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Stephen D. Miller

Publications and source records attributed to Stephen D. Miller.

At least 19 recordsLinked to original sources
arXiv2026

The $D^6 R^4$ interaction as a Poincaré series, and a related shifted convolution sum

We complete the program, initiated in a 2015 paper of Green, Miller, and Vanhove, of directly constructing the automorphic solution to the string theory $D^6 R^4$ differential equation $(Δ-12)f=-E_{3/2}^2$ for $SL(2,\Z)$. The construction is via a type of Poincaré series, and requires explicitly evaluating a particular double integral. We also show how to use double Dirichlet series to formally derive the predicted vanishing of one type of term appearing in $f$'s Fourier expansion, confirming a conjecture made by Chester, Green, Pufu, Wang, and Wen motivated by Yang-Mills theory (and later proved rigorously by Fedosova, Klinger-Logan, and Radchenko using the Gross-Zagier Holomorphic Projection Lemma.).

math.NT↗
arXiv2026

The sphere packing problem in dimension 24

Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska's function for the eight-dimensional case.

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arXiv2026

Non-unitarity outside the fundamental parallelepiped

One of the challenges of the unitary dual problem is the daunting number of possible representations to consider. This article describes an approach to narrowing the search space for minimal principal series representations, in terms of the ``fundamental parallelepiped'' (or ``FPP''): the set of linear combinations of fundamental weights with coefficients in the interval $[0,1]$. An earlier conjecture of the author, which was rooted in work of Barbasch and then subsequently vastly generalized by Vogan (and recently proven by Davis and Mason-Brown), asserts that the FPP houses all dominant infinitesimal characters for which minimal principal series have a unitarizable quotient. We present a technique to prove this ``FPP inequality'' in specific examples, demonstrated here for the split real form $E_{8(8)}$, by introducing a limiting theory of intertwining operators as $ν$ approaches $\infty$ in directions of fundamental weights. This limiting theory is inspired by Wilfried Schmid's work on variation of Hodge structure. As an application, we show that the unitary set for a particular minimal principal series consists of the closure of a single open alcove.

math.RT↗
arXiv2026

On Arthur's unitarity conjecture for split real groups

Arthur's conjectures predict the existence of some very interesting unitary representations occurring in spaces of automorphic forms. We prove the unitarity of the "Langlands element" (i.e., the one specified by Arthur) of all unipotent Arthur packets for split real groups. The proof uses Eisenstein series, Langlands' constant term formula and square integrability criterion, analytic properties of intertwining operators, and some mild arithmetic input from the theory of Dirichlet L-functions, to reduce to a more combinatorial problem about intertwining operators. This updated arXiv posting also includes some comments (in blue) concerning statements about normalized intertwining operators we quoted from the literature in Section 9.

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arXiv2026

Stochastic global optimization of continuous functions via random walks on Grassmannians

We introduce a stochastic global optimization method based on random walks on Grassmannian manifolds. To minimize a continuous objective $\ell:\mathbb{R}^d\rightarrow\mathbb{R}$, the method repeatedly samples random $k$-dimensional linear subspaces (with $k\ll d$), solves the resulting low-dimensional restrictions of these problems to these subspaces using an arbitrary black-box optimizer, and updates the iterate (which monotonically improves upon the previous iterate). Unlike classical optimization analyses that rely on convexity, smoothness, Lipschitz bounds, or Polyak-Lojasiewicz-type conditions, our convergence guarantees depend only on the geometric distribution of restricted minima across the $k$-dimensional subspaces passing through a given point in $\mathbb{R}^d$. We identify a gap parameter -- an analogue of a spectral gap for random walks -- that controls the rate at which the iterates approach the global minimum value. Finally, we argue that the same analysis yields a blind-spot robustness property: sufficiently narrow, deep dips of the loss function (small-measure regions where $\ell$ spikes downward) have limited influence on the algorithm's trajectory, since they are unlikely to be encountered by random subspace sampling.

