Second moments and the bias conjecture for the family of cubic pencils

2020/12/21 by Matija Kazalicki, Kazalicki, Matija, Bartosz Naskręcki +1 · 2 citations
Materials Science · Mathematics · #Algebraic Geometry (math.AG) #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.2012.11306

openalex publication_date 2020/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a 1-parametric family E_k of elliptic curves over ℚ and a prime p, consider the second moment sum M_2,p(E_k)=∑_k ∈ \mathbbF_p a_k,p², where a_k,p=p+1-#E_k(\mathbbF_p). Inspired by Rosen and Silverman's proof of Nagao conjecture which relates the first moment of a rational elliptic surface to the rank of Mordell-Weil group of the corresponding elliptic curve, S. J. Miller initiated the study of the asymptotic expansion of M_2,p(E_k)=p²+O(p^3/2) (which by the work of Deligne and Michel has cohomological interpretation). He conjectured that, similar to the first moment case, the largest lower-order term that does not average to 0 has a negative bias. In this paper, we provide an explicit formula for the second moment M_2,p(E_U) of E_U:y²=P(x)U+Q(x), where \textrmdeg P(x), \textrmdeg Q(x)≤ 3. For a generic choice of polynomials P(x) and Q(x) this formula is expressed in terms of the point count of a certain genus two curve. As an application, we prove that the Bias conjecture holds for the pencil of the cubics E_U.

Cited by

• Biases in Moments of the Dirichlet Coefficients in One- and Two-Parameter Families of Elliptic Curves
• Negative bias in moments of the Legendre family of elliptic curves

Related

• Lower Order Biases in Moment Expansions of One Parameter Families of Elliptic Curves
• Biases in Moments of Dirichlet Coefficients of Elliptic Curve Families
• Biases in Moments of the Dirichlet Coefficients in One- and Two-Parameter Families of Elliptic Curves
• Applications of Moments of Dirichlet Coefficients in Elliptic Curve Families
• The bias conjecture for elliptic curves over finite fields and Hurwitz class numbers in arithmetic progressions