Elliptic curves, random matrices and orbital integrals

Abstract

An isogeny class of elliptic curves over a finite field is determined by a quadratic Weil polynomial. Gekeler has given a product formula, in terms of congruence considerations involving that polynomial, for the size of such an isogeny class. In this paper, we give a new, transparent proof of this formula; it turns out that this product actually computes an adelic orbital integral which visibly counts the desired cardinality. This answers a question posed by N. Katz.

Appendix by Salim Ali Altug. V3: Clarified Section 3.3

References in corpus (3)

• Beyond Endoscopy via the Trace Formula-I: Poisson Summation and Contributions of Special Representations
• Sums of Euler products and statistics of elliptic curves
• The Discriminant of an Algebraic Torus

Cited by in corpus (6)

• Computing square-free polarized abelian varieties over finite fields
• Polarizations of abelian varieties over finite fields via canonical liftings
• Counting abelian varieties over finite fields via Frobenius densities
• Real structures on ordinary Abelian varieties
• Local heuristics and an exact formula for abelian varieties of odd prime dimension over finite fields
• On the $L$-polynomials of curves over finite fields