One- and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries

By Steven J. Miller. Compositio Mathematica, 2003

107 citations recorded by the source corpus.

Following Katz–Sarnak, Iwaniec–Luo–Sarnak and Rubinstein, we use the one- and two-level densities to study the distribution of low-lying zeros for one-parameter rational families of elliptic curves over $\mathbb{Q}(t)$. Modulo standard conjectures, for small support the densities agree with Katz and Sarnak's predictions. Further, the densities confirm that the curves' L-functions behave in a manner consistent with having r zeros at the critical point, as predicted by the Birch and Swinnerton-Dyer conjecture. By studying the two-level densities of some constant sign families, we find the first examples of families of elliptic curves where we can distinguish SO(even) from SO(odd) symmetry.

View Paper: https://www.semanticscholar.org/paper/15d59c525f718ce5d5ed9e9b2c2323be01000c22

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Citations in the corpus

On the behaviour of root numbers in families of elliptic curves (2004, unknown): “this fact is used in [5] to construct families with excess rank. The distribution of the root number is also crucial to issues of the distribution of the zeroes of L-functions of elliptic curves; see [24] for an application of the main result of the present paper. ON THE BEHAVIOUR OF ROOT NUMBERS IN”
Low-lying zeros of families of elliptic curves (2004, background): “The 1-level density has been studied for a wide variety of families of L-functions; see [R], [KS1], [ILS], [ Mil ] for example.”
Families of L -Functions and Their Symmetry (2014, methodology): “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].”
Statistics for low-lying zeros of symmetric power L-functions in the level aspect (2007, background): “Miller [28] that this is sufficient to distinguish the symmetry types).”
Local spectral equidistribution for Siegel modular forms and applications (2010, methodology): “ot try to predict whether odd or even orthogonal symmetry should occur; these could be distinguished most simply by computing the 2-level density for test functions with restricted support as done in [33] for classical modular forms (one can also attempt to study the low-lying zeros for test functio”
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, Ka Sa2, Mil1, OS2, RR, Ro, Rub1, Yo2]).”