Abstract
The 'excised ensemble', a random matrix model for the zeros of quadratic twist families of elliptic curve L-functions, was introduced by Dueñez et al (2012 J. Phys. A: Math. Theor.
45 115207) The excised model is motivated by a formula for central values of these L-functions in a paper by Kohnen and Zagier (1981 Invent. Math.
64 175–98). This formula indicates that for a finite set of L-functions
from a family of quadratic twists, the central values are all either
zero or are greater than some positive cutoff. The excised model imposes
this same condition on the central values of characteristic polynomials
of matrices from . Strangely, the cutoff on the characteristic polynomials that results in a convincing model for the L-function
zeros is significantly smaller than that which we would obtain by
naively transferring Kohnen and Zagier's cutoff to the
ensemble. In this current paper we investigate a modification to the
excised model. It lacks the simplicity of the original excised ensemble,
but it serves to explain the reason for the unexpectedly low cutoff in
the original excised model. Additionally, the distribution of central L-values
is 'choppier' than the distribution of characteristic polynomials, in
the sense that it is a superposition of a series of peaks: the
characteristic polynomial distribution is a smooth approximation to
this. The excised model did not attempt to incorporate these successive
peaks, only the initial cutoff. Here we experiment with including some
of the structure of the L-value distribution. The conclusion is
that a critical feature of a good model is to associate the correct mass
to the first peak of the L-value distribution.
Original language | English |
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Article number | 075202 |
Number of pages | 25 |
Journal | Journal of Physics A: Mathematical and Theoretical |
Volume | 49 |
Issue number | 7 |
Early online date | 8 Jan 2016 |
DOIs | |
Publication status | Published - 19 Feb 2016 |
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Professor Nina C Snaith
- School of Mathematics - Professor of Mathematical Physics
- Applied Mathematics
- Mathematical Physics
- Pure Mathematics
Person: Academic , Member