On $L$-functions of quadratic $\mathbb {Q}$-curves

Abstract:

Let $K$ be a quadratic number field and let $E$ be a $\mathbb {Q}$-curve without CM completely defined over $K$ and not isogenous to an elliptic curve over $\mathbb {Q}$. In this setting, it is known that there exists a weight $2$ newform of suitable level and character, such that $L(E,s)=L(f,s)L({}^\sigma \! f,s)$, where ${}^\sigma \! f$ is the unique Galois conjugate of $f$. In this paper, we first describe an algorithm to compute the level, the character and the Fourier coefficients of $f$. Next, we show that given an invariant differential $\omega _E$ on $E$, there exists a positive integer $Q=Q(E,\omega _E)$ such that $L(E,1)/P(E/K)\cdot Q$ is an integer, where $P(E/K)$ is the period of $E$. Assuming a generalization of Manin’s conjecture, the integer $Q$ is made effective. As an application, we verify the weak BSD conjecture for some curves of rank two, we compute the $L$-ratio of a curve of rank zero and we produce relevant examples of newforms of large level.