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A random matrix model for elliptic curve \(L\)-functions of finite conductor. (English) Zbl 1292.11101

Summary: We propose a random-matrix model for families of elliptic curve \(L\)-functions of finite conductor. A repulsion of the critical zeros of these \(L\)-functions away from the centre of the critical strip was observed numerically by S. J. Miller [Exp. Math. 15, No. 3, 257–279 (2006; Zbl 1131.11042)]; such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros (for large conductors this distribution is expected to approach the one-level density of eigenvalues of orthogonal matrices after appropriate rescaling). Our purpose here is to provide a random-matrix 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 sub-ensemble of \(\mathrm{SO}(2N)\) the excised orthogonal ensemble. The sieving-off of matrices with small values of the characteristic polynomial is akin to the discretization of the central values of \(L\)-functions implied by the formulae of Waldspurger and Kohnen-Zagier. The cut-off scale appropriate to modelling elliptic curve \(L\)-functions is exponentially small relative to the matrix size \(N\). The one-level density of the excised ensemble can be expressed in terms of that of the well-known 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 cut-off value, followed by soft repulsion on a much larger scale. Neither of these features is present in the one-level density of \(\mathrm{SO}(2N)\). When \(N \rightarrow \infty \) we recover the limiting orthogonal behaviour. Our results agree qualitatively with Miller’s discrepancy. Choosing the cut-off appropriately gives a model in good quantitative agreement with the number-theoretical data.

MSC:

11M50 Relations with random matrices
11M41 Other Dirichlet series and zeta functions
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture

Citations:

Zbl 1131.11042