Weighted Distributions of Complex Multiplication Orders in Ordinary Isogeny Classes

TL;DR

This paper develops a global arithmetic framework to analyze the distribution of endomorphism rings within ordinary elliptic isogeny classes over finite fields, connecting class field theory, isogeny graph geometry, and Deuring's correspondence.

Contribution

It introduces explicit formulas for global distributions of endomorphism rings and relates CM order existence to splitting conditions in ring class fields.

Findings

Explicit formulas for weighted class number distributions.

Distribution laws for endomorphism rings across isogeny classes.

Connection between CM orders and splitting in ring class fields.

Abstract

We develop a global arithmetic framework for studying endomorphism rings inside ordinary elliptic isogeny classes over finite fields. Let p be a prime and let I(t,p) be an ordinary isogeny class over the finite field F_p with Frobenius trace t. The discriminant Delta = t^2 - 4p can be written as Delta = v^2 D_K, where D_K is the fundamental discriminant of an imaginary quadratic field K. In this setting, the possible endomorphism rings are precisely the quadratic orders O_f = Z + f O_K, with f dividing v. Building on Deuring's correspondence, we express the distribution of these orders in terms of weighted class numbers h*(D) = h(D)/w(D), and obtain explicit formulas for global distributions across the entire isogeny class. This approach goes beyond the classical local viewpoint, where the endomorphism ring is constant along each level of an ell-isogeny volcano. In particular, we introduce weighted exact and cumulative distributions of endomorphism rings. These distributions induce canonical laws for the ell-adic valuation of conductors and recover the vertical stratification of ell-volcanoes in an averaged sense. On the global side, by varying the prime p, we relate the existence of curves with a prescribed CM order O_D to splitting conditions in the associated ring class field L_D. Using the Chebotarev density theorem, we obtain the natural density 1/(2h(D)) for primes admitting CM by O_D. This gives a horizontal distribution law complementary to the vertical conductor distribution. These results establish a unified perspective linking Deuring theory, isogeny graph geometry, and class field theory. They also provide a natural framework for quantitative and algorithmic studies of ordinary isogeny classes.