Biases in Moments of the Dirichlet Coefficients in One-Parameter Families of Elliptic Curves - Miller, Steven J.; Weng, Yan (The PUMP Journal of Undergraduate Research, 2021)
Elliptic curves arise in many important areas of modern number theory. One way to study them is to take local data, the number of solutions modulo a prime p, and create an L-function. The behavior of this global object is related to two of the seven Clay Millenial Problems: the Birch and Swinnerton-Dyer Conjecture and the Generalized Riemann Hypothesis. We study one-parameter families over Q(T). We look at the r-th moment of the series expansion of the L-function (the p-th coefficient is related to the number of solutions to the elliptic curve modulo p). Rosen and Silverman showed biases in the first moment equal the rank of the Mordell-Weil group of rational solutions. Michel proved the main term of the second moment is universal, with the lower order terms smaller by at least a factor of the square-root of the prime. 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-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-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 ranks. 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.