Modular forms of weight 1 for GL2 over CM fields

Abstract: Recall that the global Langlands program for GL2/Q in the case of Maass forms (of Laplace eigenvalue in 1/4) is still largely open, predicting that they should be in bijection with the set of 2-dimensional even Galois representations of finite image (note that even a construction of the conjectured Galois representations is not known). One difficulty that arises in the case of Maass forms is that they do not seem to arise in (algebraic) geometry (as is the case for modular forms which appear as global sections of line bundles of modular curves).

In his talk at the Clay Research Conference in 2018, Scholze defined a geometric space, similar to a locally symmetric space of GL2 over an imaginary quadratic field (or more generally any cyclic extension of Q), whose cohomology valued in a well-chosen line bundle (giving rise to a case of ''modular forms of weight 1'') may contain (some) Maass forms (of Laplace eigenvalue 1/4) of GL2/Q (via base change). In particular, in this way Maass forms may appear in (perhaps nicely behaved) geometry. Furthermore, each ''modular form of weight 1'' arising from Scholze's construction has a Galois representation attached (respecting the standard compatibilities), so one can naturally ask whether the associated Galois representation has finite image.

In this talk, I will recall Scholze's construction and then I will discuss work with McDonald and current work in progress, both of which are statements of the form ''local-global compatibility at p for completed cohomology'' and might be of independent interest. Applying these results to Scholze's construction, we can now define a space of ''modular forms of weight 1'' (i.e. choose a correct line bundle). Furthermore, these results are part of first steps in verifying finiteness of the Galois representations associated to the ''weight 1 modular forms''.

number theory

Audience: researchers in the topic

London number theory seminar

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