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Let $E/\mathbb{Q}$ be an elliptic curve and let $p\ge 5$ be a prime of good supersingular reduction. We generalize results due to Meng Fai Lim proving Kida's formula and integrality results for characteristic elements of signed Selmer…

Let $\ell$ be a prime number and let $E$ and $E'$ be $\ell$-isogenous elliptic curves defined over a finite field $k$ of characteristic $p \ne \ell$. Suppose the groups $E(k)$ and $E'(k)$ are isomorphic, but $E(K) \not \simeq E'(K)$, where…

For an elliptic curve $E$ defined over a number field $K$, the heuristic density of the set of primes of $K$ for which $E$ has cyclic reduction is given by an inclusion-exclusion sum $\delta_{E/K}$ involving the degrees of the $m$-division…

Division polynomials associated to an elliptic curve $E/K$ are polynomials $\phi_n, \psi_n^2$ that arise from the sequence of points $\{nP\}_{n \in \mathbb{N}}$ on this curve. If one wishes to study $\mathbb{Z}$--linear combination of…

Using a combination of several powerful modularity theorems and class field theory we derive a new modularity theorem for semistable elliptic curves over certain real abelian fields. We deduce that if $K$ is a real abelian field of…

Let $G$ be a commutative connected algebraic group over a number field $K$, let $A$ be a finitely generated and torsion-free subgroup of $G(K)$ of rank $r>0$ and, for $n>1$, let $K(n^{-1}A)$ be the smallest extension of $K$ inside an…

Let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j_{K,f})$,…

Given a prime power q, for every pair of positive integers m and n with m dividing the GCD of n and q-1, we construct a modular curve over F_q that parametrizes elliptic curves over F_q along with F_q-defined points P and Q of order m and…

Let $K=\mathbb{Q}(\sqrt{-p})$ be a quadratic field for an odd prime $p$. We show that there exist infinitely many primes $p$ for which no elliptic curve $E/\mathbb{Q}$ has torsion subgroup $\mathbb{Z}/2\mathbb{Z}\times…

We discuss principality of prime ideals of finite algebraic number fields $L=K(\theta)$ over an algebraic number field $K ([K:\mathbb{Q}]<\infty)$ defined by irreducible polynomials $f(x)\in \mathfrak{O}_{K}[x]$ and $f(\theta)=0$. Our main…

We construct isotrivial and non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ with an arbitrarily large set of separable integral points. As an application of this construction, we prove that there are isotrivial log-general type…

Let $p$ be a prime number, $K$ a finite unramified extension of $\mathbb{Q}_p$ and $\mathbb{F}$ a finite extension of $\mathbb{F}_p$. Using perfectoid spaces we associate to any finite-dimensional continuous representation $\overline{\rho}$…

Let E be an elliptic curve over a real quadratic field K and F/K a totally real finite Galois extension. We prove that E/F is modular.

In this note we prove that for every integer $d \geq 1$, there exists an explicit constant $B_d$ such that the following holds. Let $K$ be a number field of degree $d$, let $q > \max\{d-1,5\}$ be any rational prime that is totally inert in…

For a quadratic field $\mathcal{K}$ without rationally defined CM, we prove that there exists of a prime $p_{\mathcal{K}}$ depending only on $\mathcal{K}$ such that if $d$ is a positive integer whose minimal prime divisor is greater than…

Given an elliptic curve $E$ over a number field $K$, the $\ell$-torsion points $E[\ell]$ of $E$ define a Galois representation $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$. A famous theorem of Serre states that as long as $E$ has no Complex…

Fix distinct primes $p$ and $q$ and let $E$ be an elliptic curve defined over a number field $K$. The $(p,q)$-entanglement type of $E$ over $K$ is the isomorphism class of the group $\operatorname{Gal}(K(E[p])\cap K(E[q])/K)$. The size of…

Building on the positive solution of Pillay's conjecture we present a notion of "intrinsic" reduction for elliptic curves over a real closed field K. We compare such notion with the traditional algebro-geometric reduction and produce a…

Let $p \geq 5$ be a prime number. We find all the possible subgroups $G$ of ${\rm GL}_2 ( \mathbb{Z} / p \mathbb{Z} )$ such that there exists a number field $k$ and an elliptic curve ${\mathcal{E}}$ defined over $k$ such that the ${\rm Gal}…

In 1985, Schoof devised an algorithm to compute zeta functions of elliptic curves over finite fields by directly computing the numerators of these rational functions modulo sufficiently many primes (see \cite{schoof_1985}). If $E/K$ is an…