The moments of the coefficients of elliptic curve L-functions are related to numerous arithmetic problems. Rosen and Silverman proved a conjecture of Nagao relating the first moment of one-parameter families satisfying Tate's conjecture to the rank of the corresponding elliptic surface over Q(T); one can also construct families of moderate rank by finding families with large first moments. Michel proved that if j(T) is not constant, then the second moment of the family is of size p^2 + O(p^(3/2)); these two moments show that for suitably small support the behavior of zeros near the central point agree with that of eigenvalues from random matrix ensembles, with the higher moments impacting the rate of convergence. In his thesis, Miller noticed a negative bias in the second moment of every one-parameter family of elliptic curves over the rationals whose second moment had a calculable closed-form expression, specifically the first lower order term which does not average to zero is on average negative. This Bias Conjecture is confirmed for many families; however, these are highly non-generic families whose resulting Legendre sums can be determined. Inspired by the recent successes by Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, Alexey Pozdnyakov and others in investigations of murmurations of elliptic curve coefficients with machine learning techniques, we pose a similar problem for trying to understand the Bias Conjecture. As a start to this program, we numerically investigate the Bias Conjecture for a family whose bias is positive for half the primes. Since the numerics do not offer conclusive evidence that negative bias for the other half is enough to overwhelm the positive bias, the Bias Conjecture cannot be verified for the family.

翻译：椭圆曲线L函数系数的矩与众多算术问题相关联。Rosen和Silverman证明了Nagao的一个猜想，该猜想将满足Tate猜想的单参数族的一阶矩与对应椭圆曲面在Q(T)上的秩联系起来；通过寻找具有较大一阶矩的族，人们也可以构造中等秩的椭圆曲线族。Michel证明了若j(T)非常值，则该族的二阶矩具有p^2 + O(p^(3/2))的量级；这两个矩表明，在适当小的支撑集下，中心点附近零点的行为与随机矩阵系综的特征值行为一致，其中更高阶矩影响着收敛速度。Miller在其博士论文中发现，每个二阶矩具有可计算闭式表达式的有理数域上单参数椭圆曲线族的二阶矩都存在负偏差，具体而言，第一个不平均为零的低阶项在平均意义下为负。这一偏差猜想已在许多族中得到证实；然而，这些族都是高度非一般的，其对应的Legendre和可以确定。受到杨辉、李圭焕、Thomas Oliver、Alexey Pozdnyakov等人最近运用机器学习技术研究椭圆曲线系数"鸟群现象"取得成功的启发，我们提出了一个类似的问题，试图理解偏差猜想。作为该研究计划的开端，我们对一个在半数素数处呈现正偏差的族进行了偏差猜想的数值研究。由于数值结果未能提供决定性证据证明另一半素数的负偏差足以压倒正偏差，因此该族的偏差猜想无法得到验证。