Galois images match that of $E$.

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