In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep over the finite field with p elements, with p a prime number, obtained from an elliptic curve E over the rational number field, by the process of reduction modulo a prime for almost all p. If Np denotes the number of points on Ep and defined over the field with p elements, the conjecture gives an answer to the distribution of the second-order term for Np. That is, by Hasse's theorem on elliptic curves we have
as p → ∞, and the point of the conjecture is to predict how the O-term varies.
Let E be an elliptic curve defined over the rationals numbers without complex multiplication. Define θp as the solution to the equation
Then, for every two real numbers and for which ,
By Hasse's theorem on elliptic curves, the ratio
is between -1 and 1. Thus it can be expressed as cos θ for an angle θ; in geometric terms there are two eigenvalues accounting for the remainder and with the denominator as given they are complex conjugate and of absolute value 1. The Sato–Tate conjecture, when E doesn't have complex multiplication, states that the probability measure of θ is proportional to
Proofs and claims in progress
In 2008, Richard Taylor, joint with Laurent Clozel, Michael Harris, and Nicholas Shepherd-Barron, published 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, in a series of three papers.
Further results are conditional on improved forms of the Arthur–Selberg trace formula. Harris has a conditional proof of a result for the product of two elliptic curves (not isogenous) following from such a hypothetical trace formula. In 2011, Richard Taylor published joint work with Thomas Barnet-Lamb, David Geraghty, and Michael Harris, which claims to prove a generalized version of the Sato–Tate conjecture for an arbitrary non-CM holomorphic modular form of weight greater than or equal to two, by improving the potential modularity results of previous papers. They also assert that the prior issues involved with the trace formula have been solved by Michael Harris, and Sug Woo Shin.
There are generalisations, involving the distribution of Frobenius elements in Galois groups involved in the Galois representations on étale cohomology. In particular there is a conjectural theory for curves of genus n > 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 the compact Lie group USp(2n) = Sp(n). The Haar measure on USp(2n) then gives the conjectured distribution, and the classical case is USp(2) = SU(2).
More precise questions
There are also more refined statements. The Lang–Trotter conjecture (1976) of Serge Lang and Hale Trotter predicts the asymptotic number of primes p with a given value of ap, the trace of Frobenius that appears in the formula. For the typical case (no complex multiplication, trace ≠ 0) their formula states that the number of p up to X is asymptotically
with a specified constant c. Neal Koblitz (1988) provided detailed conjectures for the case of a prime number q of points on Ep, motivated by elliptic curve cryptography. In 1999, Chantal David and Francesco Pappalardi proved that the Lang–Trotter conjecture holds in most cases.
- In the case of an elliptic curve with complex multiplication, the Hasse–Weil L-function is expressed in terms of a Hecke L-function (a result of Max Deuring). The known analytic results on these answer even more precise questions.
- To normalise, put 2/π in front.
- It is mentioned in J. Tate, Algebraic cycles and poles of zeta functions in the volume (O. F. G. Schilling, editor), Arithmetical Algebraic Geometry, pages 93–110 (1965).
- That is, for some p where E has bad reduction (and at least for elliptic curves over the rational numbers there are some such p), the type in the singular fibre of the Néron model is multiplicative, rather than additive. In practice this is the typical case, so the condition can be thought of as mild. In more classical terms, the result applies where the j-invariant is not integral.
- Taylor, Richard (2008). "Automorphy for some l-adic lifts of automorphic mod l Galois representations. II". Publ. Math. Inst. Hautes Études Sci. 108: 183&ndash, 239. doi:10.1007/s10240-008-0015-2. MR 2470688.
- Clozel, Laurent; Harris, Michael; Taylor, Richard (2008). "Automorphy for some l-adic lifts of automorphic mod l Galois representations". Publ. Math. Inst. Hautes Études Sci. 108: 1&ndash, 181. doi:10.1007/s10240-008-0016-1. MR 2470687.
- Harris, Michael; Shepherd-Barron, Nicholas; Taylor, Richard (2010), "A family of Calabi–Yau varieties and potential automorphy", Annals of Mathematics, 171 (2): 779&ndash, 813, doi:10.4007/annals.2010.171.779, MR 2630056
- See Carayol's Bourbaki seminar of 17 June 2007 for details.
- Barnet-Lamb, Thomas; Geraghty, David; Harris, Michael; Taylor, Richard (2011). "A family of Calabi–Yau varieties and potential automorphy. II". Publ. Res. Inst. Math. Sci. 47 (1): 29&ndash, 98. doi:10.2977/PRIMS/31. MR 2827723.
- Theorem B of Barnet-Lamb et al. 2009
- Harris, M. (2011). "An introduction to the stable trace formula". In Clozel, L.; Harris, M.; Labesse, J.-P.; Ngô, B. C. The stable trace formula, Shimura varieties, and arithmetic applications. Volume I: Stabilization of the trace formula. Boston: International Press. pp. 3–47. ISBN 978-1-57146-227-5.
- Shin, Sug Woo (2011). "Galois representations arising from some compact Shimura varieties". Annals of Mathematics. 173 (3): 1645–1741. doi:10.4007/annals.2011.173.3.9.
- See p. 71 and Corollary 8.9 of Barnet-Lamb et al. 2009
- "Richard Taylor, Institute for Advanced Study: 2015 Breakthrough Prize in Mathematics".
- Katz, Nicholas M. & Sarnak, Peter (1999), Random matrices, Frobenius Eigenvalues, and Monodromy, Providence, RI: American Mathematical Society, ISBN 0-8218-1017-0
- Lang, Serge; Trotter, Hale F. (1976), Frobenius Distributions in GL2 extensions, Berlin: Springer-Verlag, ISBN 0-387-07550-X
- Koblitz, Neal (1988), "Primality of the number of points on an elliptic curve over a finite field", Pacific Journal of Mathematics, 131 (1): 157–165, doi:10.2140/pjm.1988.131.157, MR 0917870.
- "Concordia Mathematician Recognized for Research Excellence". Canadian Mathematical Society. 2013-04-15. Archived from the original on 2017-02-01. Retrieved 2018-01-15.
- Report on Barry Mazur giving context
- Michael Harris notes, with statement (PDF)
- La Conjecture de Sato–Tate [d'après Clozel, Harris, Shepherd-Barron, Taylor], Bourbaki seminar June 2007 by Henri Carayol (PDF)
- Video introducing Elliptic curves and its relation to Sato-Tate conjecture, Imperial College London, 2014 (Last 15 minutes)