Suppose that $E$ is an elliptic curve over $\mathbf{Q}$. Let $K$ be a number field and let $L(E/K, s)$ be the Hasse-Weil $L$-function of $E$ base-changed to $K$. The modularity theorem tells us that $L(E/\mathbf{Q},s)$ has an analytic continuation to the entire complex plane. My question is: does the base-changed $L$-function $L(E/K, s)$ also have an analytic continuation to the entire complex plane?
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asked Feb 26, 2023 at 17:24
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8$\begingroup$ Only in special cases. For instance, abelian extensions are ok as one can twist the modular form by any Dirichlet character and still get a modular form whose $L$-function has analytic continuation. $\endgroup$– Chris WuthrichCommented Feb 26, 2023 at 17:28
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1$\begingroup$ I was told that if $K$ is contained in a solvable extension then by base change theorem in the theory of automorphic forms, the base changed $L$-function has meromorphic continuation. $\endgroup$– François BrunaultCommented Feb 27, 2023 at 7:45
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1$\begingroup$ Just to be clear, this is expected to be true for any $K$, but it is only known in special cases. I'll leave it to an actual expert to tell us the current state of knowledge about which ones those are. $\endgroup$– David LoefflerCommented Feb 27, 2023 at 7:48
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$\begingroup$ Does the standard argument using Brauer induction allow one to prove anything about the non-abelian case? $\endgroup$– Daniel LoughranCommented Feb 27, 2023 at 10:39
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2$\begingroup$ Brauer induction + potential modularity theorems give you meromorphy, not holomorphy, of the L-function, which is why meromorphic continuation is known in much greater generality (any totally real or CM field, and $E$ doesn't have to be a base-change from $\mathbb{Q}$ but can be any elliptic curve over $K$). $\endgroup$– David LoefflerCommented Feb 28, 2023 at 15:10
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