arXiv subjects

Stephen D. Miller

Publications and source records attributed to Stephen D. Miller.

47 records · Page 3Linked to original sources

arXiv2005

Do All Elliptic Curves of the Same Order Have the Same Difficulty of Discrete Log?

The aim of this paper is to justify the common cryptographic practice of selecting elliptic curves using their order as the primary criterion. We can formalize this issue by asking whether the discrete log problem (DLOG) has the same difficulty for all curves over a given finite field with the same order. We prove that this is essentially true by showing polynomial time random reducibility of DLOG among such curves, assuming the Generalized Riemann Hypothesis (GRH). We do so by constructing certain expander graphs, similar to Ramanujan graphs, with elliptic curves as nodes and low degree isogenies as edges. The result is obtained from the rapid mixing of random walks on this graph. Our proof works only for curves with (nearly) the same endomorphism rings. Without this technical restriction such a DLOG equivalence might be false; however, in practice the restriction may be moot, because all known polynomial time techniques for constructing equal order curves produce only curves with nearly equal endomorphism rings.

math.NT↗

arXiv2004

The Highly Oscillatory Behavior of Automorphic Distributions for SL(2)

Automorphic distributions for SL(2) arise as boundary values of modular forms and, in a more subtle manner, from Maass forms. In the case of modular forms of weight one or of Maass forms, the automorphic distributions have continuous first antiderivatives. We recall earlier results of one of us on the Holder continuity of these continuous functions and relate them to results of other authors; this involves a generalization of classical theorems on Fourier series by S. Bernstein and Hardy-Littlewood. We then show that the antiderivatives are non-differentiable at all irrational points, as well as all, or in certain cases, some rational points. We include graphs of several of these functions, which clearly display a high degree of oscillation. Our investigations are motivated in part by properties of "Riemann's nondifferentiable function", also known as "Weierstrass' function".

math.NT↗

arXiv2004

Cancellation in additively twisted sums on GL(n)

In a previous paper with Schmid (math.NT/0402382) we considered the regularity of automorphic distributions for GL(2,R), and its connections to other topics in number theory and analysis. In this paper we turn to the higher rank setting, establishing the nontrivial bound sum_{n < T} a_n exp(2 pi i n alpha) = O_ε(T^{3/4+ε}) uniformly for alpha real, for a_n the coefficients of the L-function of a cusp form on GL(3,Z)\GL(3,R). We also derive an equivalence (Theorem 7.1) between analogous cancellation statements for cusp forms on GL(n,R), and the sizes of certain period integrals. These in turn imply estimates for the second moment of cusp form L-functions.

math.NT↗

arXiv2004

Distributions and Analytic Continuation of Dirichlet Series

This paper is second in a series of three papers; the first of which is "Summation Formulas, from Poisson and Voronoi to the Present" (math.NT/0304187), and the third of which is "Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)". The first paper is primarily an expository paper, while the third proves a Voronoi-style summation formula for the coefficients of a cusp form on GL(3,Z)\GL(3,R). This present paper contains the distributional machinery used in the third paper for rigorously deriving the summation formula, and also for the proof of the GL(3)xGL(1) converse theorem given in the third paper. The primary concept studied is a notion of the order of vanishing of a distribution along a closed submanifold. Applications are given to the analytic continuation of Riemann's zeta function; degree 1 and degree 2 L-functions; the converse theorem for GL(2); and a characterization of the classical Mellin transform/inversion relations on functions with specified singularities.

math.FA↗

arXiv2003

Riemann's Zeta Function and Beyond

In recent years L-functions and their analytic properties have assumed a central role in number theory and automorphic forms. In this expository article, we describe the two major methods for proving the analytic continuation and functional equations of $L$-functions: the method of integral representations, and the method of Fourier expansions of Eisenstein series. Special attention is paid to technical properties, such as boundedness in vertical strips; these are essential in applying the converse theorem, a powerful tool that uses analytic properties of L-functions to establish cases of Langlands functoriality conjectures. We conclude by describing striking recent results which rest upon the analytic properties of L-functions.

math.NT↗

arXiv2003

Summation Formulas, from Poisson and Voronoi to the Present

We give an overview of classical summation formulations, such as Poisson's and Voronoi's, and then turn to modern versions involving modular form coefficients. A new formula involving the coefficients of cusp forms on GL(3) is described, and its proof sketched, followed by applications to L-functions. The main method used is the boundary value distribution of automorphic forms.

math.NT↗

arXiv2002

A Summary of the Langlands-Shahidi Method of Constructing L-functions

These notes are from the Database of Automorphic L-functions at http://www.math.rutgers.edu/~sdmiller/l-functions . They were written up by Stephen Miller, and are based on discussions with Freydoon Shahidi, Purdue University. They are meant to serve as an introduction to the Langlands-Shahidi method of studying L-functions through the Fourier coefficients of Eisenstein series.

math.NT↗

arXiv2001

The Highest-Lowest Zero and other Applications of Positivity

The first nontrivial zeroes of the Riemann $ζ$ function are $\approx 1/2+\pm14.13472i$. We investigate the question of whether or not any other L-function has a higher lowest zero. To do so we try to quantify the notion that the L-function of a ``small'' automorphic representation (i.e. one with small level and archimedean type) does not have small zeroes, and vice-versa. We prove that many types of automorphic L-functions have a lower first zero than $ζ$'s. This is done using Weil's explicit formula with carefully-chosen test functions. When this method does not immediately show L-functions of a certain type have low zeroes, we then attempt to turn the tables and show no L-functions of that type exist. Thus the argument is a combination of proving low zeroes exist and that certain cusp forms do not. Consequently we are able to prove cohomological vanishing theorems and improve upon existing bounds on the Laplace spectrum on $L^2(\quo n)$.

math.NT↗

arXiv2000

On the Existence and Temperedness of Cusp Forms for SL(3,Z)

We develop a partial trace formula which circumvents some technical difficulties in computing the Selberg trace formula for the quotient $SL_3({\Z})\backslash SL_3({\R})/SO_3({\R})$. As applications, we establish the Weyl asymptotic law for the discrete Laplace spectrum and prove that almost all of its cusp forms are tempered at infinity. The technique shows there are non-lifted cusp forms on $SL_3({\Z})\backslash SL_3({\R})/SO_3({\R})$ as well as non-self-dual ones. A self-contained description of our proof for $SL_2({\Z})\backslash \U$ is included to convey the main new ideas. Heavy use is made of truncation and the Maass-Selberg relations.

math.NT↗

arXiv2000

Non-vanishing of the Central Derivative of Canonical Hecke L-functions

In the early 1980s, Rohrlich began a study of canonical Hecke characters, which are closely related to the simplest examples of CM elliptic curves. He and Montgomery showed the non-vanishing of the central value when the L-function has an even functional equation, and we now show the non-vanishing of the central derivative when the functional equation is odd. Using the results of Gross-Zagier and Kolyvagin-Logachev, we can apply the non-vanishing to the ranks and Shafarevitch-Tate groups of the Q-curves "A(p)" studied by Gross in his thesis. In particular, their rank is determined by a congruence condition. http://www.math.yale.edu/users/steve/milleryang

math.NT↗

arXiv1999

Landau-Siegel Zeroes and Black Hole Entropy

There has been some speculation about relations of D-brane models of black holes to arithmetic. In this note we point out that some of these speculations have implications for a circle of questions related to the generalized Riemann hypothesis on the zeroes of Dirichlet $L$-functions.

hep-th↗