Overconvergent cohomology and overconvergent modular forms

Question

I've been reading a preprint by David Hansen (with appendix by James Newton) called Universal eigenvarieties, trianguline Galois representations and p-adic Langlands functoriality. In it he talks about using overconvergent cohomology to construct eigenvarieties.

Now I was wondering if I could get a reference/ explanation as to why these eigenvarieties will be related to the ones one make coming from overconvergent modular forms (as in Kevin Buzzard's- Eigenvarieties) I don't quite see the relation between this OC cohomology and OC modular forms. I have a feeling it has something to do (as I think Hansen says on p42) with the classical Eichler--Shimura and Theorem 3.2.5 on the above paper, which is a generalization of Stevens control theorem. But it's not clear to me what the relation is.

Thank you.

Answer 1

Unfortunately there isn't a simple direct relationship between OC mod forms and OC cohomology! The miracle is that they contain the same finite-slope Hecke data. As you've guessed, one proves this by using classical Eichler-Shimura and Coleman's and Stevens's control theorems, plus $p$-adic interpolation - this is a beautiful idea of Chenevier. Very roughly, OC forms and OC cohomology both spread out to coherent sheaves $M,M'$ over spectral varieties $Z,Z' \subset W \times \mathbb{A}^1$. Now the control thms + Eichler-Shimura tell you that $Z \cap Z'$ is Zariski-dense in both $Z$ and $Z'$, so $Z=Z'$ (up to nilpotents). Now $M,M'$ both have Hecke actions and the control theorems applied a second time tell you that there's a dense set of points $z \in Z$ such that for any element $t$ of the Hecke algebra, $\det(1-tX)|M_z \in k_z[X]$ divides $\det(1-tX)|M'_z$, which turns out to imply the same divisibility for any point $z \in Z$, and from here its not a big leap to get a relation between eigenvarieties. I hope this isn't too cryptic - section 5 of the paper is largely devoted to formalizing this kind of argument and giving examples.

It's very tempting to expect some kind of direct relationship between OC forms and OC cohomology, but whatever form this takes, it should properly be $p$-adic Hodge-theoretic in nature. This is an active area of research; see e.g. Andreatta-Iovita-Stevens' paper "Overconvergent Eichler-Shimura isomorphisms" and the abstract for Andreatta's talk at Glenn Stevens's 60th birthday conference.