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We study one-parameter families of elliptic curves over $\mathbb{Q}(T)$, which are of the form $y^2=x^3+A(T)x+B(T)$, with non-constant $j$-invariant. We define the $r$\textsuperscript{th} moment of an elliptic curve to be $A_{r,E}(p) := \frac1{p} \sum_{t \bmod p} a_t(p)^r$, where $a_t(p)$ is $p$ minus the number of solutions to $y^2 = x^3 + A(t)x + B(t) \bmod p$. Rosen and Silverman showed biases in the first moment equal the rank of the Mordell-Weil group of rational solutions. Michel proved that $pA_{2,E}(p)=p^2+O(p^{3/2})$. 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., Comment: Version 2.0, 44 pages, 21 figures. Expanded version of arXiv:2102.02702 (version submitted to journal)

Let $\mathcal E: y^2 = x^3 + A(T)x + B(T)$ be a nontrivial one-parameter family of elliptic curves over $\mathbb{Q}(T)$, with $A(T), B(T) \in \mathbb Z(T)$, and consider the $k$\textsuperscript{th} moments $A_{k,\mathcal{E}}(p) := \sum_{t (p)} a_{\mathcal{E}_t}(p)^k$ of the Dirichlet coefficients $a_{\mathcal{E}_t}(p) := p + 1 - |\mathcal{E}_t (\mathbb{F}_p)|$. Rosen and Silverman proved a conjecture of Nagao relating the first moment $A_{1,\mathcal{E}}(p)$ to the rank of the family over $\mathbb{Q}(T)$, and Michel proved that if $j(T)$ is not constant then the second moment is equal to $A_{2,\mathcal{E}}(p) = p^2 + O(p^{3/2})$. Cohomological arguments show that the lower order terms are of sizes $p^{3/2}, p, p^{1/2}$, and $1$. In every case we are able to 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$(2)/\mathbb{Q}$, 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 in families of $L$-functions., Comment: Version 1.0, 40 pages, 2 appendices

By Zeckendorf's theorem, an equivalent definition of the Fibonacci sequence (appropriately normalized) is that it is the unique sequence of increasing integers such that every positive number can be written uniquely as a sum of non-adjacent elements; this is called a legal decomposition. Previous work examined the distribution of the number of summands and the spacings between them, in legal decompositions arising from the Fibonacci numbers and other linear recurrence relations with non-negative integral coefficients. Many of these results were restricted to the case where the first term in the defining recurrence was positive. We study a generalization of the Fibonacci numbers with a simple notion of legality which leads to a recurrence where the first term vanishes. We again have unique legal decompositions, Gaussian behavior in the number of summands, and geometric decay in the distribution of gaps., Comment: Version 1.0, 22 pages, 2 figures