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

Abstract

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 $A(T),B(T)\in \mathbb{Z}(T)$. Consider the k^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 Nagao’s conjecture relating the first moment to the family’s rank over $\mathbb{Q}(T)$, 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$(2)/\mathbb{Q}$, 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.

Keywords:

• Dirichlet characters
• elliptic curves
• cuspidal newforms
• L-functions
• lower order terms
• excess rank

2010 Mathematics Subject Classification:

• Primary 60B10
• 11B39
• 11B05
• Secondary 65Q30

Carnegie Mellon University, Princeton University, Williams College, the Eureka Program, the Finnerty Fund, and the Clare Boothe Luce Program of the Henry Luce Foundation. We thank to the referee for numerous helpful comments, and Matija Kazalicki and Bartosz Naskrecki for sharing their preprints.

Notes

1 Note the 1-level density is well-defined even if GRH fails, though if there are zeros off the line then we lose the spectral interpretation of the zeros. If we adjust the rescaling of the zeros slightly we can remove the big-Oh error term, but its presence does not matter for calculating the main term, and is only important when we look at the 2 or higher level densities.

2 The Satake parameters $\left|{\alpha }_{\pi ,i}\right|$ are bounded by $p^{\delta }$ for some δ; conjecturally δ = 0. There has been significant progress toward these bounds with some $\delta <1/2$; see [17, 18]. Any $\delta <1/6$ implies the $\nu \ge 3$ terms do not contribute to the main term.

3 We need to divide the p sum to have a moment, as we are averaging over p terms; some works include this division in the definition, others have it separate. We choose not to divide by p so that our sums are integer polynomials in p.

4 An elliptic surface $y^2=x^3+A(T)x+B(T)$ is rational if and only if one of the following is true: (1) $0<\max (\mathrm{deg}A,2\mathrm{deg}B)<12$; (2) $3\mathrm{deg}A=2\mathrm{deg}B=12$ and ${\text{ord}}_{T=0}{T}^{12}\Delta (T^{-1})=0$.

Additional information

Funding

This work was supported by NSF (grants nos. DMS 1347804, DMS1265673 and DMS1561945).