Dorian Goldfeld

Search for another author

Author Description

Dorian Goldfeld is an American mathematician and professor at Columbia University. He is known primarily for his work in analytic number theory and automorphic forms. Goldfeld has made major contributions to the study of zeta functions, L-functions, and elliptic curves, including results that connect modular forms to deep questions about the ranks of elliptic curves. In recognition of his research, he received a Guggenheim Fellowship in 1985 and was awarded the Cole Prize in Number Theory in 1987. He has also served on the editorial boards of various mathematical journals and is a highly regarded mentor in his field.

Ask a Question About This Mathematician

All published works (136)

Multiplicativity of Fourier Coefficients of Maass Forms for SL($n,\mathbb Z$)

2025-02-05

Dorian Goldfeld , Eric Stade , Michael Woodbury

The Fourier coefficients of a Maass form $\phi$ for SL$(n,\mathbb Z)$ are complex numbers $A_\phi(M)$, where $M=(m_1,m_2,\ldots,m_{n-1})$ and $m_1,m_2,\ldots ,m_{n-1}$ are nonzero integers. It is well known that coefficients of … The Fourier coefficients of a Maass form $\phi$ for SL$(n,\mathbb Z)$ are complex numbers $A_\phi(M)$, where $M=(m_1,m_2,\ldots,m_{n-1})$ and $m_1,m_2,\ldots ,m_{n-1}$ are nonzero integers. It is well known that coefficients of the form $A_\phi(m_1,1,\ldots,1)$ are eigenvalues of the Hecke algebra and are multiplicative. We prove that the more general Fourier coefficients $A_\phi(m_1,\ldots,m_{n-1})$ are also eigenvalues of the Hecke algebra and satisfy the multiplicativity relations $$A_\phi\big(m_1m_1',\;m_2m_2', \;\ldots\; m_{n-1}m_{n-1}'\big) = A_\phi\big(m_1,m_2,\ldots,m_{n-1})\cdot A_\phi(m_1',m_2',\ldots,m_{n-1}'\big)$$ provided the products $\prod\limits_{i=1}^{n-1} m_i$ and $\prod\limits_{i=1}^{n-1} m_i'$ are relatively prime to each other.

Chat View PDF

The shifted convolution L-function for Maass forms

2024-10-16

Dorian Goldfeld , Gerhardt Hinkle , Jeffrey Hoffstein

The first coefficient of Langlands Eisenstein series for ${\rm SL}(n,\mathbb Z)$

2023-12-21

Dorian Goldfeld , Eric Stade , Michael Woodbury

Fourier coefficients of Eisenstein series figure prominently in the study of automorphic L-functions via the Langlands–Shahidi method, and in various other aspects of the theory of automorphic forms and representations. … Fourier coefficients of Eisenstein series figure prominently in the study of automorphic L-functions via the Langlands–Shahidi method, and in various other aspects of the theory of automorphic forms and representations. In this paper, we defi

Chat View PDF

The functional equations of Langlands Eisenstein series for SL(n, ℤ)

2023-10-16

Dorian Goldfeld , Eric Stade , Michael Woodbury

Chat View PDF

Number theoretical locally recoverable codes

2023-09-16

Andrea Ferraguti , Dorian Goldfeld , Giacomo Micheli

In this paper, we give constructions for infinite sequences of finite nonlinear locally recoverable codes [Formula: see text] over a product of finite fields arising from basis expansions in algebraic … In this paper, we give constructions for infinite sequences of finite nonlinear locally recoverable codes [Formula: see text] over a product of finite fields arising from basis expansions in algebraic number fields. The codes in our sequences have increasing length and size, constant rate, fixed locality, and minimum distance going to infinity.

Chat View PDF

ℵ-structures: One-way Actions via Holomorphs and Split Extensions with Cryptographic Applications

2023-06-01

Iris Anshel , Dorian Goldfeld , Paul E. Gunnells

The first coefficient of Langlands Eisenstein series for $\hbox{SL}(n,\mathbb Z)$

2023-01-01

Dorian Goldfeld , Eric Stade , Michael Woodbury

Fourier coefficients of Eisenstein series figure prominently in the study of automorphic L-functions via the Langlands-Shahidi method, and in various other aspects of the theory of automorphic forms and representations. … Fourier coefficients of Eisenstein series figure prominently in the study of automorphic L-functions via the Langlands-Shahidi method, and in various other aspects of the theory of automorphic forms and representations. In this paper, we define Langlands Eisenstein series for ${\rm SL}(n,\mathbb Z)$ in an elementary manner, and then determine the first Fourier coefficient of these series in a very explicit form. Our proofs and derivations are short and simple, and use the Borel Eisenstein series as a template to determine the first Fourier coefficient of other Langlands Eisenstein series.

