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Abstract
We propose a randommatrix model for families of elliptic curve Lfunctions of finite conductor. A repulsion of the critical zeros of these Lfunctions away from the centre of the critical strip was observed numerically by Miller (2006 Exp. Math. 15 257–79); such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros (for large conductors this distribution is expected to approach the onelevel density of eigenvalues of orthogonal matrices after appropriate rescaling). Our purpose here is to provide a randommatrix model for Miller's surprising discovery. We consider the family of even quadratic twists of a given elliptic curve. The main ingredient in our model is a calculation of the eigenvalue distribution of random orthogonal matrices whose characteristic polynomials are larger than some given value at the symmetry point in the spectra. We call this subensemble of SO(2N) the excised orthogonal ensemble. The sievingoff of matrices with small values of the characteristic polynomial is akin to the discretization of the central values of Lfunctions implied by the formulae of Waldspurger and Kohnen–Zagier. The cutoff scale appropriate to modelling elliptic curve Lfunctions is exponentially small relative to the matrix size N. The onelevel density of the excised ensemble can be expressed in terms of that of the wellknown Jacobi ensemble, enabling the former to be explicitly calculated. It exhibits an exponentially small (on the scale of the mean spacing) hard gap determined by the cutoff value, followed by soft repulsion on a much larger scale. Neither of these features is present in the onelevel density of SO(2N). When N → ∞ we recover the limiting orthogonal behaviour. Our results agree qualitatively with Miller's discrepancy. Choosing the cutoff appropriately gives a model in good quantitative agreement with the numbertheoretical data.
Original language  English 

Article number  115207 
Number of pages  32 
Journal  Journal of Physics A: Mathematical and Theoretical 
Volume  45 
Issue number  11 
DOIs  
Publication status  Published  23 Mar 2012 
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Projects
 2 Finished

RANDOM MATRIX THEORY AND NUMBER THEORY: DISTRIBUTION OF PRIMES AND HIGHER ORDER VANISHING OF LFUNCTIONS
1/10/05 → 1/04/09
Project: Research
