arXiv subjects

Pedro Lemos

Publications and source records attributed to Pedro Lemos.

5 recordsLinked to original sources

arXiv2022

Bounds on the number of rational points of curves in families

In this note, we give an alternative proof of uniform boundedness of the number of integral points of smooth projective curves over a fixed number field with good reduction outside of a fixed set of primes. We use that due to Bertin-Romagny, the Kodaira-Parshin families constructed by Lawrence-Venkatesh can themselves be assembled into a family. We then repeat Lawrence-Venkatesh's study of the p-adic period map, together with the comparison of nearby fibres.

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arXiv2020

Residual Galois representations of elliptic curves with image contained in the normaliser of a non-split Cartan

It is known that if $p>37$ is a prime number and $E/\mathbb{Q}$ is an elliptic curve without complex multiplication, then the image of the mod $p$ Galois representation $$ \barρ_{E,p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}(E[p]) $$ of $E$ is either the whole of $\operatorname{GL}(E[p])$, or is \emph{contained} in the normaliser of a non-split Cartan subgroup of $\operatorname{GL}(E[p])$. In this paper, we show that when $p>1.4\times 10^7$, the image of $\barρ_{E,p}$ is either $\operatorname{GL}(E[p])$, or the \emph{full} normaliser of a non-split Cartan subgroup. We use this to show the following result, partially settling a question of Najman. For $d\geq 1$, let $I(d)$ denote the set of primes $p$ for which there exists an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication admitting a degree $p$ isogeny defined over a number field of degree $\leq d$. We show that, for $d\geq 1.4\times 10^7$, we have $$ I(d)=\{p\text{ prime}:p\leq d-1\}. $$

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arXiv2018

Some cases of Serre's uniformity problem

We show that if $E/\mathbb{Q}$ is an elliptic curve without complex multiplication and for which there is a prime $q$ such that the image of $\barρ_{E,q}$ is contained in the normaliser of a split Cartan subgroup of $\rm{GL}_2(\mathbb{F}_q)$, then $\barρ_{E,p}$ surjects onto $\rm{GL}_2(\mathbb{F}_p)$ for every prime $p>37$. This result complements a previous result by the author. We also prove analogue results for certain families of $\mathbb{Q}$-curves, building on results of Ellenberg (2004) and Le Fourn (2016).

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arXiv2017

Serre's Uniformity Conjecture for Elliptic Curves with Rational Cyclic Isogenies

Let $E$ be an elliptic curve over $\mathbb{Q}$ such that $\mathrm{End}_{\bar{\mathbb{Q}}}(E)=\mathbb{Z}$ and which admits a non-trivial cyclic $\mathbb{Q}$-isogeny. We prove that, for $p>37$, the residual mod $p$ Galois representation $\barρ_{E,p}:G_{\mathbb{Q}}\rightarrow\mathrm{GL}_2(\mathbb{F}_p)$ is surjective.

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arXiv2016

Residual Representations of Semistable Principally Polarized Abelian Varieties

Let $A$ be a semistable principally polarized abelian variety of dimension $d$ defined over the rationals. Let $\ell$ be a prime and let $\barρ_{A,\ell} : G_{\mathbb{Q}} \rightarrow \mathrm{GSp}_{2d}(\mathbb{F}_\ell)$ be the representation giving the action of $G_{\mathrm{Q}} :=\mathrm{Gal}(\bar{\mathrm{Q}}/\mathrm{Q})$ on the $\ell$-torsion group $A[\ell]$. We show that if $\ell \ge \max(5,d+2)$, and if image of $\barρ_{A,\ell}$ contains a transvection then $\barρ_{A,\ell}$ is either reducible or surjective. With the help of this we study surjectivity of $\barρ_{A,\ell}$ for semistable principally polarized abelian threefolds, and give an example of a genus $3$ hyperelliptic curve $C/\mathbb{Q}$ such that $\barρ_{J,\ell}$ is surjective for all primes $\ell \ge 3$, where $J$ is the Jacobian of $C$.

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