Chat View PDF

Number Theoretical Locally Recoverable Codes

2023-01-01

Andrea Ferraguti , Dorian Goldfeld , Giacomo Micheli

In this paper we give constructions for infinite sequences of finite non-linear locally recoverable codes $\mathcal C\subseteq \prod\limits^N_{i=1}\mathbb F_{q_i}$ over a product of finite fields arising from basis expansions in … In this paper we give constructions for infinite sequences of finite non-linear locally recoverable codes $\mathcal C\subseteq \prod\limits^N_{i=1}\mathbb F_{q_i}$ over a product of finite fields arising from basis expansions in algebraic number fields. The codes in our sequences have increasing length and size, constant rate, fixed locality, and minimum distance going to infinity.

Chat View PDF

The cubic Pell equation $L$-function

2023-01-01

Dorian Goldfeld , Gerhardt Hinkle

For $d \gt 1$ a cubefree rational integer, we define an $L$-function (denoted $L_d(s)$) whose coefficients are derived from the cubic theta function for $\mathbb Q(\sqrt {-3})$. The Dirichlet series … For $d \gt 1$ a cubefree rational integer, we define an $L$-function (denoted $L_d(s)$) whose coefficients are derived from the cubic theta function for $\mathbb Q(\sqrt {-3})$. The Dirichlet series defining $L_d(s)$ converges for ${\rm Re}(s) \gt 1$, and

Chat View PDF

The shifted convolution $L$-function for Maass forms

2023-01-01

Dorian Goldfeld , Gerhardt Hinkle , Jeffrey Hoffstein

Let $\phi$ be a Maass form for $\text{SL}(2,\mathbb Z)$ with Fourier coefficients $c(n), n\ne 0.$ For a positive integer $h$ the meromorphic continuation and growth in $s\in\mathbb C$ (away from … Let $\phi$ be a Maass form for $\text{SL}(2,\mathbb Z)$ with Fourier coefficients $c(n), n\ne 0.$ For a positive integer $h$ the meromorphic continuation and growth in $s\in\mathbb C$ (away from poles) of the shifted convolution $L$-function $$L_h(s,\phi)\, := \sum_{n \neq 0,-h} c(n) \overline{c(n + h)} \cdot \big|n(n + h)\big|^{-\frac{1}{2}s}$$ is obtained. For ${\rm Re}(s) > 0$ it is shown that the only poles are possible simple poles at $\frac{1}{2} \pm ir_k$, where $\tfrac14+r_k^2$ are eigenvalues of the Laplacian. As an application we obtain, for $T\to\infty$, the asymptotic formula \begin{align*} & \underset{n \neq 0,-h}{\sum_{\sqrt{|n (n + h)|}<T} } c(n) \overline{c(n + h)} \left(\text{log}\Big(\tfrac{T}{\sqrt{|n (n + h)|}}\,\Big)\right)^{\frac{3}{2} + \varepsilon} \hskip-2pt =\; f_{h,\varepsilon}(T) \cdot T^{\frac{1}{2}} \; + \; \mathcal O\left( h^{1-\varepsilon} T^\varepsilon + h^{1 + \varepsilon} T^{-2 - 2\varepsilon} \right), \end{align*} where the function $f_{h,\varepsilon}(T)$ is given as an explicit spectral sum that satisfies the bound $f_{h,\varepsilon}(T) \ll h^{\theta + \varepsilon}$. We also obtain a sharp bound for the above shifted convolution sum with sharp cutoff, i.e., without the smoothing weight $\log(*)^{\frac32+\varepsilon}$ with uniformity in the $h$ aspect. Specifically, we show that for $h < x^{\frac{1}{2} - \varepsilon}$, \[ \underset{n \neq 0,-h}{\sum_{\sqrt{|n (n + h)|} < x}} c(n) \overline{c(n + h)} \ll h^{\frac{2}{3}\theta + \varepsilon}x^{\frac{2}{3} (1 + \theta) + \varepsilon} + h^{\frac{1}{2} + \varepsilon}x^{\frac{1}{2} + 2\theta + \varepsilon}. \]

