Abstract
The 'excised ensemble', a random matrix model for the zeros of quadratic twist families of elliptic curve Lfunctions, 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 Lfunctions in a paper by Kohnen and Zagier (1981 Invent. Math.
64 175–98). This formula indicates that for a finite set of Lfunctions
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 Lfunction
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 Lvalues
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 Lvalue distribution. The conclusion is
that a critical feature of a good model is to associate the correct mass
to the first peak of the Lvalue distribution.
Original language  English 

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