Editing Sato–Tate conjecture (section)

==Proof==
In 2008, Clozel, Harris, Shepherd-Barron, and Taylor published a proof of the Sato–Tate conjecture for elliptic curves over [[totally real field]]s satisfying a certain condition: of having multiplicative reduction at some prime,<ref>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.</ref> in a series of three joint papers.<ref>{{Cite journal
| last=Taylor
| first=Richard
| title=Automorphy for some ''l''-adic lifts of automorphic mod ''l'' Galois representations. II
| journal=Publ. Math. Inst. Hautes Études Sci.
| volume=108
| year=2008
| pages=183&ndash;239
| doi=10.1007/s10240-008-0015-2
| mr=2470688
| citeseerx=10.1.1.116.9791
}}</ref><ref>{{Cite journal
| last1=Clozel
| first1=Laurent
| last2=Harris
| first2=Michael
| last3=Taylor
| first3=Richard
| title=Automorphy for some ''l''-adic lifts of automorphic mod ''l'' Galois representations
| journal=Publ. Math. Inst. Hautes Études Sci.
| volume=108
| year=2008
| pages=1&ndash;181
| doi=10.1007/s10240-008-0016-1
| mr=2470687
| citeseerx=10.1.1.143.9755
}}</ref><ref>{{Citation
| last1=Harris
| first1=Michael
| last2=Shepherd-Barron
| first2=Nicholas
| last3=Taylor
| first3=Richard
| title=A family of Calabi&ndash;Yau varieties and potential automorphy
| journal=[[Annals of Mathematics]]
| year=2010
| volume=171
| issue=2
| pages=779&ndash;813
| doi=10.4007/annals.2010.171.779
| mr=2630056
| doi-access=free
}}</ref>

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 [[isogeny|isogenous]]) following from such a hypothetical trace formula.<ref>See Carayol's Bourbaki seminar of 17 June 2007 for details.</ref> 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,<ref>{{Cite journal
| last1=Barnet-Lamb
| first1=Thomas
| last2=Geraghty
| first2=David
| last3=Harris
| first3=Michael
| last4=Taylor
| first4=Richard
| title=A family of Calabi&ndash;Yau varieties and potential automorphy. II
| year=2011
| pages=29&ndash;98
| journal=Publ. Res. Inst. Math. Sci.
| volume=47
| issue=1
| mr=2827723
| doi=10.2977/PRIMS/31
| doi-access=free
}}</ref> by improving the potential modularity results of previous papers.<ref>Theorem B of {{harvnb|Barnet-Lamb|Geraghty|Harris|Taylor|2011}}</ref> The prior issues involved with the trace formula were solved by [[Michael Harris (mathematician)|Michael Harris]],<ref>{{cite book |last=Harris |first=M. |chapter=An introduction to the stable trace formula |editor-first=L. |editor-last=Clozel |editor2-first=M. |editor2-last=Harris |editor3-first=J.-P. |editor3-last=Labesse |editor4-first=B. C. |editor4-last=Ngô |title=The stable trace formula, Shimura varieties, and arithmetic applications |volume=I: Stabilization of the trace formula |location=Boston |publisher=International Press |year=2011 |pages=3–47 |isbn=978-1-57146-227-5 }}</ref> and [[Sug Woo Shin]].<ref>{{cite journal |first=Sug Woo |last=Shin |title=Galois representations arising from some compact Shimura varieties |journal=[[Annals of Mathematics]] |volume=173 |issue=3 |pages=1645–1741 |year=2011 |doi=10.4007/annals.2011.173.3.9 |doi-access=free }}</ref><ref>See p. 71 and Corollary 8.9 of {{harvnb|Barnet-Lamb|Geraghty|Harris|Taylor|2011}}</ref>

In 2015, Richard Taylor was awarded the [[Breakthrough Prize in Mathematics]] "for numerous breakthrough results in (...) the Sato–Tate conjecture."<ref>{{cite web |title=Richard Taylor, Institute for Advanced Study: 2015 Breakthrough Prize in Mathematics |url=https://breakthroughprize.org/?controller=Page&action=laureates&p=3&laureate_id=59 }}</ref>