Derivatives of random matrix characteristic polynomials with applications to elliptic curves

Snaith, Nina C

doi:10.1088/0305-4470/38/48/007

Cited by 13 publications

(16 citation statements)

References 38 publications

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“…The philosophy of Keating and Snaith suggests that the moments of L (E d , 1) should behave as the moments of the derivative of characteristic polynomials of SO(2N +1) evaluated at 1 [18]. More precisely, this implies that we should have…”

Section: Rank 1 Casementioning

confidence: 99%

Moments of the Orders of Tate–shafarevich Groups

Delaunay

2005

Int. J. Number Theory

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We give some conjectures for the moments of the orders of the Tate-Shafarevich groups of elliptic curves belonging to a family of quadratic twists. These conjectures follow from the predictions on L-functions given by the random matrix theory [12,5] and from the Birch and Swinnerton-Dyer conjecture. Furthermore, including the Cohen-Lenstra type heuristics for Tate-Shafarevich groups, we obtain some conjectural estimates for the regulator of rank 1 elliptic curves in a family of quadratic twists.

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Section: Rank 1 Casementioning

confidence: 99%

Moments of the Orders of Tate–shafarevich Groups

Delaunay

2005

Int. J. Number Theory

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“…An interesting extension of this is to find a random matrix model for elliptic curve L-functions of a given order of vanishing at the critical point. The first steps in this direction have been taken by Snaith [Sna05] and Miller/Dueñez [Mil06], but it is clear from Miller's numerical computations that there is a still simpler problem concerning the zero statistics of families of rank zero curves that is far from being understood. This problem is the main motivation for the work we shall report on here.…”

Section: Introductionmentioning

confidence: 99%

Lower order terms for the one-level density of elliptic curve L-functions

Huynh

Keating

Snaith

2009

Journal of Number Theory

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It is believed that, in the limit as the conductor tends to infinity, correlations between the zeros of elliptic curve L-functions averaged within families follow the distribution laws of the eigenvalues of random matrices drawn from the orthogonal group. For test functions with restricted support, this is known to be the true for the one-and two-level densities of zeros within the families studied to date. However, for finite conductor Miller's experimental data reveal an interesting discrepancy from these limiting results. Here we use the L-functions ratios conjectures to calculate the 1-level density for the family of even quadratic twists of an elliptic curve L-function for large but finite conductor. This gives a formula for the leading and lower order terms up to an error term that is conjectured to be significantly smaller. The lower order terms explain many of the features of the zero statistics for relatively small conductor and model the very slow convergence to the infinite conductor limit. However, our main observation is that they do not capture the behaviour of zeros in the important region very close to the critical point and so do not explain Miller's discrepancy. This therefore implies that a more accurate model for statistics near to this point needs to be developed.

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“…We note that an orthogonal circular random matrix ensemble with fixed degenerate eigenvalue at 1 was considered by Snaith in [26] in conjectural relation to number theoretic questions on Lfunctions of elliptic curves. More general Jacobi circular ensembles were studied recently in [11].…”

Section: Introductionmentioning

confidence: 99%

Painlevé IV and degenerate Gaussian unitary ensembles

Chen

Feigin

2006

J. Phys. A: Math. Gen.

123

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We consider those Gaussian Unitary Ensembles where the eigenvalues have prescribed multiplicities, and obtain joint probability density for the eigenvalues. In the simplest case where there is only one multiple eigenvalue t , this leads to orthogonal polynomials with the Hermite weight perturbed by a factor that has a multiple zero at t. We show through a pair of ladder operators, that the diagonal recurrence coefficients satisfy a particular Painlevé IV equation for any real multiplicity. If the multiplicity is even they are expressed in terms of the generalized Hermite polynomials, with t as the independent variable. † ychen@ic.ac.uk

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Derivatives of random matrix characteristic polynomials with applications to elliptic curves

Cited by 13 publications

References 38 publications

Moments of the Orders of Tate–shafarevich Groups

Moments of the Orders of Tate–shafarevich Groups

Lower order terms for the one-level density of elliptic curve L-functions

Painlevé IV and degenerate Gaussian unitary ensembles

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