Lower-Order Biases in the Second Moment of Dirichlet Coefficients in Families of L-Functions

Let epsilon : y(2) = x(3) + A(T)x + B(T) be a nontrivial one-parameter family of elliptic curves over Q(T), with A(T), B(T) is an element of Z(T). Consider the kth moments A(k,epsilon)(p) := Sigma(t(p)) a epsilon(t)(p)(k) of the Dirichlet coefficients a epsilon(t) (p) := p + 1 - vertical bar epsilon(t)(F-p)vertical bar. Rosen and Silverman proved Nagao's conjecture relating the first moment to the family's rank over Q(T), and Michel proved if j(T) is not constant then the second moment equals p(2) + O(p(3/2)). Cohomological arguments show the lower order terms are of sizes p(3/2), p, p(1/2) and 1. In every case, we can analyze in closed form, the largest lower order term in the second moment expansion that does not average to zero is on average negative, though numerics suggest this may fail for families of moderate rank. We prove this Bias Conjecture for several large classes of families, including families with rank, complex multiplication, and constant j(T)-invariant. We also study the analogous Bias Conjecture for families of Dirichlet characters, holomorphic forms on GL(2)/Q, and their symmetric powers and Rankin-Selberg convolutions. We identify all lower order terms in large classes of families, shedding light on the arithmetic objects controlling these terms. The negative bias in these lower order terms has implications toward the excess rank conjecture and the behavior of zeros near the central point.