Sato-Tate distributions

be the trace of Frobenius. By a theorem of Hasse, each normalized trace xp := ap/ p p is a real number in the interval [−2, 2]. The xp vary with p in an apparently unpredictable way, and in the absence of any other information, one might suppose that they should be uniformly distributed over [−2, 2]. A few experiments quickly dispels this notion (here is a typical example), however, the distribution of the xp does appear to be converging to something. Remarkably, with just a few exceptions, it does not seem to matter which elliptic curve we use (here is an extreme example), the picture always looks the same asymptotically. This was observed some fifty years ago by Mikio Sato and John Tate, who independently conjectured that the semicircular distribution visible in two of the three linked examples above is the limiting distribution of the xp for every elliptic curve E/Q without complex multiplication (this means End(EQ) = Z, which is typically the case). Thanks to recent work by Richard Taylor and others [5, 13, 34], the Sato-Tate conjecture is now a celebrated theorem. The Frobenius traces ap also appear as coefficients in the L-series of the elliptic curve. One can ask similar questions about other L-functions, such as those attached to modular forms (with rational coefficients), algebraic curves, abelian varieties, Galois representations, or more generally, any motivic L-function. Almost all of these more general questions remain open, but a rich theory and a precise set of conjectures has arisen around them that suggest deep connections between the analytic and arithmetic aspects of these L-functions (this may be viewed as part of the Langlands program). The goal of this course is to introduce Sato-Tate distributions, both from an analytic perspective (as distributions of normalized Euler factors of L-functions), and an arithmetic perspective (as distributions of normalized Frobenius polynomials), and to describe the generalized SatoTate conjecture, which postulates that in each case these distributions are governed by the Haar measure of a certain compact Lie group, the Sate-Tate group (of the L-function or motive).