Chat View PDF

The cubic Pell equation L-function

2022-01-01

Dorian Goldfeld , Gerhardt Hinkle

For $d > 1$ a cubefree rational integer, we define an $L$-function (denoted $L_d(s)$) whose coefficients are derived from the cubic theta function for $\mathbb Q\left(\sqrt{-3}\right)$. The Dirichlet series defining … For $d > 1$ a cubefree rational integer, we define an $L$-function (denoted $L_d(s)$) whose coefficients are derived from the cubic theta function for $\mathbb Q\left(\sqrt{-3}\right)$. The Dirichlet series defining $L_d(s)$ converges for $\text{Re}(s) > 1$, and its coefficients vanish except at values corresponding to integral solutions of $mx^3 - dny^3 = 1$ in $\mathbb Q\left(\sqrt{-3}\right)$, where $m$ and $n$ are squarefree. By generalizing the methods used to prove the Takhtajan-Vinogradov trace formula, we obtain the meromorphic continuation of $L_d(s)$ to $\text{Re}(s) > \frac{1}{2}$ and prove that away from its poles, it satisfies the bound $L_d(s) \ll |s|^{\frac{7}{2}}$ and has a possible simple pole at $s = \frac{2}{3}$, possible poles at the zeros of a certain Appell hypergeometric function, with no other poles. We conjecture that the latter case does not occur, so that $L_d(s)$ has no other poles with $\text{Re}(s) > \frac{1}{2}$ besides the possible simple pole at $s = \frac{2}{3}$.

Chat View PDF

An Asymptotic Orthogonality Relation for ${\rm GL}(n, \mathbb R)$

2022-01-01

Dorian Goldfeld , Eric Stade , Michael Woodbury

Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on GL(1)) was used … Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on GL(1)) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Asymptotic orthogonality relations for GL$(n)$, with $n\le 3$, and applications to number theory, have been considered by various researchers over the last 45 years. Recently, the authors of the present work have derived an explicit asymptotic orthogonality relation, with a power savings error term, for GL$(4,\mathbb R)$. Here we we extend those results to GL$(n,\mathbb R)$ $(n\ge2)$. For $n\le 5$ our results are unconditional. In particular, the case $n=5$ represents a new result. The key new ingredient for the proof of the case $n=5$ is the theorem of Kim-Shahidi that functorial products of cusp forms on GL(2)$\times$GL(3) are automorphic on GL(6). For $n>5$ our results are conditional on two conjectures, both of which have been verified in various special cases. The first of these conjectures regards lower bounds for Rankin-Selberg L-functions, and the second concerns recurrence relations for Mellin transforms of GL$(n,\mathbb R)$ Whittaker functions. Our methods assume the Ramanujan conjecture at the infinite place for Maass cusp forms, but this assumption can be removed with a weakening in our error term. Central to our proof is an application of the Kuznetsov Trace formula, and a detailed analysis, utilizing a number of novel techniques, of the various entities -- Hecke-Maass cusp forms, Langlands Eisenstein series, spherical principal series Whittaker functions and their Mellin transforms, and so on -- that arise in this application.

Chat View PDF

Preface

2021-09-17

Dorian Goldfeld , Ambrus Pál , Federico Pellarin +1 more

Automorphic Representations and L-Functions for GL(n)

2021-07-15

Dorian Goldfeld , Hervé Jacquet

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

An orthogonality relation for (with an appendix by Bingrong Huang)

2021-01-01

Dorian Goldfeld , Eric Stade , Michael Woodbury

Abstract Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on $\mathrm {GL}(1)$ … Abstract Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on $\mathrm {GL}(1)$ ) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Orthogonality relations for $\mathrm {GL}(2)$ and $\mathrm {GL}(3)$ have been worked on by many researchers with a broad range of applications to number theory. We present here, for the first time, very explicit orthogonality relations for the real group $\mathrm {GL}(4, \mathbb R)$ with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula.

Chat View PDF

The algebraic theory of fractional jumps

2020-10-26

Dorian Goldfeld , Giacomo Micheli

In this paper, we start by briefly surveying the theory of fractional jumps and transitive projective maps. Then we show some new results on the absolute jump index, on projectively … In this paper, we start by briefly surveying the theory of fractional jumps and transitive projective maps. Then we show some new results on the absolute jump index, on projectively primitive polynomials, and on compound constructions.