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arXiv2023

On the convergence of Kac-Moody Eisenstein series

Let $G$ be a representation-theoretic Kac--Moody group associated to a nonsingular symmetrizable generalized Cartan matrix. We first consider Kac-Moody analogs of Borel Eisenstein series (induced from quasicharacters on the Borel), and prove they converge almost everywhere inside the Tits cone for arbitrary spectral parameters in the Godement range. We then use this result to show the full absolute convergence everywhere inside the Tits cone (again for spectral parameters in the Godement range) for a class of Kac-Moody groups satisfying a certain combinatorial property, in particular for rank-2 hyperbolic groups.

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arXiv2022

Universal optimality of the $E_8$ and Leech lattices and interpolation formulas

We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently shown optimality as sphere packings, and broadly generalizes it to allow for long-range interactions. The proof uses sharp linear programming bounds for energy. To construct the optimal auxiliary functions used to attain these bounds, we prove a new interpolation theorem, which is of independent interest. It reconstructs a radial Schwartz function $f$ from the values and radial derivatives of $f$ and its Fourier transform $\widehat{f}$ at the radii $\sqrt{2n}$ for integers $n\ge1$ in $\mathbb{R}^8$ and $n \ge 2$ in $\mathbb{R}^{24}$. To prove this theorem, we construct an interpolation basis using integral transforms of quasimodular forms, generalizing Viazovska's work on sphere packing and placing it in the context of a more conceptual theory.

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arXiv2021

Generating cryptographically-strong random lattice bases and recognizing rotations of $\mathbb{Z}^n$

Lattice-based cryptography relies on generating random bases which are difficult to fully reduce. Given a lattice basis (such as the private basis for a cryptosystem), all other bases are related by multiplication by matrices in $GL(n,\mathbb{Z})$. We compare the strengths of various methods to sample random elements of $GL(n,\mathbb{Z})$, finding some are stronger than others with respect to the problem of recognizing rotations of the $\mathbb{Z}^n$ lattice. In particular, the standard algorithm of multiplying unipotent generators together (as implemented in Magma's RandomSLnZ command) generates instances of this last problem which can be efficiently broken, even in dimensions nearing 1,500. Likewise, we find that the random basis generation method in one of the NIST Post-Quantum Cryptography competition submissions (DRS) generates instances which can be efficiently broken, even at its 256-bit security settings. Other random basis generation algorithms (some older, some newer) are described which appear to be much stronger.

cs.CR↗
arXiv2020

Coppersmith's lattices and "focus groups": an attack on small-exponent RSA

We present a principled technique for reducing the lattice and matrix size in some applications of Coppersmith's lattice method for finding roots of modular polynomial equations. Motivated by ideas from machine learning, it relies on extrapolating patterns from the actual behavior of Coppersmith's attack for smaller parameter sizes, which can be thought of as "focus group" testing. When applied to the small-exponent RSA problem, our technique reduces lattice dimensions and consequently running times, and hence can be applied to a wider range of exponents. Moreover, in many difficult examples our attack is not only faster but also more successful in recovering the RSA secret key. We include a discussion of subtleties concerning whether or not existing metrics (such as enabling condition bounds) are decisive in predicting the true efficacy of attacks based on Coppersmith's method. Finally, indications are given which suggest certain lattice basis reduction algorithms (such as Nguyen-Stehlé's L2) may be particularly well-suited for Coppersmith's method.

cs.CR↗
arXiv2020

A template method for Fourier coefficients of Langlands Eisenstein series

This paper introduces the template method for computing the first coefficient of Langlands Eisenstein series on $\GL(n,\mathbb R)$ and more generally on Chevalley groups over the adele ring of $\mathbb Q.$ In brief, the first coefficient of Borel Eisenstein series can be used as a template to compute the first coefficient of more general Eisenstein series by elementary linear algebra calculations.

