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

Abstract

Let $K$ be a quadratic number field of discriminant $Δ_K$, let $E$ be a $\mathbb Q$-curve without CM completely defined over $K$ and let $ω_E$ be an invariant differential on $E$. Let $L(E,s)$ be the $L$-function of $E$. In this setting, it is known that $L(E,s)$ possesses an analytic continuation to $\mathbb C$. The period of $E$ can be written (up to a power of $2$) as the product of the Tamagawa numbers of $E$ with $Ω_E/\sqrt{|Δ_K|}$, where $Ω_E$ is a quantity, independent of $ω_E$, which encodes the real periods of $E$ when $K$ is real and the covolume of the period lattice of $E$ when $K$ is imaginary. In this paper we compute, under the generalized Manin conjecture, an effective nonzero integer $Q=Q(E,ω_E)$ such that if $L(E,1)\neq 0$ then $L(E,1)\cdot Q\cdot\sqrt{|Δ_K|}/Ω_E$ is an integer. Computing $L(E,1)$ up to sufficiently high precision, our result allows us to prove that $L(E,1)=0$ whenever this is the case and to compute the $L$-ratio $L(E,1)\cdot\sqrt{|Δ_K|}/Ω_E$ when $L(E,1)\neq 0$. An important ingredient is an algorithm to compute a newform $f$ of weight $2$ level $Γ_1(N)$ such that $L(E,s)=L(f,s)\cdot L({}^{σ\!} f,s)$, for ${}^{σ\!} f$ the unique Galois conjugate of $f$. As an application of these results, we verify the validity of the weak BSD conjecture for some $\mathbb Q$-curves of rank $2$ and we will compute the $L$-ratio of a curve of rank $0$.