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Jack Buttcane

Publications and source records attributed to Jack Buttcane.

At least 19 recordsLinked to original sources
arXiv2025

Bessel Functions on GL(n), II -- The case $n=4$

The purpose of this article is to verify the conjectures of the previous paper in the particular case of $GL(4)$. We accomplish this in general, but observe two failures of the conjectures: First, that the Strong Interchange of Integrals conjecture is perhaps false for a single Weyl element $w_{2,2}$, though we prove the Weak Interchange of Integrals still holds. Second, again for a single Weyl element $w_{2,1,1}$ and its conjugate $w_{1,1,2}$, it appears that the space of solutions to the Bessel differential equations may not be spanned by the Frobenius series solutions. We discuss what refinements, namely to the Asymptotics Theorem, would be necessary to uniquely identify the Bessel functions for such Weyl elements, and prove them in the exceptional cases for $GL(4)$.

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arXiv2023

Bessel functions on GL(n), I

In the context of the Kuznetsov trace formula, we outline the theory of the Bessel functions on $GL(n)$ as a series of conjectures designed as a blueprint for the construction of Kuznetsov-type formulas with given ramification at infinity. We are able to prove one of the conjectures at full generality on $GL(n)$ and most of the conjectures in the particular case of the long Weyl element; as with previous papers, we give some unconditional results on Archimedean Whittaker functions, now on $GL(n)$ with arbitrary weight. We expect the heuristics here to apply at the level of real reductive groups. In an appendix, we make good progress toward series and integral representations of $GL(4)$ Bessel functions by proving several of the conjectures for $GL(4)$.

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arXiv2022

On Sums of hyper-Kloosterman Sums

A formula of Kuznetsov allows one to interpret a smooth sum of Kloosterman sums as a sum over the spectrum of $GL(2)$ automorphic forms. In this paper, we construct a similar formula for the first hyper-Kloosterman sums using $GL(3)$ automorphic forms, resolving a long-standing problem of Bump, Friedberg and Goldfeld. Along the way, we develop what are apparently new bounds for the order derivatives of the classical $J$-Bessel function, and we conclude with a discussion of the original method of Bump, Friedberg and Goldfeld.

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arXiv2019

The arithmetic Kuznetsov formula on $GL(3)$, II: The general case

We obtain the last of the standard Kuznetsov formulas for $SL(3,\Bbb{Z})$. In the previous paper, we were able to exploit the relationship between the positive-sign Bessel function and the Whittaker function to apply Wallach's Whittaker expansion; now we demonstrate the expansion of functions into Bessel functions for all four signs, generalizing Wallach's theorem for $SL(3)$. As applications, we again consider the Kloosterman zeta functions and smooth sums of Kloosterman sums. The new Kloosterman zeta functions pose the same difficulties as we saw with the positive-sign case, but for the negative-sign case, we obtain some analytic continuation of the unweighted zeta function and give a sort of reflection formula that exactly demonstrates the obstruction when the moduli are far apart. The completion of the remaining sign cases means this work now both supersedes the author's thesis and completes the work started in the original paper of Bump, Friedberg and Goldfeld.

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arXiv2019

Kuznetsov, Petersson and Weyl on $GL(3)$, II: The generalized principal series forms

This paper initiates the study by analytic methods of the generalized principal series Maass forms on $GL(3)$. These forms occur as an infinite sequence of one-parameter families in the two-parameter spectrum of $GL(3)$ Maass forms, analogous to the relationship between the holomorphic modular forms and the spherical Maass cusp forms on $GL(2)$. We develop a Kuznetsov trace formula attached to these forms at each weight and use it to prove an arithmetically-weighted Weyl law, demonstrating the existence of forms which are not self-dual. Previously, the only such forms that were known to exist were the self-dual forms arising from symmetric-squares of $GL(2)$ forms. The Kuznetsov formula developed here should take the place of the $GL(2)$ Petersson trace formula for theorems "in the weight aspect". As before, the construction involves evaluating the Archimedian local zeta integral for the Rankin-Selberg convolution and proving a form of Kontorovich-Lebedev inversion.

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arXiv2018

The arithmetic Kuznetsov formula on $GL(3)$, I: The Whittaker case

The original formulae of Kuznetsov for $SL(2,\mathbb{Z})$ allowed one to study either a spectral average via Kloosterman sums or to study an average of Kloosterman sums via a spectral interpretation. In previous papers, we have developed the spectral Kuznetsov formulae at the minimal weights for $SL(3,\mathbb{Z}))$, and in these formulae, the big-cell Kloosterman sums occur with weight functions attached to four different integral kernels, according to the choice of signs of the indices. These correspond to the $J$- and $K$-Bessel functions in the case of $GL(2)$. In this paper, we demonstrate a linear combination of the spherical and weight one $SL(3,\mathbb{Z})$ Kuznetsov formulae that isolates one particular integral kernel, which is the spherical $GL(3)$ Whittaker function. Using the known inversion formula of Wallach, we give the first arithmetic Kuznetsov formula for $SL(3,\mathbb{Z})$ and use it to study smooth averages and the Kloosterman zeta function attached to this particular choice of signs.

