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. :)