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S3, Archimedes, SU2, and the semicircle

Archimedes death, by Edouard Vimont (1846 - 1930)

This post is about the semicircle distribution, and the way it appears in special unitary 2×2 matrices. The link passes through the sphere S3 and the Archimedes theorem.

Semicircle. The semicircle distribution on the interval [2σ,2σ], σ>0, is dμSCσ(x)=4σ2x22πσ21x[2σ,2σ](x)dx. It has mean 0 and variance σ2. It is a symmetric Beta[2σ,2σ](32,32) distribution since 4σ2x2=(2σx)321(2σ+x)321. It is the projection on R of the uniform distribution on the disc of radius 2σ.

Archimedes and projection of S3. The famous extended Archimedes theorem on the sphere and the cylinder states that if (U1,,Un+2) is uniformly distributed on the unit sphere Sn+1={xRn+2:x21++x2n+2=1} of Rn+2, then its projection (U1,,Un) on Rn is uniformly distributed on the unit ball {xRn:x21++x2n1}. In particular, the projection on R2 of the uniform distribution on S3 is the uniform distribution on the unit disc. Also its projection on R is the semicircle distribution μSC12.

Let us retain that if a random vector is uniformly distributed on S3 then each of its 4 coordinates are distributed according to the semicircle distribution μSC12.

Note that S3 can also be understood as the quaternionic unit sphere via |x|=x21+x22+x23+x24wherex=x1+x2i+x3j+x4kHR4, just like S1 and S2 are the unit spheres of R and C respectively.

S3 is diffeomorphic to SU(2). The special unitary group SU(2) is the set of 2×2 unitary matrices with determinant equal to 1. Let us write USU(2) as U=(acbd),a,b,c,dC. The condition UU(2) gives |a|2+|b|2=1, |c|2+|d|2=1, and (c,d)(a,b), hence (c,d)=eiθ(¯b,¯a), for some phase θ[0,2π). Next, the unit determinant condition reads 1=adbc=(|a|2+|b|2)eiθ, hence θ=0. We have obtained the parametrization U=(a¯bb¯a),a,bC,|a|2+|b|2=1. In other words SU(2) is diffeomorphic to S3, which is also the quaternionic unit sphere. The uniform probability measure on S3 gives the uniform probability measure on SU(2), which is the normalized Haar measure on this group. Also, combining with what we have already seen on S3, if U is a random matrix distributed according to the normalized Haar measure on SU(2), then U11 follows the semicircle distribution μSC12.

Eigenvalues of SU(2). If USU(2), then its eigenvalues λ1 and λ2 belong to the unit circle of the complex plane, while 1=det(U)=λ1λ2, hence λ1=eiθ and λ2=eiθ, for some θ[0,π). Thus, using the fact that U22=¯U11, we get 2U11=Tr(U)=2cos(θ). It follows that if U is a random matrix distributed according to the normalized Haar measure on SU(2), then Tr(U)=2cos(θ) follows the semicircle distribution μSC1.

Conjugacy classes of SU(2). Two elements U,VSU(2) have same conjugacy class iff U=WVW for some WSU(2). This is equivalent to say that they have the same eigenvalues {e±iθ}, in other words, the same trace, as seen previously. If we parametrize the set of conjugacy classes by the trace, then it follows the semicircle distribution μSC1.

Elliptic curves and Sato-Tate conjecture. Formulated independently by Mikio Sato (1928 - 2023) and John Torrence Tate Jr (1925 - 2019) around 1960, it states that for any fixed elliptic curve over the rationals, without complex multiplication, denoting N its conductor or complexity, Np the number of its points mod p, Pn the set of prime numbers n not dividing N, and |Pn| its cardinal, we have the asymptotic equipartition 1|Pn|pPnδapweaknμSC1whereap=(p+1)Npp. The weak (or narrow) convergence is with respect to continuous and bounded test functions, namely the convergence in law of probability theory. The link with SU(2) is via the interpretation of ap as a normalized trace of a Frobenius endomorphism. This conjecture was essentially proved around 2008 by Laurent Clozel (1953 - ), Michael Harris (1954 - ), Nicholas Shepherd-Barron (1955 - ), and Richard Taylor (1962 - ).

Elliptic curves are fascinating objects from algebraic number theory, connected to modular form by the Shimura-Taniyama correspondence, after Gorō Shimura (1930 - 2019) and Yutaka Taniyama (1927 - 1958), at the heart of the proof by Andrew Wiles (1953 - ) and Richard Taylor (1962 - ) of the last theorem of Pierre de Fermat (196? - 1665). They are also used in cryptography - in particular for cryptocurrencies such as Bitcoin - an application that motivated a conjecture formulated in 1976 by Serge Lang (1927 - 2005) and Hale Trotter (1931 - 2022).

Further reading.

  • On this blog
    Archimedes theorem on sphere and cylinder
    March 18, 2024
  • Henri Carayol
    La conjecture de Sato-Tate, d'après Clozel, Harris, Shepherd-Barron, Taylor
    Séminaire Bourbaki, Exposé n°977 (2007)
  • Michael Harris
    The Sato-Tate conjecture
    Introductory course for doctoral students, available online (2007)

Final comment. Henri Carayol (1953 - ) is a French number‑theorist at Strasbourg. He should not be confused with Michel Charles Henri Carayol (1934 - 2003), the CEA engineer‑physicist who is one of the fathers of the French hydrogen bomb.

Photo of Statue of Pierre de Fermat (160? - 1665) at Beaumont-de-Lomagne, Tarn-et-Garonne, by Alexandre Falguière (1831 - 1900), funded by Théodore Despeyrous (1815 - 1883)
Statue of Pierre de Fermat (160? - 1665) at Beaumont-de-Lomagne, Tarn-et-Garonne, by Alexandre Falguière (1831 - 1900), funded by Théodore Despeyrous (1815 - 1883)
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