2012/03/31 by Francesc Fité, Andrew Sutherland, Andrew V. Sutherland
Computer Science · Mathematics · #Abelian group #Algebraic Geometry and Number Theory #Algebraic number field #Coding theory and cryptography #Conjecture #Cryptography and Residue Arithmetic #Distribution (mathematics) #Elliptic curve #Euler characteristic #Group (periodic table) #Order (exchange) #Surface (topology) #math.AG #math.NT #msc:11G10 #msc:11G20 #msc:11M50 #msc:14G10 #msc:14K15
paper · pdf · doi:10.2140/ant.2014.8.543
published as Algebra and Number Theory 8 (2014), 543-585 · minor edits, 42 pages
arxiv created 2012/11/20 · openalex publication_date 2014/05/31 · arxiv updated 2014/06/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We determine the limiting distribution of the normalized Euler factors of an abelian surface [math] defined over a number field [math] when [math] is [math] -isogenous to the square of an elliptic curve defined over [math] with complex multiplication. As an application, we prove the Sato–Tate conjecture for Jacobians of [math] -twists of the curves [math] and [math] , which give rise to 18 of the 34 possibilities for the Sato–Tate group of an abelian surface defined over [math] . With twists of these two curves, one encounters, in fact, all of the [math] possibilities for the Sato–Tate group of an abelian surface that is [math] -isogenous to the square of an elliptic curve with complex multiplication. Key to these results is the twisting Sato–Tate group of a curve, which we introduce in order to study the effect of twisting on the Sato–Tate group of its Jacobian.