Low-Lying Zeros of Maass Form L-Functions

Abstract

The Katz-Sarnak density conjecture states that the scaling limits of the distributions of zeros of families of automorphic L-functions agree with the scaling limits of eigenvalue distributions of classical subgroups of the unitary groups U (N ). This conjecture is often tested by way of computing particular statistics, such as the one-level density, which evaluates a test function with compactly supported Fourier transform at normalized zeros near the central point. Iwaniec, Luo, and Sarnak studied the one-level densities of cuspidal newforms of weight k and level N . They showed in the limit as kN → ∞ that these families have one-level densities agreeing with orthogonal type for test functions with Fourier transform supported in (−2, 2). Exceeding (−1, 1) is important as the three orthogonal groups are indistinguishable for support up to (−1, 1) but are distinguishable for any larger support. We study the other family of GL 2 automorphic forms over Q: Maass forms. To facilitate the analysis, we use smooth weight functions in the Kuznetsov formula which, among other restrictions, vanish to order 2M at the origin. For test functions with Fourier transform supported inside −2 + 2 2M +1 , 2 − 2 2M +1 , we unconditionally prove the one-level density of the low-lying zeros of level 1 Maass forms, as the eigenvalues tend to infinity, agrees only with that of the scaling limit of orthogonal matrices.2010 Mathematics Subject Classification. 11M26 (primary), 11M41, 15A52 (secondary).

Weighted Sato–Tate Vertical Distribution of the Satake Parameter of Maass Forms on PGL(N)

“…We predict that Conjecture 6.1 can be derived from the Kuznetsov trace formula. For N = 2, Proposition 4.1 of [8] gives a version of Conjecture 6.1 and numerous similar identities are obtained for various cases on GL (2). The works of [13] and [5] establish versions of Conjecture 6.1 for N = 3.…”

“…There are numerous applications of the orthogonality relation. The orthogonality relations with error terms for N = 2, 3 are applied to studying the symmetry types of the low-lying zeroes of families of L-functions in [1], [2], [15], and [13]. For N = 2, it is also applied to Sato-Tate distribution of Hecke eigenvalues in [8], [18], and [19] and to p-adic Plancherel distribution of Hecke eigenvalues in [17].…”

“…Let m 1 , m 2 , n 1 , n 2 be positive integers and let P = m 1 m 2 n 1 n 2 . Let θ ≤ 7/64 be a bound towards the Ramanujan conjecture on GL (2). For T ≫ 1, we define…”

“…1 − x 2 4 dx, when |x| ≤ 2, 0, otherwise, called the Sato-Tate measure for GL (2), or the semi-circle measure. The Sato-Tate conjecture is a more refined statement about the statistics of the Hecke eigenvalues, stating that if ϕ is a non-CM holomorphic modular form of weight k ≥ 2, then a ϕ (p)/p k−1 2 is an equidistributed sequence as p → ∞ with respect to the Sato-Tate measure µ ∞ .…”

One-level density for holomorphic cusp forms of arbitrary level

“…While the Fourier transforms of the densities of the orthogonal groups all equal δ 0 (y)+1/2 in (−1, 1), they are mutually distinguishable for larger support (and are distinguishable from the unitary and symplectic cases for any support). There is now an enormous body of work showing the 1-level densities of many families (such as Dirichlet L-functions, elliptic curves, cuspidal newforms, Maass forms, number field L-functions, and symmetric powers of GL 2 automorphic representations) agree with the scaling limits of a random matrix ensemble; see [AAILMZ,AM,DM1,FiMi,FI,Gao,GK,Gü,HM,HR,ILS,KS1,KS2,Mil,MilPe,OS1,OS2,RR,Ro,Rub1,Rub2,ShTe,Ya,Yo] for some examples, and [DM1,DM2,ShTe] for discussions on how to determine the underlying symmetry. For additional readings on connections between random matrix theory, nuclear physics and number theory see Con,CFKRS,FM,For,KeSn1,KeSn2,KeSn3,Meh] We concentrate on extending the results of Iwaniec, Luo, and Sarnak in [ILS].…”