Extending the support of $1$- and $2$-level densities for cusp form $L$-functions under square-root cancellation hypotheses

The Katz-Sarnak philosophy predicts that the behavior of zeros near the central point in families of $L$-functions agrees with that of eigenvalues near 1 of random matrix ensembles. Under GRH, Iwaniec, Luo and Sarnak showed agreement in the one-level densities for cuspidal newforms with the support of the Fourier transform of the test function in $(-2, 2)$. They increased the support further under a square-root cancellation conjecture, showing that a ${\rm GL}(1)$ estimate led to additional agreement between number theory and random matrix theory. We formulate a two-dimensional analog and show it leads to improvements in the two-level density. Specifically, we show that a square-root cancellation of certain classical exponential sums over primes increases the support of the test functions such that the main terms in the $1$- and $2$-level densities of cuspidal newforms averaged over bounded weight $k$ (and fixed level $1$) converge to their random matrix theory predictions. We also conjecture a broad class of such exponential sums where we expect improvement in the case of arbitrary $n$-level densities, and note that the arguments in [ILS] yield larger support than claimed.