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SUMMARY:Arav Karighattam (MIT)
DTSTART:20261112T210000Z
DTEND:20261112T223000Z
DTSTAMP:20261112T133745Z
UID:STAGE/169
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/STAGE/169/">
 The elliptic gamma function I</a>\nby Arav Karighattam (MIT) as part of ST
 AGE\n\nLecture held in Room 2-132 in the MIT Simons Building.\n\nAbstract\
 nRecall that the exponential function is the classical example of a singly
  periodic entire function in one complex variable.  Last talk\, we learned
  about theta functions\, which are entire functions in one complex paramet
 er $z$ and one parameter $\\tau$ in the upper half-plane\, satisfying fasc
 inating periodicity conditions (with respect to two periods) and also modu
 lar-like conditions.  Special values of modifications of theta functions c
 an produce units in abelian extensions of imaginary quadratic fields (for 
 instance\, see Rubin\, 1999). These functions were generalized by (Ruijsen
 aars\, 1994) to what are known as elliptic gamma functions\, in one comple
 x variable $z$ and two variables $\\tau\, \\sigma$ in the upper half-plane
 \; these functions were studied further by (Felder-Varchenko\, 2000).  The
  importance of the elliptic gamma function for us in STAGE comes from the 
 conjectural relationship of special values of variants of this function to
  units in abelian extensions of complex cubic fields (Bergeron-Charollois-
 Garcia 2024)\, the subject of future talks. \n\nIn this talk\, following (
 Felder-Varchenko\, 2000)\, I will explain the theory of elliptic gamma fun
 ctions\, their periodicity conditions (with three periods\, 1\, $\\tau$\, 
 and $\\sigma$) involving a variant of the theta function\, and significant
 ly more complicated modular identities.   I will also explain how these sy
 stems of identities can be understood in the context of "automorphic forms
 " associated to elements of certain cohomology groups of $\\text{SL}_3(\\m
 athbb{Z})\\ltimes\\Z^3$.   Time permitting\, I will also discuss the conti
 nuous extension of the elliptic gamma function to a region where $\\tau$ a
 nd/or $\\sigma$ lie on a subset of the real line satisfying a certain Diop
 hantine approximation property.\n\nMain reference: G. Felder\, A. Varchenk
 o\, <a href="https://www.sciencedirect.com/science/article/pii/S0001870800
 919510"><i>The Elliptic Gamma Function and $\\text{SL}_3(\\mathbb{Z})\\lti
 mes\\Z^3$</i></a>\, Adv. Math. <b>156</b> (2000)\, pp. 44-76\n
LOCATION:https://researchseminars.org/talk/STAGE/169/
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