Kimball Martin, Thomas Pharis
We conjecture that, for a fixed prime $p$, rational elliptic curves with higher rank tend to have more points mod $p$. We show that there is an analogous bias for modular forms with respect to root numbers, and conjecture that the order of the rank bias for elliptic curves is greater than that of the root number bias for modular forms.
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We establish a conditional equivalence between quantitative unboundedness of the analytic rank of elliptic curves over $\mathbb Q$ and the existence of highly biased elliptic curve prime number races. We show that conditionally on a Riemann Hypothesis and on a hypothesis on the multiplicity of the zeros of $L(E,s)$, large analytic ranks translate into an extreme Chebyshev bias. Conversely, we show under a certain linear independence hypothesis on zeros of $L(E,s)$ that if highly biased elliptic curve prime number races do exist, then the Riemann Hypothesis holds for infinitely many elliptic cu…
Negative bias in moments of the Legendre family of elliptic curves
该研究针对S. J. Miller的负偏差猜想，确定勒让德椭圆曲线族高阶矩的偏差，证明低阶项平均为零或负，计算了第三、第四矩，所得赫维茨类数相关结果或具其他应用价值。
The bias conjecture for elliptic curves over finite fields and Hurwitz class numbers in arithmetic progressions
本文研究了Frobenius迹位于算术级数的有限域椭圆曲线的二阶矩偏差猜想，证明偏差对正密度级数为正、对正密度级数为负，与经典猜想相反，并给出了相关矩量的显式公式。
A New Aspect of Chebyshev's Bias for Elliptic Curves over Function Fields
本文研究了函数域上非常数椭圆曲线的素数竞赛，证明了其 Chebyshev 偏差方向取决于椭圆曲线的秩是否大于零，主要创新在于利用函数域上的 Deep Riemann Hypothesis 证明了中心处部分 Euler 乘积的收敛性。
On the average congruence class bias for cyclicity and divisibility of the groups of $\mathbb{F}_p$-points of elliptic curves
本文改进了Banks和Shparlinski关于椭圆曲线模素数约化点群循环性与可除性的平均密度结果，将素数限制在算术级数中，并研究不同同余类间的统计偏差。
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…
Divisibility Biases in the Orders of Elliptic Curve Reductions
本文利用Chebotarev密度定理，分析了椭圆曲线约化阶数被m整除的素数密度，证明对于Serre曲线该密度约为1/φ(m)且大于1/m，并验证了Banks-Shparlinski平均密度的正确性。
Elliptic curves arise in many important areas of modern number theory. One way to study them is take local data, the number of solutions modulo $p$, and create an $L$-function. The behavior of this global object is related to two of the seven Clay Millennial Problems: the Birch and Swinnerton-Dyer Conjecture and the Generalized Riemann Hypothesis. We study one-parameter families 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-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)…
Opposing Average Congruence Class Biases in the Cyclicity and Koblitz Conjectures for Elliptic Curves
本文改编Zywina方法提出算术级数中Koblitz猜想，细化Jones定理，发现循环性与Koblitz常数在同余类上平均偏差相反，并提供Magma计算代码。
By considering a one-parameter family of elliptic curves defined over $\mathbb{Q}$, we might ask ourselves if there is any bias in the distribution (or parity) of the root numbers at each specialization. From the work of Helfgott, we know (at least conjecturally) that the average root number of an elliptic curve defined over $\mathbb{Q}(T)$ is zero as soon as there is a place of multiplicative reduction over $\mathbb{Q}(T)$ other than $-$deg. Recently, Helfgott's work was extended by Desjardins, where she relaxes some of Helfgott's hypotheses and is able to provide unconditional results on the…
Applications of Moments of Dirichlet Coefficients in Elliptic Curve Families
本文研究椭圆曲线L函数系数二阶矩的负偏差猜想，受机器学习启发对半数素数呈正偏差的族进行数值检验，发现现有证据不足以验证该猜想。
The Bias Conjecture for One-Parameter, Non-Isotrivial Elliptic Curve Families
本文研究Miller偏差猜想，针对具有非常数j不变量的单参数椭圆曲线族，通过分解二阶矩的上同调分量并利用Sato-Tate猜想及Chebotarev密度定理，证明p项系数平均为负，从而确立偏差猜想。
In this article, we propose a new probabilistic model for the distribution of ranks of elliptic curves in families of fixed Selmer rank, and compare the predictions with previous results, and with the databases of curves over the rationals that we have at our disposal. In addition, we document a phenomenon we refer to as Selmer bias that seems to play an important role in the data and in our models.
