Centro de Ciencias de Benasque Pedro Pascual

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Sato-Tate in higher dimension

2011, Jul 25 -- Aug 06

Xavier Guitart (UPC, Barcelona)
Joan-C. Lario (UPC, Barcelona)
Marc Masdeu (Columbia University, New York)

Emil Artin conjectured in his 1924 thesis that the number of points of an elliptic curve over the field of p elements is close to p+1, in the sense that the difference between these two quantities (i.e. the error term) is bounded by twice the square root of p. This fact was proved by Hasse in 1933 and it motivated a great deal of fundamental research (e.g. it is an analogous of the Riemann Hypothesis and it inspired Weil's conjectures), as well as finding many applications in fields such as cryptography or coding theory.

For an elliptic curve over Q there is an error term for each prime number p, obtained by considering the reduced curve mod p. A next step towards controlling the error terms was taken over forty years ago by Sato and Tate. They independently formulated a conjecture which predicts the probability distribution of the error terms (scaled by the maximum allowed by Hasse's bound) as p ranges through the primes. The Sato-Tate conjecture can also be stated for elliptic curves over arbitrary number fields.

Following a strategy designed by Jean-Pierre Serre, in 2006 Richard Taylor joint with Laurent Clozel, Michael Harris and Nicholas Shepherd-Barron announced the final step of a proof of the Sato–Tate conjecture for elliptic curves over totally real fields satisfying a certain condition: of having multiplicative reduction at some prime. Later on, Richard Taylor joint with Thomas Barnet-Lamb, David Geraghty, and Michael Harris (2008) posted an article which proves a generalized version of the Sato–Tate conjecture for an arbitrary non-CM holomorphic modular form of weight greater than or equal to two.

There are generalisations, involving the distribution of Frobenius elements occuring in Galois representations on étale cohomology. In particular there is a conjectural Sato-Tate theory for abelian varieties of higher dimension g>1. Under the random matrix model developed by Nick Katz and Peter Sarnak, there is a conjectural correspondence between (unitarized) characteristic polynomials of Frobenius elements and conjugacy classes in certain compact Lie subgroups of USp(2g). The Haar measure on these subgroups gives the conjectured distributions.

The workshop is intended to provide a fruitful atmosphere for discussions and joint research work on Sato-Tate distributions: generalizations to higher dimension, variants, applications, and related topics including computational aspects.

Further Information.

This session has received financial support from the following institutions:

  • logo CSIC
  • Ministerio de educación y ciencia
  • Ayuntamiento de Benasque
  • Gobierno de Aragón
  • Universidad de Zaragoza


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