2021/03/05 by Steven J. Miller, Yan Weng, Miller, Steven J. +1
Mathematics · #11B05 (primary) 65Q30 (secondary) #11B39 #60B10 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2103.03942
openalex publication_date 2021/03/05 · openalex created_date 2021/03/15 · openalex updated_date 2026/07/28
We study one-parameter families of elliptic curves over ℚ(T), which are of the form y2=x3+A(T)x+B(T), with non-constant j-invariant. We define the r\textsuperscriptth moment of an elliptic curve to be Ar,E(p) := \frac1p ∑t \bmod p at(p)r, where at(p) is p minus the number of solutions to y2 = x3 + A(t)x + B(t) \bmod p. Rosen and Silverman showed biases in the first moment equal the rank of the Mordell-Weil group of rational solutions. Michel proved that pA2,E(p)=p2+O(p3/2). Based on several special families where computations can be done in closed form, Miller in his thesis conjectured that the largest lower-order term in the second moment that does not average to 0 is on average negative. He further showed that such a negative bias has implications in the distribution of zeros of the elliptic curve L-function near the central point. To date, evidence for this conjecture is limited to special families. In this paper, we explore the first and second moments of some one- and two-parameter families of elliptic curves, looking to see if the biases persist and exploring the consequence these have on fundamental properties of elliptic curves. We observe that in all of the one- and two-parameter families where we can compute in closed form that the first term that does not average to zero in the second-moment expansion of the Dirichlet coefficients has a negative average. In addition to studying some additional families where the calculations can be done in closed form, we also systematically investigate families of various rank. These are the first general tests of the conjecture; while we cannot in general obtain closed form solutions, we discuss computations which support or contradict the conjecture. We then generalize to higher moments, and see evidence that the bias continues in the even moments.