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• W2963749457 abstract "Let $K$ be a quadratic number field of discriminant $Delta_K$, let $E$ be a $mathbb Q$-curve without CM completely defined over $K$ and let $omega_E$ 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 $Omega_E/sqrt{|Delta_K|}$, where $Omega_E$ is a quantity, independent of $omega_E$, 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 $Q=Q(E,omega_E)$ such that if $L(E,1)neq 0$ then $L(E,1)cdot Qcdotsqrt{|Delta_K|}/Omega_E$ 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 $L(E,1)cdotsqrt{|Delta_K|}/Omega_E$ when $L(E,1)neq 0$. An important ingredient is an algorithm to compute a newform $f$ of weight $2$ level $Gamma_1(N)$ such that $L(E,s)=L(f,s)cdot L({}^{sigma!} f,s)$, for ${}^{sigma!} f$ 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 $0$." @default.
• W2963749457 created "2019-07-30" @default.
• W2963749457 creator A5071321653 @default.
• W2963749457 creator A5074382116 @default.
• W2963749457 date "2017-06-13" @default.
• W2963749457 modified "2023-09-24" @default.
• W2963749457 title "On $L$-functions of quadratic $mathbb {Q}$-curves" @default.
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• W2963749457 doi "https://doi.org/10.1090/mcom/3217" @default.
• W2963749457 hasPublicationYear "2017" @default.
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