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SUMMARY:Jelena Ivančić (Max Planck Institute for Mathematics\, Universit
 y of Bonn)
DTSTART:20261028T160000Z
DTEND:20261028T170000Z
DTSTAMP:20261107T050916Z
UID:LNTS/203
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LNTS/203/">M
 odular forms of weight 1 for GL2 over CM fields</a>\nby Jelena Ivančić (
 Max Planck Institute for Mathematics\, University of Bonn) as part of Lond
 on number theory seminar\n\nLecture held in Imperial College London\, Huxl
 ey 139.\n\nAbstract\nRecall that the global Langlands program for GL2/Q in
  the case of Maass forms (of Laplace eigenvalue in 1/4) is still largely o
 pen\, predicting that they should be in bijection with the set of 2-dimens
 ional even Galois representations of finite image (note that even a constr
 uction of the conjectured Galois representations is not known). One diffic
 ulty that arises in the case of Maass forms is that they do not seem to ar
 ise in (algebraic) geometry (as is the case for modular forms which appear
  as global sections of line bundles of modular curves).\n\nIn 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 f
 ield (or more generally any cyclic extension of Q)\, whose cohomology valu
 ed 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 ap
 pear in (perhaps nicely behaved) geometry.\nFurthermore\, each ''modular f
 orm of weight 1'' arising from Scholze's construction has a Galois represe
 ntation attached (respecting the standard compatibilities)\, so one can na
 turally ask whether the associated Galois representation has finite image.
 \n\nIn this talk\, I will recall Scholze's construction and then I will di
 scuss work with McDonald and current work in progress\, both of which are 
 statements of the form ''local-global compatibility at p for completed coh
 omology'' and might be of independent interest. Applying these results to 
 Scholze's construction\, we can now define a space of ''modular forms of w
 eight 1'' (i.e. choose a correct line bundle). Furthermore\, these results
  are part of first steps in verifying finiteness of the Galois representat
 ions associated to the ''weight 1 modular forms''.\n
LOCATION:https://researchseminars.org/talk/LNTS/203/
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