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SUMMARY:A class number formula for Picard modular surfaces - Shrenik Shah 
 (Columbia University)
DTSTART:20170530T143000Z
DTEND:20170530T153000Z
UID:TALK72869@talks.cam.ac.uk
CONTACT:G. Rosso
DESCRIPTION:The original class number formula of Dirichlet connected the r
 esidue of the Dedekind zeta function of a number field at s=1 to various a
 rithmetic invariants of the number field\, including a transcendental quan
 tity called the regulator\, which measures the covolume of a lattice gener
 ated by logarithms of units.  Beilinson formulated a conjectural generaliz
 ation of this formula\, again connecting a regulator\, defined in terms of
  "higher" units\, to the special value of a "motivic" L-function.  We stud
 y a particular case of this conjecture\, namely for the motive of the midd
 le degree cohomology of the smooth compactified Picard modular surfaces X 
 attached to the unitary group GU(2\,1) for an imaginary quadratic extensio
 n E/Q.  We will concretely describe a construction of suitable elements in
  the motivic cohomology H_M^3(X\,Q(2)).  We then explain how to compute th
 eir regulator as an element of Deligne cohomology by interpreting this map
  via the pairing of these classes against automorphic differential forms\,
  and show that the regulator is non-vanishing when predicted. As a consequ
 ence\, we obtain a higher "class number formula" involving a non-critical 
 L-value of the degree 6 Standard L-function\, a Whittaker period\, and the
  regulator. One interesting aspect of this work is that we must account fo
 r endoscopic forms via a period\, which is predicted by the conjecture. Th
 is is joint work with Aaron Pollack.
LOCATION:MR13
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