Derivatives of random matrix characteristic polynomials with applications to elliptic curves

Abstract

The value distribution of derivatives of characteristic polynomials of matrices from SO(N) is calculated at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. We consider subsets of matrices from SO(N) that are constrained to have $n$ eigenvalues equal to 1, and investigate the first non-zero derivative of the characteristic polynomial at that point. The connection between the values of random matrix characteristic polynomials and values of $L$-functions in families has been well-established. The motivation for this work is the expectation that through this connection with $L$-functions derived from families of elliptic curves, and using the Birch and Swinnerton-Dyer conjecture to relate values of the $L$-functions to the rank of elliptic curves, random matrix theory will be useful in probing important questions concerning these ranks.

References in corpus (8)

• 1- and 2-Level Densities for Rational Families of Elliptic Curves: Evidence for the Underlying Group Symmetries
• Autocorrelation of Random Matrix Polynomials
• Random matrix theory and discrete moments of the Riemann zeta function
• A Hybrid Euler-Hadamard product formula for the Riemann zeta function
• Secondary terms in the number of vanishings of quadratic twists of elliptic curve L-functions
• Discretisation for odd quadratic twists
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Cited by in corpus (9)

• Painleve IV and degenerate Gaussian Unitary Ensembles
• Roots of the derivative of the Riemann zeta function and of characteristic polynomials
• A Random Matrix Model for Elliptic Curve L-Functions of Finite Conductor
• Discretisation for odd quadratic twists
• A new approach to the characteristic polynomial of a random unitary matrix
• Modeling families of L-functions
• Investigations of Zeros Near the Central Point of Elliptic Curve L-Functions
• Regulators of rank one quadratic twists
• Conditional Haar measures on classical compact groups