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I'm curious about how the L-functions of elliptic curves behave as the elliptic curves vary in families. In other words, if we regard the L-function $L(E_{/K},s)$ of an elliptic curve $E$ over a number field $K$ as a function on the moduli space of elliptic curves over $K$, $\mathcal{M}_{E/K}$, in the first variable, what can be said about its properties, both for fixed values of the second variable $s$, and as a two-variable function on $\mathcal{M}_{E/K} \times \mathbb{C}$?

The question has a natural extension to other moduli spaces of arithmetic varieties.

References to where it is discussed in detail would be appreciated.

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    $\begingroup$ The moduli space is usually taken with isomorphisms over $\bar K$, but the $L$-function depends on which twist you take. So a meaningful question would be to ask how the $L$-function depends on the coefficients of a Weierstrass equation. $\endgroup$ – Chris Wuthrich Mar 12 at 9:03
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    $\begingroup$ Since the $L$-function is of arithmetic nature, there will not be nice properties with respect to the analytic structure of $\mathcal{M}_{E/K}(\mathbb{C})$ e.g. it certainly won't extend to a continuous function. I think it's the same for the associated rigid $p$-adic analytic variety, because you can only control the Euler factor at $p$ of the $L$-function. But we could look at other kind of topology or structure on the moduli space. $\endgroup$ – François Brunault Mar 12 at 9:58

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