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

摘要: Let $K$ be a quadratic number field of discriminant $\Delta_K$, let $E$ $\mathbb Q$-curve without CM completely defined over and $\omega_E$ an invariant differential on $E$. $L(E,s)$ the $L$-function In this setting, it is known that possesses analytic continuation to C$. The period can written (up power $2$) as product Tamagawa numbers with $\Omega_E/\sqrt{|\Delta_K|}$, where $\Omega_E$ quantity, independent $\omega_E$, which encodes real periods when covolume lattice imaginary. paper we compute, under generalized Manin conjecture, effective nonzero integer $Q=Q(E,\omega_E)$ such if $L(E,1)\neq 0$ then $L(E,1)\cdot Q\cdot\sqrt{|\Delta_K|}/\Omega_E$ integer. Computing $L(E,1)$ up sufficiently high precision, our result allows us prove $L(E,1)=0$ whenever case compute $L$-ratio $L(E,1)\cdot\sqrt{|\Delta_K|}/\Omega_E$ 0$. An important ingredient algorithm newform $f$ weight $2$ level $\Gamma_1(N)$ $L(E,s)=L(f,s)\cdot L({}^{\sigma\!} f,s)$, for ${}^{\sigma\!} f$ unique Galois conjugate $f$. As application these results, verify validity weak BSD conjecture some Q$-curves rank will curve $0$.