Levent Alpoge

Low-Lying Zeros of Maass Form L-Functions

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 -> infinity 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.

Maass Waveforms and Low-Lying Zeros

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 \(\mathrm{GL}_{2}/\mathbb{Q}\) 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 \(\mathop{\mathrm{supp}}(\hat{\phi }) \subseteq (-3/2,3/2)\) 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.

Square-root cancellation for the signs of Latin squares

Let L(n) be the number of Latin squares of order n, and let Leven(n) and Lodd(n) be the number of even and odd such squares, so that L(n)=Leven(n)+Lodd(n). The Alon-Tarsi conjecture states that Leven(n) ≠ Lodd(n) when n is even (when n is odd the two are equal for very simple reasons). In this short note we prove that

van der Waerden and the Primes

\left| {{L^{even}}\left( n \right) - {L^{odd}}\left( n \right)} \right| \leqslant L{\left( n \right)^{\frac{1}{2} + o\left( 1 \right)}}

Proof of a conjecture of Stanley–Zanello

|Leven(n)−Lodd(n)|⩽L(n)12+o(1), thus establishing the conjecture that the number of even and odd Latin squares, while conjecturally not equal in even dimensions, are equal to leading order asymptotically. Two proofs are given: both proceed by applying a differential operator to an exponential integral over SU(n). The method is inspired by a recent result of Kumar-Landsberg.

Self-conjugate core partitions and modular forms

Given a formal language L specified in various ways, we consider the problem of determining if L is nonempty. If L is indeed nonempty, we find upper and lower bounds on the length of the shortest string in L.

The second moment of the number of integral points on elliptic curves is bounded

Abstract In this note we prove the infinitude of the primes via an application of van der Waerdens theorem.