Low-lying zeros of families of elliptic curves

By M. Young. 2004

130 citations recorded by the source corpus.

The random matrix model predicts that many statistics associated to zeros of a family of L-functions can be modeled (or predicted) by the distribution of eigenval ues of large random matrices in one of the classical linear groups. If the statistics of a family of L-functions are modeled by the eigenvalues of the group G, then we say that G is the symmetry group (or symmetry type) associated to the family. The statistic of interest to us in this work is the density of zeros near the central point (also known as the 1-level density). The random matrix model predicts that the distribution of these zeros should be modeled by the eigenvalues nearest 1 for one of the symmetry types G (unitary, symplectic, and orthogonal). All of the different groups G have distinct behavior in this regard. Therefore, computing the 1-level density gives a theoretical way to predict the symmetry type of a family. The 1-level density has been studied for a wide variety of families of L-functions; see [R], [KSI], [ILS], [Mil] for example. It is standard to assume the Generalized Riemann Hypothesis (GRH) to study the 1-level density and we do so throughout this work. In particular, it is necessary to use GRH for the application of obtaining a bound on the average analytic rank from a density theorem. In some cases the use of GRH improves the range of the density theorem, which translates to an improved bound on the average rank. Besides these crucial applications of GRH, we have freely assumed GRH even when it could be removed with extra work since it simplifies arguments in some non essential places. It is especially interesting to investigate the 1-level density for families of L functions attached to elliptic curves over the rationals since zeros at the central point have important arithmetic information (by the conjecture of Birch and Swinnerton Dyer). These investigations have been the main focus of this work.

Subfields: Algebra and Number Theory, Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d7d2ec9d6e785799971a326e4606ee40d3312ce7

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Citations in the corpus

Mean values with cubic characters (2008, background): “in Theorem 1.5. Another potential application of Theorem 1.5 is in the study of the low-lying zeros of the family of all elliptic curves (which has applications to the average rank of the family). In [Y] the second named author related the problem to a certain complicated character sum which contain”
The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1 (2013, methodology): “(Conditional on GRH and BSD, a finite upper bound of 2.3 on the average rank was demonstrated by Brumer [13]; improved conditional upper bounds of 2.0 and 1.79 were given by Heath-Brown [32] and more recently by Young [42], respectively.)”
Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves (2010, unknown): “79 by Young [41].”
Families of L -Functions and Their Symmetry (2014, background): “The upshot is that Conjecture 2 is verified for families of quadratic twists in [86], for F in [5, 108] and under the above assumptions for 1-parameter families in [79].”
Average Ranks of Elliptic Curves: Tension between Data and Conjecture (2007, background): “The best known results (conditional on a Generalized Riemann Hypothesis and a Parity Principle analogue) on nonvanishing of even parity L-functions appear in the work of Young [YouB], and results about average (analytic) ranks and their relation to random matrix theory appear in [ YouA ].”
A Symplectic Test of the L-Functions Ratios Conjecture (2007, background): “weight k level N cuspidal newforms, symmetric powers of GL(2) L-functions, and certain families of GL(4) and GL(6) L-functions (see [DM1, FI, Gu, HR, HM, ILS, KaSa2, Mil1, OS2, RR, Ro, R ub1 , Yo2]).”