LOWER-ORDER BIASES IN SECOND MOMENTS OF FOURIER COEFFICIENTS IN FAMILIES OF L-FUNCTIONS MEGUMI ASADA, RYAN CHEN, EVA FOURAKIS, YUJIN KIM, ANDREW KWON, JARED DUKER LICHTMAN, BLAKE MACKALL, STEVEN J. MILLER, ERIC WINSOR, KARL WINSOR, JIANING YANG, AND KEVIN YANG ABSTRACT. Let E : 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 ) ∈ Z(T ), and consider the k th moments A k,E (p) := ∑ t(p) a Et (p) k of the Fourier coefficients a Et (p) := p +1 − |E t (F p )|. Rosen and Silverman proved a conjecture of Nagao relating the first moment A k,E (p) to the rank of the family over Q(T ), and Michel proved that if j (T ) is not constant then the second moment is equal to A k,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, the largest lower order term in the second moment expansion that does not average to zero is on average negative. 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 in families of L-functions. CONTENTS 1. Introduction 2 1.1. Bias in Families of L-Functions 2 1.2. Bias in Elliptic Curve Families 4 1.3. Outline 5 2. Linear one-parameter families of elliptic curves 7 3. Elliptic curve families of constant j (T )−invariant 11 3.1. Counting Points Preliminaries 11 3.2. Moments of E r (T ): y 2 = x 3 − T r Ax 12 3.3. Moments of E r : y 2 = x 3 − T r B 14 3.4. Computing the k th moment for a family of any constant j (T )−invariant 15 4. GL(1) Families (Dirichlet Characters) 16 4.1. Preliminaries: Primes in Arithmetic Progression 16 4.2. Characters of Prime Level 16 Date: February 28, 2020. 2010 Mathematics Subject Classification. 60B10, 11B39, 11B05 (primary) 65Q30 (secondary). Key words and phrases. Dirichlet characters, elliptic curves, cuspidal newforms, L-functions, lower order terms, excess rank. This work was supported by NSF grants DMS 1347804, DMS1265673 and DMS1561945, Carnegie Mellon Uni- versity and Williams College, the Eureka Program, the Finnerty Fund, and the Clare Boothe Luce Program of the Henry Luce Foundation. We thank the referee for numerous helpful comments. 1