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arXiv · 1510.07068

Elliptic curves, random matrices and orbital integrals

Jeff Achter·Julia Gordon·Salim Ali Altug
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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.

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Jeff Achter, Julia Gordon, Salim Ali Altug. 2015-10-23. Elliptic curves, random matrices and orbital integrals. https://doi.org/10.2140/pjm.2017.286.1

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