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On an elliptic curve over $\mathbb{Q}$, we can associate a canonical Neron model and with it a Neron differential, whose coefficients in some natural coordinates yield the Dirichlet coefficients of the L-function (or the coefficients of the associated modular form, as you prefer). Is there a similar story for K3 surfaces? I envision a canonical integral model and a section of the canonical bundle such that it is related to the automorphy and thus the L-function of the weight-2 part of cohomology in a simple way.

Maybe there's even a similar story for more general Calabi-Yau varieties? I am more skeptical of this.

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    $\begingroup$ There is certainly no Neron model of a K3 surface in the sense of an integral model that satisfies the Neron lifting property. One aspect of this is the existence of nonconstant morphisms from $\mathbb{P}^1$ to the generic fiber; the automorphism group of $\mathbb{P}^1$ is too large to allow the Neron lifting property. $\endgroup$
    – Jason Starr
    Commented May 26, 2018 at 9:43
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    $\begingroup$ There are relations between singular K3 surfaces and modular forms of weight 3, see work of Schütt. If I recall correctly non-singular K3 are related to Sym^2 of weight 2 modular forms. There are partial results in higher dimension, see the survey "Modularity of Calabi-Yau varieties" by Yui. See also the work of Livné $\endgroup$
    – François Brunault
    Commented May 26, 2018 at 11:19

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