Maass Waveforms and Low-Lying Zeros

Levent Alpoge (), Nadine Amersi (), Geoffrey Iyer (), Oleg Lazarev (), Steven J. Miller () and Liyang Zhang ()
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Levent Alpoge: Princeton University, Department of Mathematics
Nadine Amersi: University College London, Department of Mathematics
Geoffrey Iyer: UCLA, Department of Mathematics
Oleg Lazarev: Stanford University, Department of Mathematics
Steven J. Miller: Williams College, Department of Mathematics & Statistics
Liyang Zhang: Yale University, Department of Mathematics

A chapter in Analytic Number Theory, 2015, pp 19-55 from Springer

Abstract: Abstract The Katz–Sarnak Density Conjecture states that the behavior of zeros of a family of L-functions near the central point (as the conductors tend to zero) agrees with the behavior of eigenvalues near 1 of a classical compact group (as the matrix size tends to infinity). Using the Petersson formula, Iwaniec, Luo, and Sarnak proved that the behavior of zeros near the central point of holomorphic cusp forms agrees with the behavior of eigenvalues of orthogonal matrices for suitably restricted test functions ϕ. We prove similar results for families of cuspidal Maass forms, the other natural family of GL 2 ∕ ℚ $$\mathrm{GL}_{2}/\mathbb{Q}$$ L-functions. For suitable weight functions on the space of Maass forms, the limiting behavior agrees with the expected orthogonal group. We prove this for supp ( ϕ ̂ ) ⊆ ( − 3 ∕ 2 , 3 ∕ 2 ) $$\mathop{\mathrm{supp}}(\hat{\phi }) \subseteq (-3/2,3/2)$$ when the level N tends to infinity through the square-free numbers; if the level is fixed the support decreases to being contained in (−1, 1), though we still uniquely specify the symmetry type by computing the 2-level density.

Keywords: Classical Compact Groups; Petersson Formula; Maass Forms; Cusp Forms; Kuznetsov Trace Formula (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-22240-0_2

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