Lower Order Biases in Moment Expansions of One Parameter Families of Elliptic Curves
本文创建数据库与框架系统研究椭圆曲线单参数族二阶矩的低阶偏差，在 $p \leq 250,000$ 范围内发现多处潜在违背 Miller 猜想的情况，并提出数据驱动的新猜想。
On the Identification of Elliptic Curves That Admit Infinitely Many Twists Satisfying the Birch-Swinnerton-Dyer Conjecture
本文基于Burungale-Skinner-Tian-Wan的研究，提出一个显式算法，应用于LMFDB数据库，识别出所有导数不超过500,000的椭圆曲线，这些曲线有无限多个二次扭曲满足强Birch-Swinnerton-Dyer猜想，并提供了Radziwiłł和Soundararajan猜想的数值证据。
The Katz-Sarnak philosophy predicts that statistics of zeros of families of L-functions are strikingly universal. However, subtle arithmetical differences between families of the same symmetry type can be detected by calculating lower-order terms of the statistics of interest. In this paper we calculate lower-order terms of the 1-level density of some families of elliptic curves. We show that there are essentially two different effects on the distribution of low-lying zeros. First, low-lying zeros are more numerous in families of elliptic curves E with relatively large numbers of points (mod p…
We study the prime number race for elliptic curves over the function field of a proper, smooth and geometrically connected curve over a finite field. This constitutes a function field analogue of prior work by Mazur, Sarnak and the second author. In this geometric setting we can prove unconditional results whose counterparts in the number field case are conditional on a Riemann Hypothesis and a linear independence hypothesis on the zeros of the implied L-functions. Notably we show that in certain natural families of elliptic curves, the bias generically dissipates as the conductor grows. This …
Let $\mathcal{X} : y^2 = f(x)$ be a hyperelliptic curve over $\mathbb{Q}(T)$ of genus $g\geq 1$. Assume that the jacobian of $\mathcal{X}$ over $\mathbb{Q}(T)$ has no subvariety defined over $\mathbb{Q}$. Denote by $\mathcal{X}_t$ the specialization of $\mathcal{X}$ to an integer $T=t$, let $a_{\mathcal{X}_t}(p)$ be its trace of Frobenius, and $A_{\mathcal{X},r}(p) = \frac{1}{p}\sum_{t=1}^p a_{\mathcal{X}_t}(p)^r$ its $r$-th moment. The first moment is related to the rank of the jacobian $J_\mathcal{X}\left(\mathbb{Q}(T)\right)$ by a generalization of a conjecture of Nagao: $$\lim_{X \to \inft…
These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at Zhejiang University in July, 2008. Their goal is to introduce and motivate basic concepts and constructions (such as orbifolds and stacks) important in the study of moduli spaces of curves and abelian varieties through the example of elliptic curves. The reason for working with elliptic curves is that most constructions are elementary and explicit in this case. All four approaches to moduli spaces of curves -- complex analytic, topological, algebro-geometric, and number theoretic -- are con…
We present a new algorithm for computing the endomorphism ring of an ordinary abelian surface over a finite field which is subexponential and generalizes an algorithm of Bisson and Sutherland for elliptic curves. The correctness of this algorithm only requires the heuristic assumptions required by the algorithm of Biasse and Fieker which computes the class group of an order in a number field in subexponential time. Thus we avoid the multiple heuristic assumptions on isogeny graphs and polarized class groups which were previously required. The output of the algorithm is an ideal in the maximal …
Second moments and the bias conjecture for the family of cubic pencils
本文研究1参数椭圆曲线族的二阶矩和，给出三次束族二阶矩的显式公式（通过亏格2曲线点数计数表示），并证明偏差猜想对该三次束族成立。
The Katz-Sarnak density conjecture states that, in the limit as the conductors tend to infinity, the behavior of normalized zeros near the central point of families of L-functions agree with the N -> oo scaling limits of eigenvalues near 1 of subgroups of U(N). Evidence for this has been found for many families by studying the n-level densities; for suitably restricted test functions the main terms agree with random matrix theory. In particular, all one-parameter families of elliptic curves with rank r over Q(T) and the same distribution of signs of functional equations have the same limiting …
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