The Highest-Lowest Zero and other Applications of Positivity

The first nontrivial zeroes of the Riemann ζ function are ≈ 1 2 ± 14.13472i. We investigate the question of whether or not any other Lfunction has a higher lowest zero. To do so we try to quantify the notion that the L-function of a “small” automorphic representation (i.e. one with small level and archimedean type) does not have small zeroes, and vice-versa. We prove that many types of automorphic L-functions have a lower first zero than ζ’s (see Theorems 1.1 and 1.2). This is done using Weil’s explicit formula with carefully-chosen test functions. When this method does not immediately show L-functions of a certain type have low zeroes, we then attempt to turn the tables and show no L-functions of that type exist. Thus the argument is a combination of proving low zeroes exist and that certain cusp forms do not. Consequently we are able to prove vanishing theorems and improve upon existing bounds on the Laplace spectrum on L(SLn(Z)\SLn(R)/SOn(R)). These in turn can be used to show that SL68(Z)\SL68(R)/SO68(R) has a discrete, non-constant, non-cuspidal eigenvalue outside the range of the continuous spectrum on L2(SL68(R)/SO68(R)), but that this never happens for SLn(Z)\SLn(R)/SOn(R) in lower rank. Another application is to cuspidal cohomology: we show there are no cuspidal harmonic forms on SLn(Z)\SLn(R)/SOn(R) for n < 27.