JournalsjemsVol. 23, No. 9pp. 2949–3016

Equidistribution on Kuga–Sato varieties of torsion points on CM elliptic curves

  • Ilya Khayutin

    Northwestern University, Evanston, USA
Equidistribution on Kuga–Sato varieties of torsion points on CM elliptic curves cover
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Abstract

A connected Kuga–Sato variety Wr\mathbf{W}^r parametrizes tuples of rr points on elliptic curves (with level structure). A special point of Wr\mathbf{W}^r is a tuple of torsion points on a CM elliptic curve. A sequence of special points is strict if any CM elliptic curve appears at most finitely many times and no relation between the points in the tuple is satisfied infinitely often. The genus orbit of a special point is the Gal(Qˉ/Qab)\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}^{\mathrm{ab}})-orbit. We show that genus orbits of special points in a strict sequence equidistribute in Wr(C)\mathbf{W}^r(\mathbb{C}), assuming a congruence condition at two fixed primes. A genus orbit can be very sparse in the full Galois orbit. In particular, the number of torsion points on each elliptic curve in a genus orbit is not bounded below by the torsion order. A genus orbit corresponds to a toral packet in an extension of SL2\mathbf{SL}_2 by a vector representation. These packets also arise in the study by Aka, Einsiedler and Shapira of grids orthogonal to lattice points on the 2-sphere. As an application we establish their joint equidistribution assuming two split primes.

Cite this article

Ilya Khayutin, Equidistribution on Kuga–Sato varieties of torsion points on CM elliptic curves. J. Eur. Math. Soc. 23 (2021), no. 9, pp. 2949–3016

DOI 10.4171/JEMS/1067