arXiv subjects

Zoë X. Batterman

Publications and source records attributed to Zoë X. Batterman.

4 recordsLinked to original sources

arXiv2024

A Random Matrix Model for a Family of Cusp Forms

The Katz-Sarnak philosophy states that statistics of zeros of $L$-function families near the central point as the conductors tend to infinity agree with those of eigenvalues of random matrix ensembles as the matrix size tends to infinity. While numerous results support this conjecture, S. J. Miller observed that for finite conductors, very different behavior can occur for zeros near the central point in elliptic curve $L$-function families. This led to the creation of the excised model of Dueñez, Huynh, Keating, Miller, and Snaith, whose predictions for quadratic twists of a given elliptic curve are well fit by the data. The key ingredients are relating the discretization of central values of the $L$-functions to excising matrices based on the value of the characteristic polynomials at 1 and using lower order terms (in statistics such as the one-level density and pair-correlation) to adjust the matrix size. We extended this model for a family of twists of an $L$-function associated to a given holomorphic cuspidal newform of odd prime level and arbitrary weight. We derive the corresponding "effective" matrix size for a given form by computing the one-level density and pair-correlation statistics for a chosen family of twists, and we show there is no repulsion for forms with weight greater than 2 and principal nebentype. We experimentally verify the accuracy of the model, and as expected, our model recovers the elliptic curve model.

math.NT↗

arXiv2024

A Survey of a Random Matrix Model for a Family of Cusp Forms

The Katz-Sarnak philosophy states that statistics of zeros of $L$-function families near the central point as the conductors tend to infinity agree with those of eigenvalues of random matrix ensembles as the matrix size tends to infinity. While numerous results support this conjecture, S. J. Miller observed that for finite conductors, very different behavior can occur for zeros near the central point in elliptic curve families. This led to the excised model of Dueñez, Huynh, Keating, Miller, and Snaith, whose predictions for quadratic twists of a given elliptic curve are beautifully fit by the data. The key ingredients are relating the discretization of central values of the $L$-functions to excising matrices based on the value of the characteristic polynomials at 1 and using lower order terms (in statistics such as the one-level density and pair-correlation) to adjust the matrix size. We discuss recent successes by the authors in extending this model to a family of quadratic twists of finite conductor of a given holomorphic cuspidal newform of level an odd prime level. In particular, we predict very little repulsion for forms with weight greater than 2.

math.NT↗

arXiv2023

The Hamming Distance and the Fell Topology on AF Algebras

We introduce a new metric on the ideal space of an AF algebra that metrizes the Fell topology. The novelty of this metric lies in the use of a Hamming distance type metric in its construction. Furthermore, this metric captures more of the ideal structure of AF algebras in comparison to known metrics on the Fell topology of an AF algebra. We explicitly test this on the C*-algebra of complex-valued continuous functions on a quantized interval by comparing our new metric with the dual Hausdorff distance on the ideals of this C*-algebras induced by the Hausdorff distance on the closed subsets of the quantized interval.

math.OA↗

arXiv2023

The Reversed Zeckendorf Game

Zeckendorf proved that every natural number $n$ can be expressed uniquely as a sum of non-consecutive Fibonacci numbers, called its Zeckendorf decomposition. Baird-Smith, Epstein, Flint, and Miller created the Zeckendorf game, a two-player game played on partitions of $n$ into Fibonacci numbers which always terminates at a Zeckendorf decomposition, and proved that Player 2 has a winning strategy for $n\geq 3$. Since their proof was non-constructive, other authors have studied the game to find a constructive winning strategy, and lacking success there turned to related problems. For example, Cheigh, Moura, Jeong, Duke, Milgrim, Miller, and Ngamlamai studied minimum and maximum game lengths and randomly played games. We explore a new direction and introduce the reversed Zeckendorf game, which starts at the ending state of the Zeckendorf game and flips all the moves, so the reversed game ends with all pieces in the first bin. We show that Player 1 has a winning strategy for $n = F_{i+1} + F_{i-2}$ and solve various modified games.

math.CO↗