The Bias Conjecture for One-Parameter, Non-Isotrivial Elliptic Curve Families
Abstract
We investigate Miller's bias conjecture for general one-parameter families of elliptic curves over ℚ with non-constant j-invariants. Utilizing methods developed by Michel, we view the second moment as the sum of distinct cohomological components: a main p2 term, a p3/2 term arising from the first cohomology over the projective line, and terms of order p or lower corresponding to singular fibers. By assuming the generalized Sato-Tate conjecture, we employ representation and motive theory to demonstrate that the coefficient of the p3/2 term averages to zero in the limit. We then independently analyze the lower-order contributions without assuming Sato-Tate. By applying the Chebotarev density theorem and analyzing Galois representations, we show that the contributions to the order p term from additive singular fibers also average to zero. Consequently, the only remaining average contribution to the p term is −Bpp, where Bp denotes the number of ``bad'' points where the discriminant vanishes. Because Bp averages to a strictly positive value, the overall coefficient of the p term averages to a strictly negative value in the limit, thereby establishing the bias conjecture for these families.
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BibTeX
@misc{chen2026bias,
title = {{The Bias Conjecture for One-Parameter, Non-Isotrivial Elliptic Curve Families}},
author = {Lucas Chen and Joshua Im and Steven J. Miller and Devayani Pradhan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.38664},
eprint = {2609.38664},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38664},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Bias Conjecture for One-Parameter, Non-Isotrivial Elliptic Curve Families
AU - Lucas Chen
AU - Joshua Im
AU - Steven J. Miller
AU - Devayani Pradhan
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2609.38664
UR - https://arxiv.org/abs/2609.38664
AB - We investigate Miller's bias conjecture for general one-parameter families of elliptic curves over ℚ with non-constant j-invariants. Utilizing methods developed by Michel, we view the second moment as the sum of distinct cohomological components: a main p^2 term, a p^3/2 term arising from the first cohomology over the projective line, and terms of order p or lower corresponding to singular fibers. By assuming the generalized Sato-Tate conjecture, we employ representation and motive theory to demonstrate that the coefficient of the p^3/2 term averages to zero in the limit. We then independently analyze the lower-order contributions without assuming Sato-Tate. By applying the Chebotarev density theorem and analyzing Galois representations, we show that the contributions to the order p term from additive singular fibers also average to zero. Consequently, the only remaining average contribution to the p term is −B_pp, where B_p denotes the number of ``bad'' points where the discriminant vanishes. Because B_p averages to a strictly positive value, the overall coefficient of the p term averages to a strictly negative value in the limit, thereby establishing the bias conjecture for these families.
ER -