Sato–Tate conjecture
Field | Arithmetic geometry |
---|---|
Conjectured by | Mikio Sato John Tate |
Conjectured in | c. 1960 |
First proof by | Laurent Clozel Thomas Barnet-Lamb David Geraghty Michael Harris Nicholas Shepherd-Barron Richard Taylor |
First proof in | 2011 |
In
independently posed the conjecture around 1960.If Np denotes the number of points on the elliptic curve Ep defined over the finite field with p elements, the conjecture gives an answer to the distribution of the second-order term for Np. By Hasse's theorem on elliptic curves,
as , and the point of the conjecture is to predict how the
The original conjecture and its generalization to all
Statement
Let E be an elliptic curve defined over the rational numbers without complex multiplication. For a prime number p, define θp as the solution to the equation
Then, for every two real numbers and for which
Details
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
This is due to Mikio Sato and John Tate (independently, and around 1960, published somewhat later).[3]
Proof
In 2008, Clozel, Harris, Shepherd-Barron, and Taylor published a proof of the Sato–Tate conjecture for elliptic curves over
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.[8] In 2011, Barnet-Lamb, Geraghty, Harris, and Taylor proved a generalized version of the Sato–Tate conjecture for an arbitrary non-CM holomorphic modular form of weight greater than or equal to two,[9] by improving the potential modularity results of previous papers.[10] The prior issues involved with the trace formula were solved by Michael Harris,[11] and Sug Woo Shin.[12][13]
In 2015, Richard Taylor was awarded the Breakthrough Prize in Mathematics "for numerous breakthrough results in (...) the Sato–Tate conjecture."[14]
Generalisations
There are generalisations, involving the distribution of
Under the random matrix model developed by
Refinements
There are also more refined statements. The Lang–Trotter conjecture (1976) of Serge Lang and Hale Trotter states the asymptotic number of primes p with a given value of ap,[16] 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.
See also
References
- 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).
- 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-invariantis not integral.
- MR 2470688.
- MR 2470687.
- MR 2630056
- ^ See Carayol's Bourbaki seminar of 17 June 2007 for details.
- MR 2827723.
- ^ Theorem B of Barnet-Lamb et al. 2011
- ISBN 978-1-57146-227-5.
- .
- ^ See p. 71 and Corollary 8.9 of Barnet-Lamb et al. 2011
- ^ "Richard Taylor, Institute for Advanced Study: 2015 Breakthrough Prize in Mathematics".
- ISBN 978-0-8218-1017-0
- ISBN 978-0-387-07550-1
- 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.
- ^ David, Chantal; Pappalardi, Francesco (1999-01-01). "Average Frobenius distributions of elliptic curves". International Mathematics Research Notices. 199 (4): 165–183.
External links
- 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)