Chat View PDF

Ironwood meta key agreement and authentication protocol

2020-04-13

Iris Anshel , Derek Atkins , Dorian Goldfeld +1 more

<p style='text-indent:20px;'>Number theoretic public-key solutions, currently used in many applications worldwide, will be subject to various quantum attacks, making them less attractive for longer-term use. Certain group theoretic constructs are … <p style='text-indent:20px;'>Number theoretic public-key solutions, currently used in many applications worldwide, will be subject to various quantum attacks, making them less attractive for longer-term use. Certain group theoretic constructs are now showing promise in providing quantum-resistant cryptographic primitives, and may provide suitable alternatives for those looking to address known quantum attacks. In this paper, we introduce a new protocol called a <i>Meta Key Agreement and Authentication Protocol</i> (MKAAP) that has some characteristics of a public-key solution and some of a shared-key solution. Specifically, it has the deployment benefits of a public-key system, allowing two entities that have never met before to authenticate without requiring real-time access to a third-party, but does require secure provisioning of key material from a trusted key distribution system (similar to a symmetric system) prior to deployment. We then describe a specific MKAAP instance, the Ironwood MKAAP, discuss its security, and show how it resists certain quantum attacks such as Shor's algorithm or Grover's quantum search algorithm. We also show Ironwood implemented on several "internet of things" (IoT devices), measure its performance, and show how it performs significantly better than ECC using fewer device resources.

Chat View PDF

A template method for Fourier coefficients of Langlands Eisenstein series

2020-01-01

Dorian Goldfeld , Stephen D. Miller , Michael Woodbury

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 … 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.

Chat View PDF

The Algebraic Theory of Fractional Jumps

2020-01-01

Dorian Goldfeld , Giacomo Micheli

In this paper we start by briefly surveying the theory of Fractional Jumps and transitive projective maps. Then, we give an efficient construction of a fractional jump of a projective … In this paper we start by briefly surveying the theory of Fractional Jumps and transitive projective maps. Then, we give an efficient construction of a fractional jump of a projective map and we extend the compound generator construction for the Inversive Congruential Generator to Fractional jump sequences. In addition, we provide new results on the absolute jump index, on projectively primitive polynomials, and on the explicit description of fractional jump generators.

Chat View PDF

Alexander Smith Wins the First David Goss Prize in Number Theory

2019-11-15

Dorian Goldfeld , Simone Munaò

the former editor-in-chief of the Journal of Number Theory (JNT), passed away in 2017 the former editor-in-chief of the Journal of Number Theory (JNT), passed away in 2017

Defeating the Hart et al, Beullens-Blackburn, Kotov-Menshov-Ushakov, and Merz-Petit Attacks on WalnutDSA(TM).

2019-01-01

Iris Anshel , Derek Atkins , Dorian Goldfeld +1 more

Super-positivity of a family of L-functions

2019-01-01

Dorian Goldfeld , Bingrong Huang

Zhiwei Yun and Wei Zhang introduced the notion of “super-positivity of self dual L-functions” which specifies that all derivatives of the completed L-function (including Gamma factors and power of the … Zhiwei Yun and Wei Zhang introduced the notion of “super-positivity of self dual L-functions” which specifies that all derivatives of the completed L-function (including Gamma factors and power of the conductor) at the central value $s = 1/2$ should be n

Chat View PDF

An orthogonality relation for GL(4,R)

2019-01-01

Dorian Goldfeld , Eric Stade , Michael Woodbury +1 more

Chat View PDF

Super-positivity of a family of L-functions in the level aspect

2018-03-13

Dorian Goldfeld , Bingrong Huang

Chat View PDF

Super-positivity of a family of L-functions in the level aspect

2018-01-01

Dorian Goldfeld , Bingrong Huang

An automorphic self dual L-function has the super-positivity property if all derivatives of the completed L-function at the central point $s=1/2$ are non-negative and all derivatives at a real point … An automorphic self dual L-function has the super-positivity property if all derivatives of the completed L-function at the central point $s=1/2$ are non-negative and all derivatives at a real point $s > 1/2$ are positive. In this paper we prove that at least 12% of L-functions associated to Hecke basis cusp forms of weight $2$ and large prime level $q$ have the super-positivity property. It is also shown that at least 49% of such L-functions have no real zeros on $ \Re(s) > 0$ except possibly at $s = 1/2.$

Chat View PDF

A Standard Zero Free Region for Rankin–Selberg L-Functions

2017-04-06

Dorian Goldfeld , Xiaoqing Li

A standard zero free region is obtained for Rankin–Selberg L-functions |$L(s, f\times \widetilde{f})$|⁠, where |$f$| is an almost everywhere tempered Maass form on |$GL(n)$| and |$f$| is not necessarily self … A standard zero free region is obtained for Rankin–Selberg L-functions |$L(s, f\times \widetilde{f})$|⁠, where |$f$| is an almost everywhere tempered Maass form on |$GL(n)$| and |$f$| is not necessarily self dual. The method is based on the theory of Eisenstein series generalizing a work of Sarnak.

