Moments of characteristic polynomials and their derivatives for SO(2N) and USp(2N) and their application to one-level density in families of elliptic curve L-functions

Using the ratios theorems, we calculate the leading order terms in N for the following averages of the characteristic polynomial and its derivative: ΛA(1)rΛA′(eiϕ)ΛA(eiϕ)SO(2N) and ΛA(1)rΛA′(eiϕ)ΛA(eiϕ)USp(2N). Our expression, derived for integer r, permits analytic continuation in r and we conjecture that this agrees with the above averages for non-integer exponents. We use this result to obtain an expression for the one level density of the “excised ensemble,” a subensemble of SO(2N), to next-to-leading order in N. We then present the analogous calculation for the one level density of quadratic twists of elliptic curve L-functions, taking into account a number theoretical bound on the central values of the L-functions. The method we use to calculate the above random matrix averages uses the contour integral form of the ratios theorems, which are a key tool in the growing literature on averages of characteristic polynomials and their derivatives, and as we evaluate the next-to-leading term for large matrix size N, this leads to some interesting multi-dimensional contour integrals, defined at the end of Sec. II A 3, which might be useful in other work.