Lower-Order Biases in Elliptic Curve Fourier Coefficients in Families

Abstract

and consider the k th moments A k,E (p) := t mod 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 1,E (p) to the rank of the family over Q(T), and Michel proved that the second moment is equal to A 2,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 unusual distributions of functional equation signs. We also 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 of elliptic curve L-functions.

Second moments and the bias conjecture for the family of cubic pencils

“…Essentially, the proof of (2.1) in [MMRW16] translates to the proof of a total point count on the threefold M defined in (1.4).…”

“…The terms f i (p) where i < 4 are called the lower order terms and i is the degree of that term. In [ACF + 18] and [MMRW16] the authors studied the second moments M 2,p (F k ) of the families of elliptic curves F k . Most of their examples are pencils of cubics (1.3)…”

“…The formulation of the Bias conjecture proposed in the previous work (cf. [ACF + 18], [HKL + 20], [MMRW16]) was not precise enough since it did not uniquely define the stratification of M 2,p used in a definition of bias -it only specified the exponents of sequences in the stratification. That is not enough to have a well-defined bias.…”

Lower-Order Biases Second Moments of Dirichlet Coefficients in Families of $L$-Functions

“…In this and the next section we amass more evidence for the Bias conjecture by demonstrating negative bias in additional one-parameter families of elliptic curves. See [MMRW,Mi1,Mi3] for earlier calculations on the subject. The families studied are amenable to direct calculation; thus these are not generic families but ones chosen so that the resulting Legendre sums are tractable.…”

Determining Optimal Test Functions for Bounding the Average Rank in Families of 𝐿-Functions

“…The prime example is that of whether or not their is excess rank in families of elliptic curves (see [BMSW] for a nice summary of data and conjectures); while earlier investigations indicated that such bias might persist, later studies [W] went far enough to see the average rank drop, and new random matrix models have been introduced that have the correct limiting behavior and successfully model the observed behavior for small conductors [DHKMS1,DHKMS2]. There are now many results on lower order terms in families, such as [HKS,MMRW,Mil2,Yo1], and the hope is that the methods of this paper can be extended to include these to refine estimates for finite conductors.…”