Philippe Michel: Simultaneous reductions of CM elliptic curves (NTWS 041)
Channel: Number Theory Web Seminar
Date: 04/06/2023

Abstract: Let $E$ be an elliptic curve with CM by the imaginary quadratic order $O_D$ of negative discriminant $D$. Given $p$ a prime ; if $p$ is inert or ramified in the quadratic field generated by $\sqrt D$ then $E$ has supersingular reduction at a(ny) fixed place above $p$. By a variant of Duke’s equidistribution theorem, as $D$ grows along such discriminants, the proportion of CM elliptic curves with CM by $O_D$ whose reduction at such place is a given supersingular curve converge to a natural (non-zero) limit. A further step is to fix several (distinct) primes $p_1,\cdots,p_s$ and to look for the proportion of CM curves whose reduction above each of these primes is prescribed. In this talk, we will explain how a powerful result of Einsiedler and Lindenstrauss classifying joinings of rank $2$ actions on products of locally homogeneous spaces implies that as $D$ grows along adequate subsequences of negative discriminants, this proportion converge to the product of the limits for each individual $p_i$ (a sort of asymptotic Chinese Reminder Theorem for reductions of CM elliptic curves if you wish). This is joint work with M. Aka, M. Luethi and A.Wieser. If time permits, we will also describe a further refinement -- obtained with the additional collaboration of R. Menares — of these equidistribution results for the formal groups attached to these curves.

Link to slides: https://drive.google.com/file/d/1O3x12rO3BokyOhYJjM8gxraYjwSLTYTI

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Thursday, October 8, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, October 9, 2020 (2am AEDT, 4am NZDT)