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SUMMARY:At Moller Institute: The arithmetic of families of Calabi-Yau mani
 folds: black holes and modularity - Philip Candelas (University of Oxford)
 \, Xenia de la Ossa (University of Oxford)
DTSTART:20220816T170000Z
DTEND:20220816T180000Z
UID:TALK177404@talks.cam.ac.uk
DESCRIPTION:The main goal of this talk is to explore questions of common i
 nterest for physicists\, number theorists and geometers\,&nbsp\; in the co
 ntext of the arithmetic of Calabi Yau 3-folds.\nThe main quantities of int
 erest in the arithmetic context are the numbers of points of the manifold 
 considered as a variety over a finite field. We are interested in the comp
 utation of these numbers and their dependence on the moduli of the variety
 . The surprise for a physicist is that the numbers of points over a finite
  field are also given by expressions that involve the periods of a manifol
 d. The number of points are encoded in the local zeta function\, about whi
 ch much is known in virtue of the Weil conjectures. In these talks we disc
 uss a number of interesting topics related to the zeta function\, the corr
 esponding L-function\,&nbsp\; and the appearance of modularity for one par
 ameter families of Calabi-Yau manifolds.&nbsp\; We will discuss&nbsp\; on 
 an example for which the quartic numerator of the zeta function of a one p
 arameter family factorises into two quadrics at special values of the para
 meter which satisfy an algebraic equation with coefficients in Q (so indep
 endent of any particular prime)\, and for which the underlying manifold is
  smooth. We note that these factorisations are due to a splitting of the H
 odge structure and that these special values of the parameter are rank two
  black hole attractor points in the sense of type IIB supergravity. Modula
 r groups and modular forms arise in relation to these attractor points. To
  our knowledge\, the rank two attractor points that were found by the appl
 ication of these number theoretic techniques\, provide the first explicit 
 examples of such attractor points for Calabi-Yau manifolds.&nbsp\; Time pe
 rmitting\, we will describe this scenario also for the mirror manifold in 
 type IIA supergravity.
LOCATION:No Room Required
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