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arXiv2019

Kissing numbers and transference theorems from generalized tail bounds

We generalize Banaszczyk's seminal tail bound for the Gaussian mass of a lattice to a wide class of test functions. From this we obtain quite general transference bounds, as well as bounds on the number of lattice points contained in certain bodies. As applications, we bound the lattice kissing number in $\ell_p$ norms by $e^{(n+ o(n))/p}$ for $0 < p \leq 2$, and also give a proof of a new transference bound in the $\ell_1$ norm.

math.MG↗
arXiv2019

Stopping time signatures for some algorithms in cryptography

We consider the normalized distribution of the overall running times of some cryptographic algorithms, and what information they reveal about the algorithms. Recent work of Deift, Menon, Olver, Pfrang, and Trogdon has shown that certain numerical algorithms applied to large random matrices exhibit a characteristic distribution of running times, which depends only on the algorithm but are independent of the choice of probability distributions for the matrices. Different algorithms often exhibit different running time distributions, and so the histograms for these running time distributions provide a time-signature for the algorithms, making it possible, in many cases, to distinguish one algorithm from another. In this paper we extend this analysis to cryptographic algorithms, and present examples of such algorithms with time-signatures that are indistinguishable, and others with time-signatures that are clearly distinct.

cs.CR↗
arXiv2019

A Spectral reciprocity formula and non-vanishing for L-functions on GL(4)xGL(2)

We develop a reciprocity formula for a spectral sum over central values of L-functions on GL(4)xGL(2). As an application we show that for any self-dual cusp form Pi for SL(4,Z), there exists a Maass form pi for SL(2,Z) such that L(1/2, Pi x pi) is nonvanishing. An important ingredient is a "balanced" Voronoi summation formula involving Kloosterman sums on both sides, which can also be thought of as the functional equation of a certain double Dirichlet series involving Kloosterman sums and GL(4) Hecke eigenvalues.

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arXiv2018

On the nonexistence of automorphic eigenfunctions of exponential growth on SL(3,Z)\SL(3,R)/SO(3,R)

It is well-known that there are automorphic eigenfunctions on SL(2,Z)\SL(2,R)/SO(2,R) -- such as the classical $j$-function -- that have exponential growth and have exponentially growing Fourier coefficients (e.g., negative powers of $q=e^{2πi z}$, or an I-Bessel function). We show that this phenomenon does not occur on the quotient SL(3,Z)\SL(3,R)/SO(3,R) and eigenvalues in general position (a removable technical assumption). More precisely, if such an automorphic eigenfunction has at most exponential growth, it cannot have non-decaying Whittaker functions in its Fourier expansion. This confirms part of a conjecture of Miatello and Wallach, who assert all automorphic eigenfunctions on this quotient (among other rank $\ge$ 2 examples) always have moderate growth. We additionally confirm their conjecture under certain natural hypotheses, such as the absolute convergence of the eigenfunction's Fourier expansion.

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arXiv2017

The balanced Voronoi formulas for GL(n)

In this paper we show how the GL(N) Voronoi summation formula of [MiSc2] can be rewritten to incorporate hyper-Kloosterman sums of various dimensions on both sides. This generalizes a formula for GL(4) with ordinary Kloosterman sums on both sides that was considered by Xiaoqing Li and the first-named author, and later by the second-named author in [Zho].

math.NT↗
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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arXiv2016

Entirety of cuspidal Eisenstein series on loop groups

In this paper, we prove the entirety of loop group Eisenstein series induced from cusp forms on the underlying finite dimensional group, by demonstrating their absolute convergence on the full complex plane. This is quite in contrast to the finite-dimensional setting, where such series only converge absolutely in a right half plane (and have poles elsewhere coming from $L$-functions in their constant terms). Our result is the $\Q$-analog of a theorem of A. Braverman and D. Kazhdan from the function field setting, who previously showed the analogous Eisenstein series are finite sums.

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