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arXiv2018

Kuznetsov, Petersson and Weyl on GL(3), I: The principal series forms

The Kuznetsov and Petersson trace formulae for $GL(2)$ forms may collectively be derived from Poincaré series in the space of Maass forms with weight. Having already developed the spherical spectral Kuznetsov formula for $GL(3)$, the goal of this series of papers is to derive the spectral Kuznetsov formulae for non-spherical Maass forms and use them to produce the corresponding Weyl laws; this appears to be the first proof of the existence of such forms not coming from the symmetric-square construction. Aside from general interest in new types of automorphic forms, this is a necessary step in the development of a theory of exponential sums on $GL(3)$. We take the opportunity to demonstrate a sort of minimal method for developing Kuznetsov-type formulae, and produce auxillary results in the form of generalizations of Stade's formula and Kontorovich-Lebedev inversion. This first paper is limited to the non-spherical prinicpal series forms as there are some significant technical details associated with the generalized principal series forms, which will be handled in a separate paper. The best analog of this type of form on $GL(2)$ is the forms of weight one which sometimes occur on congruence subgroups.

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arXiv2018

Higher Weight on GL(3), II - The cusp forms

The purpose of this paper is to collect and make explicit the results of Gel'fand, Graev and Piatetski-Shapiro and Miyazaki for the $GL(3)$ cusp forms which are non-trivial on $SO(3,\mathbb{R})$. We give new descriptions of the spaces of cusp forms of minimal $K$-type and from the Fourier-Whittaker expansions of such forms give a complete and completely explicit spectral expansion for $L^2(SL(3,\mathbb{Z})\backslash PSL(3,\mathbb{R}))$, accounting for multiplicities, in the style of Duke, Friedlander and Iwaniec's paper on Artin $L$-functions. We directly compute the Jacquet integral for the Whittaker functions at the minimal $K$-type, improving Miyazaki's computation. The primary tool will be the study of the differential operators coming from the Lie algebra on vector-valued cusp forms.

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arXiv2018

Subconvexity for L-functions of non-spherical cusp forms on GL(3)

Let f be a cusp form for SL(3, Z) associated with a generalized principal series representation of minimal weight d, spectral parameter r and associated L-function L(s, f). For $r \asymp d \asymp T$ the subconvexity bound $L(1/2, f) \ll T^{3/4 - 1/140000}$ is proved.

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arXiv2017

Applications of the Kuznetsov formula on GL(3): the level aspect

We develop an explicit Kuznetsov formula on GL(3) for congruence subgroups. Applications include a Lindelof on average type bound for the sixth moment of GL(3) L-functions in the level aspect, an automorphic large sieve inequality, density results for exceptional eigenvalues and density results for Maass forms violating the Ramanujan conjecture at finite places.

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arXiv2017

Weights, raising and lowering operators, and K-types for automorphic forms on SL(3,R)

We give a fully explicit description of Lie algebra derivatives (generalizing raising and lowering operators) for representations of SL(3,R) in terms of a basis of Wigner functions. This basis is natural from the point of view of principal series representations, as well as computations in the analytic theory of automorphic forms (e.g., with Whittaker functions). The method is based on the Clebsch-Gordan multiplication rule for Wigner functions, and applies to other Lie groups whose maximal compact subgroup is isogenous to a product of SU(2) and U(1) factors. As an application, we give a complete and explicit description of the K-type structure of certain cohomological representations.

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arXiv2017

Higher weight on GL(3), I: The Eisenstein series

The purpose of this paper is to collect and make explicit the results of Langlands, Bump, Miyazaki and Manabe, Ishii and Oda for the $GL(3)$ Eisenstein series and Whittaker functions which are non-trivial on $SO(3,\mathbb{R})$. The final goal for the series of papers is a complete and completely explicit spectral expansion for $L^2(SL(3,\mathbb{Z})\backslash SL(3,\mathbb{R}))$ in the style of Duke, Friedlander and Iwaniec's paper on Artin L-functions. We derive a number of new results on the Whittaker functions and Eisenstein series, and give new, concrete proofs of the functional equations and spectral expansion in place of the general constructions of Langlands.

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arXiv2016

Plancherel distribution of Satake parameters of Maass cusp forms on $GL_3$

We prove an equidistribution result for the Satake parameters of Maass cusp forms on $GL_3$ with respect to the $p$-adic Plancherel measure by using an application of the Kuznetsov trace formula. The techniques developed in this paper deal with the removal of arithmetic weight $L(1,F,Ad)^{-1}$ in the Kuznetsov trace formula on $GL_3$.

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arXiv2016

A mean value of a triple product of $L$-functions

Luo has proven an optimal upper bound for the $L^4$-norm of dihedral Maass forms of large eigenvalue, by bounding a mean value of triple product $L$-functions. Motivated by this result, we study a mean value of $L$-functions having similar shape, and obtain for it an asymptotic with power savings. Our work may be helpful in eventually obtaining an asymptotic for the $L^4$-norm.

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arXiv2015

On the subconvexity problem for L-functions on GL(3)

Let f be a cusp form for the group SL(3, Z) with Langlands parameter mu and associated L-function L(s, f). If mu is in generic position, i.e. away from the Weyl chamber walls and away from the self-dual forms, we prove the subconvexity bound L(1/2, f) << || mu || ^{3/4 - 1/120000}.

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arXiv2014

The Spectral Kuznetsov Formula on SL(3)

The $SL(3)$ Kuznetsov formula exists in several versions, and has been employed with some success to study automorphic forms on $SL(3)$. In each version, the weight functions on the geometric side are given by multiple integrals with complicated oscillating factors; this is the primary obstruction to its use. By describing them as solutions to systems of differential equations, we give power series and Mellin-Barnes integral representations of minimal dimension for these weight functions. This completes the role of harmonic analysis on symmetric spaces on the geometric side of the Kuznetsov formula, so that further study may be done through classical analytic techniques.

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