Determining optimal test functions for 2-level densities

Katz and Sarnak conjectured a correspondence between the n -level density statistics of zeros from families of L -functions with eigenvalues from random matrix ensembles. In many cases the sums of smooth test functions, whose Fourier transforms are finitely supported, over scaled zeros in a family converge to an integral of the test function against a density W_n, G depending on the symmetry G of the family (unitary, symplectic or orthogonal). This integral bounds the average order of vanishing at the central point of the corresponding family of L -functions. We can obtain better estimates on this vanishing in two ways. The first is to do more number theory, and prove results for larger n and greater support; the second is to do functional analysis and obtain better test functions to minimize the resulting integrals. We pursue the latter here when n=2 , minimizing 1/Φ (0, 0)∫ _ℝ^2 W_2,G (x, y) Φ (x, y) dx dy over test functions Φ : ℝ^2 → [0, ∞ ) with compactly supported Fourier transform. We study a restricted version of this optimization problem, imposing that our test functions take the form ϕ (x) ψ (y) for some fixed admissible ψ (y) and suppϕ⊆ [-1, 1] . Extending results from the 1-level case, namely the functional analytic arguments of Iwaniec, Luo and Sarnak and the differential equations method introduced by Freeman and Miller, we explicitly solve for the optimal ϕ for appropriately chosen fixed test function ψ . The solution allows us to deduce strong estimates for the proportion of newforms of rank 0 or 2 in the case of 𝖲𝖮(𝖾𝗏𝖾𝗇) , rank 1 or 3 in the case of 𝖲𝖮(𝗈𝖽𝖽) , and rank at most 2 for , , and ; our estimates are a significant strengthening of the best known estimates obtained with the 1-level density. As a representative example, the previous best 1-level analysis yields a lower bound of 0.7839 for vanishing to order at most 2 for the 𝖲𝖮(𝖾𝗏𝖾𝗇) family of cuspidal newforms, and our 2-level work improves this to 0.952694. We conclude by discussing further improvements on estimates by the method of iteration.