Chat View PDF

Ironwood Meta Key Agreement and Authentication Protocol

2017-02-08

Iris Anshel , Derek Atkins , Dorian Goldfeld +1 more

Number theoretic public-key solutions, currently used in many applications worldwide, will be subject to various quantum attacks, making them less attractive for longer-term use. Certain group theoretic constructs are now … Number theoretic public-key solutions, currently used in many applications worldwide, will be subject to various quantum attacks, making them less attractive for longer-term use. Certain group theoretic constructs are now showing promise in providing quantum-resistant cryptographic primitives, and may provide suitable alternatives for those looking to address known quantum attacks. In this paper, we introduce a new protocol called a Meta Key Agreement and Authentication Protocol (MKAAP) that has some characteristics of a public-key solution and some of a shared-key solution. Specifically it has the deployment benefits of a public-key system, allowing two entities that have never met before to authenticate without requiring real-time access to a third-party, but does require secure provisioning of key material from a trusted key distribution system (similar to a symmetric system) prior to deployment. We then describe a specific MKAAP instance, the Ironwood MKAAP, discuss its security, and show how it resists certain quantum attacks such as Shor's algorithm or Grover's quantum search algorithm. We also show Ironwood implemented on several ``internet of things'' (IoT devices), measure its performance, and show how it performs significantly better than ECC using fewer device resources.

Chat View PDF

Arthur’s Truncated Eisenstein Series for SL(2, Z) and the Riemann Zeta Function: A Survey

2017-01-01

Dorian Goldfeld

On an Additive Prime Divisor Function of Alladi and Erdős

2017-01-01

Dorian Goldfeld

Chat View PDF

Ironwood Meta Key Agreement and Authentication Protocol

2017-01-01

Iris Anshel , Derek Atkins , Dorian Goldfeld +1 more

Chat View PDF

On an additive prime divisor function of Alladi and Erdos

2016-05-29

Dorian Goldfeld

This paper discusses the additive prime divisor function $A(n) := \sum\limits_{p^\alpha || n} \alpha \, p$ which was introduced by Alladi and Erd\H os in 1977. It is shown that … This paper discusses the additive prime divisor function $A(n) := \sum\limits_{p^\alpha || n} \alpha \, p$ which was introduced by Alladi and Erd\H os in 1977. It is shown that $A(n)$ is uniformly distributed (mod $q$) for any fixed integer $q > 1.$

Chat View PDF

Defeating the Ben-Zvi, Blackburn, and Tsaban Attack on the Algebraic Eraser

2016-01-19

Iris Anshel , Derek Atkins , Dorian Goldfeld +1 more

The Algebraic Eraser Diffie-Hellman (AEDH) protocol was introduced in 2005 and published in 2006 by Anshel-Anshel-Goldfeld-Lemieux as a protocol suitable for use on platforms with constrained computational resources, such as … The Algebraic Eraser Diffie-Hellman (AEDH) protocol was introduced in 2005 and published in 2006 by Anshel-Anshel-Goldfeld-Lemieux as a protocol suitable for use on platforms with constrained computational resources, such as FPGAs, ASICs, and wireless sensors. It is a group-theoretic cryptographic protocol that allows two users to construct a shared secret via a Diffie-Hellman-type scheme over an insecure channel. Building on the refuted 2012 permutation-based attack of Kalka-Teichner-Tsaban, in 2015 Ben-Zvi-Blackburn-Tsaban (BBT) presented a heuristic attack that attempts to recover the AEDH shared secret. In their paper BBT reference the AEDH protocol as presented to ISO for certification (ISO 29167-20) by SecureRF. The ISO draft contains two profiles using the Algebraic Eraser. One profile is unaffected by this attack; the second profile is subject to their attack provided the attack runs in real time. This is not the case in most practical deployments. The BBT attack is simply a targeted attack that does not attempt to break the method, system parameters, or recover any private keys. Rather, its limited focus is to recover the shared secret in a single transaction. In addition, the BBT attack is based on several conjectures that are assumed to hold when parameters are chosen according to standard distributions, which can be mitigated, if not avoided. This paper shows how to choose special distributions so that these conjectures do not hold making the BBT attack ineffective for braid groups with sufficiently many strands. Further, the BBT attack assumes that certain data is available to an attacker, but there are realistic deployment scenarios where this is not the case, making the attack fail completely. In summary, the BBT attack is flawed (with respect to the SecureRF ISO draft) and, at a minimum, over-reaches as to its applicability.

