What question did this study set out to answer?

This research aims to establish the conditions under which families of elliptic curves exhibit equidistribution of their Frobenius traces in the context of the motivic rigidity conjecture.

February 20, 2026Open Access

Density Thresholds for Equidistribution in Prime-Indexed Geometric Families: A Function Field Model for the Motivic Rigidity Conjecture

Key Points

This research aims to establish the conditions under which families of elliptic curves exhibit equidistribution of their Frobenius traces in the context of the motivic rigidity conjecture.
Utilized a function field model over F_q(t) to analyze elliptic curves parametrized by irreducible polynomials.
Applied the Deligne equidistribution theorem to show unconditional equidistribution under specific conditions.
Examined the relationship between the density of irreducible polynomials and monodromy group spectral gaps.
Identified a density threshold for irreducible polynomials, which must exceed the spectral gap for equidistribution.
Confirmed that this threshold is satisfied for all degrees d ≥ 1 in function field scenarios.
Demonstrated that this density remains problematic for rational primes due to a deficit under effective equidistribution thresholds.

Abstract

This paper provides a function field model for the motivic rigidity conjecture formulated in 1. Over Fq (t), we study families of elliptic curves Ef: y² = x (x+f) (x+f+g) parametrized by monic irreducible polynomials f of degree d, and establish unconditional equidistribution of their Frobenius traces via the Deligne equidistribution theorem when SL (2) monodromy holds. The key result is a density threshold: equidistribution requires the density of irreducible polynomials (approximately 1/d by the Prime Polynomial Theorem) to exceed the spectral gap of the monodromy group, which is satisfied for all d ≥ 1 in the function field setting. Over Z, the analogous density 1/log x for rational primes falls below effective equidistribution thresholds, identifying the density deficit as a fundamental obstruction to transferring function field results to number fields.

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