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SUMMARY:Multiplicity one for the mod p cohomology of Shimura curves
DTSTART;VALUE=DATE-TIME:20170614T083000Z
DTEND;VALUE=DATE-TIME:20170614T093000Z
DTSTAMP;VALUE=DATE-TIME:20210417T110755Z
UID:indico-event-2330@indico.math.cnrs.fr
DESCRIPTION:At present\, the mod $p$ (and $p$-adic) local Langlands corres
pondence is only well understood for the group $\\mathrm{GL}_2(\\mathbb{Q}
_p)$. One of the main difficulties is that little is known about supersing
ular representations besides this case\, and we do know that there is no s
imple one-to-one correspondence between representations of $\\mathrm{GL}_2
(K)$ with two-dimensional representations of $\\mathrm{Gal}(\\overline{K}/
K)$\, at least when $K/\\mathbb{\\mathbb{Q}}_p$ is (non-trivial) finite un
ramified.\n\nÂ \n\nHowever\, the Buzzard-Diamond-Jarvis conjecture and the
mod $p$ local-global compatibility for $\\mathrm{GL}_2/\\mathbb{Q}$ sugge
st that this hypothetical correspondence may be realized in the cohomology
of Shimura curves with characteristic $p$ coefficients (cut out by some m
odular residual global representation $\\bar{r}$). Moreover\, the work of
Gee\, Breuil and Emerton-Gee-Savitt show that\, to get information about t
he $\\mathrm{GL}_2(K)$-action on the cohomology\, one could instead study
the geometry of certain Galois deformation rings of the $p$-component of $
\\bar{r}$.\n\nIn a work in progress with Haoran Wang\, we push forward the
ir analysis of the structure of potentially Barsotti-Tate deformation ring
s and\, as an application\, we prove a multiplicity one result of the coho
mology at full congruence level when $\\bar{r}$ is reducible generic \\emp
h{non-split} at $p$. (The semi-simple case was previously proved by Le-Mor
ra-Schraen and by ourselves.)\n\nhttps://indico.math.cnrs.fr/event/2330/
LOCATION:IHES Centre de confĂ©rences Marilyn et James Simons
URL:https://indico.math.cnrs.fr/event/2330/
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