Chat View PDF

A standard zero free region for Rankin-Selberg L-functions

2016-01-01

Dorian Goldfeld , Xiaoqing Li

A standard zero free region is obtained for Rankin Selberg L-functions $L(s, f\times \widetilde{f})$ where $f$ is an almost everywhere tempered Maass form on $GL(n)$ and $f$ is not necessarily … A standard zero free region is obtained for Rankin Selberg L-functions $L(s, f\times \widetilde{f})$ where $f$ is an almost everywhere tempered Maass form on $GL(n)$ and $f$ is not necessarily self dual. The method is based on the theory of Eisenstein series generalizing a work of Sarnak.

Chat View PDF

Super-positivity of a family of L-functions

2016-01-01

Dorian Goldfeld , Bingrong Huang

Zhiwei Yun and Wei Zhang introduced the notion of "super-positivity of self dual L-functions" which specifies that all derivatives of the completed L-function (including Gamma factors and power of the … Zhiwei Yun and Wei Zhang introduced the notion of "super-positivity of self dual L-functions" which specifies that all derivatives of the completed L-function (including Gamma factors and power of the conductor) at the central value $s = 1/2$ should be non-negative. They proved that the Riemann hypothesis implies super-positivity for self dual cuspidal automorphic L-functions on $GL(n)$. Super-positivity of the Riemann zeta function was established by P\'olya in 1927 and since then many other cases have been found by numerical computation. In this paper we prove, for the first time, that there are infinitely many L-functions associated to modular forms for $SL(2, \mathbb{Z})$ each of which has the super-positivity property. Our proof also establishes that all derivatives of the completed L-function at any real point $\sigma > 1/2$ must be positive.

Chat View PDF

Defeating the Ben-Zvi, Blackburn, and Tsaban Attack on the Algebraic Eraser

2016-01-01

Iris Anshel , Derek Atkins , Dorian Goldfeld +1 more

The Algebraic Eraser Diffie–Hellman (AEDH) protocol was introduced in 2005 and published in 2006 by I. Anshel, M. Anshel, D. Goldfeld, and S. Lemieux as a protocol suitable for use … The Algebraic Eraser Diffie–Hellman (AEDH) protocol was introduced in 2005 and published in 2006 by I. Anshel, M. Anshel, D. Goldfeld, and S. Lemieux as a protocol suitable for use on platforms with constrained computational resources, such as FPGAs, ASICs, and wireless sensors. It is a group-theoretic cryptographic protocol that allows two users to construct a shared secret via a Diffie–Hellman-type scheme over an insecure channel. Building on the refuted 2012 permutation-based attack of Kalka–Teichner–Tsaban (KKT), Ben-Zvi, Blackburn, and Tsaban (BBT) present a heuristic attack, published November 13, 2015, that attempts to recover the AEDH shared secret. In their paper BBT reference the AEDH protocol as presented to ISO for certification (ISO 29167-20) by SecureRF. The ISO 29167-20 draft contains two profiles using the Algebraic Eraser. One profile is unaffected by this attack; the second profile is subject to their attack provided the attack runs in real time. This is not the case in most practical deployments. The BBT attack is simply a targeted attack that does not attempt to break the method, system parameters, or recover any private keys. Rather, its limited focus is to recover the shared secret in a single transaction. In addition, the BBT attack is based on several conjectures that are assumed to hold when parameters are chosen according to standard distributions, which can be mitigated, if not avoided. This paper shows how to choose special distributions so that these conjectures do not hold making the BBT attack ineffective for braid groups with sufficiently many strands. Further, the BBT attack assumes that certain data is available to an attacker, but there are realistic deployment scenarios where this is not the case, making the attack fail completely. In summary, the BBT attack is flawed (with respect to the SecureRF ISO draft) and, at a minimum, over-reaches as to its applicability.

Chat View PDF

Defeating the Ben-Zvi, Blackburn, and Tsaban Attack on the Algebraic Eraser

2016-01-01

Iris Anshel , Derek Atkins , Dorian Goldfeld +1 more

Chat View PDF

On an additive prime divisor function of Alladi and Erdos

2016-01-01

Dorian Goldfeld

This paper discusses the additive prime divisor function $A(n) := \sum\limits_{p^α|| n} α\, p$ which was introduced by Alladi and Erd\H os in 1977. It is shown that $A(n)$ is … This paper discusses the additive prime divisor function $A(n) := \sum\limits_{p^α|| n} α\, p$ which was introduced by Alladi and Erd\H os in 1977. It is shown that $A(n)$ is uniformly distributed (mod $q$) for any fixed integer $q > 1.$

Chat View PDF

First moments of Fourier coefficients of GL(r) cusp forms

2014-12-08

Dorian Goldfeld , Jyoti Sengupta

A converse theorem for double Dirichlet series and Shintani zeta functions

2014-04-01

Nikolaos Diamantis , Dorian Goldfeld

A converse theorem for double Dirichlet series is established. As an application, we show that certain zeta functions introduced by Shintani are actually Weyl group multiple Dirichlet series associated to … A converse theorem for double Dirichlet series is established. As an application, we show that certain zeta functions introduced by Shintani are actually Weyl group multiple Dirichlet series associated to metaplectic Eisenstein series on $GL(2)$.

