arXiv subjects

Vishal Muthuvel

Publications and source records attributed to Vishal Muthuvel.

5 recordsLinked to original sources

arXiv2026

Inducing Whittaker Functions from Higher Ranks

We construct a family of Whittaker functions for $SL(m,\mathbb{Z})$ induced directly from Whittaker functions for $SL(n,\mathbb{Z})$, for any $2 \leq m<n$. Given Jacquet's Whittaker function $W_{α,N}^{(n)}$ on the generalized upper half-plane $\mathfrak{h}^n$, we show that the function $V_{α,N}^{(m)}:\mathfrak{h}^m\to\mathbb{C}$ defined by restricting $W_{α,N}^{(n)}$ to the block-diagonal embedding $\mathfrak{h}^m\hookrightarrow\mathfrak{h}^n$ is a Whittaker function for $SL(m,\mathbb{Z})$, provided the Langlands parameters $α=(α_i)_{1\leq i\leq n}$ satisfy $\sum_{i=1}^mα_i = m(m-n)/2$. Under this condition, the induced function carries Langlands parameters $\bigl(α_i+\frac{n-m}{2}\bigr)_{1\leq i\leq m}$ and inherits the first $m-1$ entries of the character tuple of $W_{α,N}^{(n)}$. This result complements the propagation formulas of Ishii and Stade, which relate Whittaker functions on $GL(n,\mathbb{R})$ to those on $GL(n-1,\mathbb{R})$ and $GL(n-2,\mathbb{R})$. In contrast, our construction passes directly from $GL(n,\mathbb{R})$ to $GL(m,\mathbb{R})$ for any $m < n$ in a single step.

math.NT↗

arXiv2026

Explicit Formulas for the Casimir Eigenvalues of $SL(n,\mathbb{Z})$-Maass Forms

Maass forms for $SL(n,\mathbb{Z})$ are defined to be eigenfunctions of the Casimir operators $\mathcal{D}_{m,n}$ of orders $1 \leq m \leq n$ for $GL(n,\mathbb{R})$. For any $1 \leq m \leq n$ and Maass form $ϕ$ for $SL(n,\mathbb{Z})$, we provide a formula for the eigenvalue of $\mathcal{D}_{m,n}$ associated with $ϕ$ in terms of the Langlands parameters of $ϕ$. In the case $m=2$, we recover the formula for the Laplace eigenvalue of a Maass form due to Terras, the Casimir differential operator of order $2$ being the Laplacian. Our proof takes a graph-theoretic approach, relating the action of every elementary differential operator of order $m$ for $GL(n,\mathbb{R})$ to the partitions of a directed, edge-ordered graph with $m$ edges and at most $m$ vertices.

math.NT↗

arXiv2026

Centered Moments of Weighted One-Level Densities of $GL(2)$ $L$-Functions

Katz and Sarnak conjectured that the behavior of zeros near the central point of any family of $L$-functions is well-modeled by the behavior of eigenvalues near $1$ of some classical compact group (either the symplectic, unitary, or even, odd, or full orthogonal group). In 2018, Knightly and Reno proved that the symmetry group can vary depending on how the $L$-functions in the family are weighted. They observed both orthogonal and symplectic symmetry in the one-level densities of families of cuspidal newform $L$-functions for different choices of weights. We observe the same dependence of symmetry on weights in the $n^{\text{th}}$ centered moments of these one-level densities, for smooth test functions whose Fourier transforms are supported in $\left(-\frac{1}{2n}, \frac{1}{2n}\right)$. To treat the new terms that emerge in our $n$-level calculations when $n>1$, i.e., the cross terms that emerge from $n$-fold products of primes rather than individual primes, we generalize Knightly and Reno's weighted trace formula from primes to arbitrary positive integers. We then perform a delicate analysis of these cross terms to distinguish their contributions to the main and error terms of the $n^{\text{th}}$ centered moments. The final novelty here is an elementary combinatorial trick that we use to rewrite the main number theoretic terms arising from our analysis, facilitating comparisons with random matrix theory.

math.NT↗

arXiv2025

Breaking Universality in the Lower Order Terms in the 1-level and 2-level Density of Holomorphic Cusp Newforms

The Katz-Sarnak density conjecture states that, as the analytic conductor $R \to \infty$, the distribution of the normalized low-lying zeros (those near the central point $s = 1/2$) converges to the scaling limits of eigenvalues clustered near 1 of subgroups of $U(N)$. There is extensive evidence supporting this conjecture for many families, including the family of holomorphic cusp newforms. Interestingly, there are very few choices for the main term of the limiting behavior. In 2009, S. J. Miller computed lower-order terms for the 1-level density of families of elliptic curve $L$-functions and compared to cuspidal newforms of prime level; while the main terms agreed, the lower order terms depended on the arithmetic of the family. We extend his work by identifying family-dependent lower-order correction terms in the weighted 1-level and 2-level densities of holomorphic cusp newforms up to $O\left(1/\log^4 R\right)$ error, sharpening Miller's $O\left(1/\log^3 R\right)$ error. We consider cases where the level is prime or when the level is a product of two, not necessarily distinct, primes. We show that the rates at which the prime factors of the level tend to infinity lead to different lower-order terms, breaking the universality of the main behavior.

math.NT↗

arXiv2023

Units of hyperelliptic curves over $\mathbb{F}_2$

We study unit groups of rings of the form $\mathbb{F}_2[x,y]/(y^2 + gy + h)$, for $g, h \in \mathbb{F}_2[x]$ -- in particular, the question of (non)triviality of such unit groups. Up to automorphisms of $\mathbb{F}_2[x,y]$ we classify such rings into 3 distinct types. For 2 of the types we show that the unit group is always trivial, and conjecture that the unit group is always nontrivial for the 3rd type. We provide support for this conjecture both theoretically and computationally, via an algorithm that has been used to compute units in large degrees.

math.AC↗