Low-lying zeroes of L-functions attached to modular forms

Abstract: We study the family of L-functions attached to Hecke newforms of weight k and level N and their low-lying zeroes. First, we recall the Density Conjecture of Katz and Sarnak and how it predicts the behaviour of the low-lying zeroes of any natural family of L-functions. Then, we review some basic theory of modular forms as an appropriate background to the subsequent investigations. Next, we follow the article [ILS00] by Iwaniec, Luo and Sarnak in their treatment of the 1-level density of our family at hand. From them we recover that the Density Conjecture holds for bounded support of ϕ when kN -->∞ and N is squarefree, conditional on the Generalized Riemann Hypothesis. Also, following Miller [Mil09] we find a term of lower order when k is fixed and N -->∞ through the primes. Lastly, we study the 1-level density through the Ratios Conjecture. The prediction of the Ratios Conjecture allows any compact support of ϕ, as well as agreeing with the explicit calculations down to a power-saving error term.