Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On $L$-functions of quadratic $\mathbb{Q}$-curves   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 $L(E,1)\cdot Q\cdot\sqrt{|\Delta_K|}/\Omega_E$ 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 $L(E,1)\cdot\sqrt{|\Delta_K|}/\Omega_E$ 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 $0$.

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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: cs
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: ao-sci
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The categories of this paper are: supr-con
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The categories of this paper are: gr-qc
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The categories of this paper are: astro-ph