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This summer, I was very fortunate and honored to attend the conference "Iwasawa 2023" at the University of Cambridge as a young Ph.D. student on Iwasawa theory. There, one of the speakers, might be Professor David Loeffler, mentioned that on the automorphic side, "integration against Eisenstein series can be regarded as a cup product in the coherent cohomology".

I am very interested in such (might be geometric?) interpretation of the integrations against Eisenstein series, so I wonder where I can find the precise meaning of the above "slogan"? More specifically, I would like to know the references explaining

  • What is the "coherent cohomology", what does it mean and how can it be used?
  • How to regard the integration against Eisenstein series as a cup product in the coherent cohomology?

Of course, it would be of great help if one could explain these in the answer of this post. I am so sorry if this post is not appropriate for this site, and thank you so much for paying attention, commenting, or answering. :)

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    $\begingroup$ Loeffler's reference is arxiv.org/abs/2303.16114 , section 5.1 $\endgroup$
    – Carlo Beenakker
    Commented Sep 19, 2023 at 13:06
  • $\begingroup$ @CarloBeenakker Thank you so much! That helps a lot! $\endgroup$
    – Hetong Xu
    Commented Sep 20, 2023 at 0:28
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    $\begingroup$ For what it's worth, the particular paper I was referring to was a different one, arxiv.org/abs/2307.07004. But the one from Carlo's link does indeed use the same methods. $\endgroup$
    – David Loeffler
    Commented Sep 21, 2023 at 6:44

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Yes, that does indeed sound like something I might have said :)

I was referring to some extremely powerful theorems, originally due to Michael Harris, which show that:

  • The cohomology groups of automorphic vector bundles on toroidal compactifications of Shimura varieties are isomorphic to certain spaces of (usually non-holomorphic) automorphic forms. (These vector bundles are coherent sheaves, hence "coherent cohomology".)
  • Via this isomorphism, the Serre duality pairing on coherent cohomology groups is given by integration of automorphic forms.

The general results are all set out in Harris' paper "Automorphic forms of $\overline{\partial}$-cohomology type as coherent cohomology classes" (J. Diff. Geom. 1990); the relation between Serre duality and integrals is Proposition 3.8.

To get some feel for how this is applied in practice I recommend reading some other papers of Harris from around the same time, e.g. this one with Kudla: Arithmetic automorphic forms for the nonholomorphic discrete series of GSp(2).

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    $\begingroup$ Thanks for your answer! And for your wonderful inspiring talk at Cambridge! :) $\endgroup$
    – Hetong Xu
    Commented Sep 22, 2023 at 0:53

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