# Arithmetic of strongly modular Q-curves and the density of coprime m-tuples of algebraic integers

Ferragutti, Andrea. Arithmetic of strongly modular Q-curves and the density of coprime m-tuples of algebraic integers. 2016, University of Zurich, Faculty of Science.

## Abstract

The main objects of study of this thesis are $\mathbb{Q}$-curves. The first chapter is devoted to giving an introduction to the theory of modular and strongly modular elliptic curve over $\overline{\mathbb{Q}}$. We will review the fundamental work of Ribet on $\mathbb{Q}$-curves as quotients of abelian varieties of ${GL_2}$-type and that of Guitart and Quer which characterize strongly modular elliptic curves.
In the second chapter we address the following problem: given a quadratic $\mathbb{Q}$-curve $\mathit{E}$ completely defined over a quadratic field $\mathit{K}$, how can one prove that $\mathit{L}$($\mathit{E}$, 1) = 0, when this is the case? The answer will be given by exhibiting, under the generalized Manin conjecture, an effective integer $\mathit{Q}$, depending on $\mathit{E}$ and on the choice of an invariant differential $\omega_{\mathit{E}}$ on $\mathit{E}$, such that $\mathit{L}$($\mathit{E}$,1)$\cdot$$\mathit{Q}$$\cdot$$\sqrt{|\Delta_{\mathit{K}}|}/\Omega_{\mathit{E}} is an integer. Here \Delta_{\mathit{K}} is the discriminant of \mathit{K} and \Omega_{\mathit{E}} is a quantity related to the infinite part of the period of \mathit{E}. An important ingredient is an algorithm to compute a newform \mathit{f} of level \Gamma_{1}(\mathit{N}) such that \mathit{L}(\mathit{E}, \mathit{s}) = \mathit{L}(\mathit{f}, \mathit{s}) \mathit{L}(^{\sigma}\mathit{f}, \mathit{s}) for ^{\sigma}\mathit{f} the unique Galois conjugate of \mathit{f}. The third chapter is dedicated to studying strongly modular twists of \mathbb{Q}-curves. In the first part, we find necessary and sufficient conditions for the existence of strongly modular twists of quadratic \mathbb{Q}-curves over their minimal field of complete definition. We also show how to characterize completely primitive twists, which are elliptic curves not isogenous to the base change of a curve over a smaller field. In the second part, we prove that a \mathbb{Q}-curve is geometrically isomorphic to a strongly modular one if and only if it is geometrically isomorphic to a \mathbb{Q}-curve whose minimal field of complete definition is abelian over \mathbb{Q}. Finally, in chapter four we study a different problem. A classical theorem, originally due to Mertens and Cesàro (independently), states that the natural density of the set of coprime \mathit{m}-tuples of integers is 1/\zeta(\mathit{m}), where \zeta(\mathit{s}) is the Riemann zeta function. Given a number field \mathit{K} with ring of integers \mathcal{O}, we introduce a notion of density (depending on the choice of a \mathbb{Z}-basis for \mathcal{O}) for subsets of \mathcal{O} which generalizes the notion of natural density for subsets of \mathbb{Z}. We then show that the density of the set of coprime m-tuples of algebraic integers in \mathbb{Z} is 1/\zeta_{\mathit{K}}(\mathit{m}), where \zeta_{\mathit{K}}(\mathit{s}) is the Dedekind zeta function of \mathit{K}. In particular, the density of this set does not depend on the choice of a \mathbb{Z}-basis for \mathcal{O}. ## Abstract The main objects of study of this thesis are \mathbb{Q}-curves. The first chapter is devoted to giving an introduction to the theory of modular and strongly modular elliptic curve over \overline{\mathbb{Q}}. We will review the fundamental work of Ribet on \mathbb{Q}-curves as quotients of abelian varieties of {GL_2}-type and that of Guitart and Quer which characterize strongly modular elliptic curves. In the second chapter we address the following problem: given a quadratic \mathbb{Q}-curve \mathit{E} completely defined over a quadratic field \mathit{K}, how can one prove that \mathit{L}(\mathit{E}, 1) = 0, when this is the case? The answer will be given by exhibiting, under the generalized Manin conjecture, an effective integer \mathit{Q}, depending on \mathit{E} and on the choice of an invariant differential \omega_{\mathit{E}} on \mathit{E}, such that \mathit{L}(\mathit{E},1)\cdot$$\mathit{Q}$$\cdot$$\sqrt{|\Delta_{\mathit{K}}|}$/$\Omega_{\mathit{E}}$ is an integer. Here $\Delta_{\mathit{K}}$ is the discriminant of $\mathit{K}$ and $\Omega_{\mathit{E}}$ is a quantity related to the infinite part of the period of $\mathit{E}$. An important ingredient is an algorithm to compute a newform $\mathit{f}$ of level $\Gamma_{1}$($\mathit{N})$ such that $\mathit{L}(\mathit{E}, \mathit{s})$ = $\mathit{L}(\mathit{f}, \mathit{s})$ $\mathit{L}(^{\sigma}\mathit{f}, \mathit{s})$ for $^{\sigma}\mathit{f}$ the unique Galois conjugate of $\mathit{f}$.
The third chapter is dedicated to studying strongly modular twists of $\mathbb{Q}$-curves. In the first part, we find necessary and sufficient conditions for the existence of strongly modular twists of quadratic $\mathbb{Q}$-curves over their minimal field of complete definition. We also show how to characterize completely primitive twists, which are elliptic curves not isogenous to the base change of a curve over a smaller field. In the second part, we prove that a $\mathbb{Q}$-curve is geometrically isomorphic to a strongly modular one if and only if it is geometrically isomorphic to a $\mathbb{Q}$-curve whose minimal field of complete definition is abelian over $\mathbb{Q}$.
Finally, in chapter four we study a different problem. A classical theorem, originally due to Mertens and Cesàro (independently), states that the natural density of the set of coprime $\mathit{m}$-tuples of integers is 1/$\zeta(\mathit{m})$, where $\zeta(\mathit{s})$ is the Riemann zeta function. Given a number field $\mathit{K}$ with ring of integers $\mathcal{O}$, we introduce a notion of density (depending on the choice of a $\mathbb{Z}$-basis for $\mathcal{O}$) for subsets of $\mathcal{O}$ which generalizes the notion of natural density for subsets of $\mathbb{Z}$. We then show that the density of the set of coprime m-tuples of algebraic integers in $\mathbb{Z}$ is 1/$\zeta_{\mathit{K}}(\mathit{m})$, where $\zeta_{\mathit{K}}(\mathit{s})$ is the Dedekind zeta function of $\mathit{K}$. In particular, the density of this set does not depend on the choice of a $\mathbb{Z}$-basis for $\mathcal{O}$.