Fourier expansions of GL(2) newforms at various cusps

2013-02-26

Dorian Goldfeld , Joseph Hundley , Min Lee

Chat View PDF

On the GL(3) Kuznetsov Formula with applications to Symmetry Types of families of $L$-functions

2012-03-29

Dorian Goldfeld , Alex Kontorovich

We present an explicit approach to the GL(3) Kuznetsov formula. As an application, for a restricted class of test functions, we obtain the low-lying zero densities for the following three … We present an explicit approach to the GL(3) Kuznetsov formula. As an application, for a restricted class of test functions, we obtain the low-lying zero densities for the following three families: cuspidal GL(3) Maass forms phi, the symmetric square family sym^2 phi on GL(6), and the adjoint family Ad phi on GL(8). Hence we can identify their symmetry types; they are: unitary, unitary, and symplectic, respectively.

Chat View PDF

Number theory, analysis and geometry : in memory of Serge Lang

2012-01-01

Dorian Goldfeld , Jay Jorgenson , Peter Jones +3 more

Defeating the Kalka--Teicher--Tsaban linear algebra attack on the Algebraic Eraser

2012-01-01

Dorian Goldfeld , Paul E. Gunnells

The Algebraic Eraser (AE) is a public key protocol for sharing information over an insecure channel using commutative and noncommutative groups; a concrete realization is given by Colored Burau Key … The Algebraic Eraser (AE) is a public key protocol for sharing information over an insecure channel using commutative and noncommutative groups; a concrete realization is given by Colored Burau Key Agreement Protocol (CBKAP). In this paper, we describe how to choose data in CBKAP to thwart an attack by Kalka--Teicher--Tsaban.

Chat View PDF

On the determination of the Plancherel measure for Lebedev–Whittaker transforms on \mathop GL(n)

2012-01-01

Dorian Goldfeld , Alex Kontorovich

We give a simple derivation of the Plancherel measure for Lebedev-Whittaker transforms on GL(n). We give a simple derivation of the Plancherel measure for Lebedev-Whittaker transforms on GL(n).

Chat View PDF

On the GL(3) Kuznetsov Formula with applications to Symmetry Types of families of $L$-functions

2012-01-01

Dorian Goldfeld , Alex Kontorovich

Chat View PDF

Multiple Dirichlet Series, L-functions and Automorphic Forms

2012-01-01

Daniel Bump , Solomon Friedberg , Dorian Goldfeld

Preface.- Introduction: Multiple Dirichlet Series.- A Crystal Description for Symplectic Multiple Dirichlet Series.- Metaplectic Whittaker Functions and Crystals of Type B.- Metaplectic Ice.- Littelmann patterns and Weyl Group Multiple Dirichlet … Preface.- Introduction: Multiple Dirichlet Series.- A Crystal Description for Symplectic Multiple Dirichlet Series.- Metaplectic Whittaker Functions and Crystals of Type B.- Metaplectic Ice.- Littelmann patterns and Weyl Group Multiple Dirichlet Series of Type D.- Toroidal Automorphic Forms, Waldspurger Periods and Double Dirichlet Series.- Natural Boundaries and Integral Moments of L-functions.- A Trace Formula of Special Values of Automorphic L-functions.- The Adjoint L-function of SU(2,1).- Symplectic Ice.- On Witten Multiple Zeta-Functions Associated with Semisimple Lie Algebras III.- A Pseudo Twin-Prime Theorem.- Principal Series Representations of Metaplectic Groups over Local Fields.- Two-Dimensional Adelic Analysis and Cuspidal Automorphic Representations of GL(2).

Chat View PDF

Natural Boundaries and Integral Moments of L-Functions

2012-01-01

Adrian Diaconu , Paul Garrett , Dorian Goldfeld

It is shown, under some expected technical assumption, that a large class of multiple Dirichlet series which arise in the study of moments of L-functions have natural boundaries. As a … It is shown, under some expected technical assumption, that a large class of multiple Dirichlet series which arise in the study of moments of L-functions have natural boundaries. As a remedy, we consider a new class of multiple Dirichlet series whose elements have nice properties: a functional equation and meromorphic continuation. This class suggests a notion of integral moments of L-functions.

Number Theory, Analysis and Geometry

2011-12-15

Dorian Goldfeld , Jay Jorgenson , Peter B. Jones +3 more

On the Kontorovich-Lebedev Transform

2011-12-01

Dorian Goldfeld , Alex Kontorovich , Eric Stade

Moments for L-Functions for GL r×GL r-1

2011-10-19

Adrian Diaconu , Paul Garrett , Dorian Goldfeld

Chat View PDF

Common Coauthors

Coauthor	Papers Together
Joseph Hundley	16
Paul E. Gunnells	11
Jeffrey Hoffstein	9
Iris Anshel	9
Eric Stade	8
Michael Woodbury	8
Solomon Friedberg	5
Adrian Diaconu	5
Bingrong Huang	4
Alex Kontorovich	4
Daniel Bump	4
Gerhardt Hinkle	4
Derek Atkins	4
Giacomo Micheli	4
Derek Atkins	3
Gautam Chinta	3
Michael Anshel	3
Alexander Lubotzky	3
Daniel Lieman	2
Xiaoqing Li	2
Min Lee	2
Hervé Jacquet	2
Nikolaos Diamantis	2
Paul Garrett	2
László Pyber	2
Dinakar Ramakrishnan	2
Andrea Ferraguti	2
Jay Jorgenson	2
Carlo Viola	2
John Tate	2
Kenneth A. Ribet	2
Vladimir Shpilrain	1
Shengxuming Zhang	1
Peter B. Jones	1
Cormac O’Sullivan	1
Simone Munaò	1
X. Li	1
Federico Pellarin	1
Peter Jones	1
Ambrus Pál	1
Andrzej Schinzel	1
Barry Mazur	1
L. Pyber	1
S. J. Patterson	1
Peter Sarnak	1
Nikolay Nikolov	1
Alex Kontorovich	1
Lejla Smajlović	1
Stephen D. Miller	1
Lothar Gerritzen	1

Commonly Cited References

Multiplicative Number Theory

1980-01-01

H. Davenport

Zeta Functions of Simple Algebras

1972-01-01

Roger Godement , Hervé Jacquet

Chat View PDF

ArchimedeanL-factors onGL(n) ×GL(n) and generalized Barnes integrals

2002-12-01

Eric Stade

Automorphic Forms on GL (2)

1970-01-01

H. Jacquet , R. P. Langlands

Chat View PDF

Topics in Classical Automorphic Forms

1997-10-10

Henryk Iwaniec

Introduction The classical modular forms Automorphic forms in general The Eisenstein and the Poincare series Kloosterman sums Bounds for the Fourier coefficients of cusp forms Hecke operators Automorphic $L$-functions Cusp … Introduction The classical modular forms Automorphic forms in general The Eisenstein and the Poincare series Kloosterman sums Bounds for the Fourier coefficients of cusp forms Hecke operators Automorphic $L$-functions Cusp forms associated with elliptic curves Spherical functions Theta functions Representations by quadratic forms Automorphic forms associated with number fields Convolution $L$-functions Bibliography Index.

On the Functional Equations Satisfied by Eisenstein Series

1976-01-01

R. P. Langlands

Chat View PDF

Applications of the Kuznetsov formula on GL(3)

2013-02-05

Valentin Blomer

We develop a fairly explicit Kuznetsov formula on GL(3) and discuss the analytic behavior of the test functions on both sides. Applications to Weyl's law, exceptional eigenvalues, a large sieve … We develop a fairly explicit Kuznetsov formula on GL(3) and discuss the analytic behavior of the test functions on both sides. Applications to Weyl's law, exceptional eigenvalues, a large sieve and L-functions are given.

Chat View PDF

Short expressions of permutations as products and cryptanalysis of the Algebraic Eraser

2012-03-22

Arkadius Kalka , Mina Teicher , Boaz Tsaban

Chat View PDF

Mean values of L-funtions associated to elliptic, fermat and other curves at the centre of the critical strip

1979-08-01

Dorian Goldfeld , Carlo Viola

Introduction to the Arithmetic Theory of Automorphic Functions

1972-10-01

Marvin Tretkoff , Goro Shimura

Rankin-Selberg Convolutions

1983-04-01

H. Jacquet , I. I. Piatetskii-Shapiro , J. A. Shalika

Properties of Eisenstein series formed with modular symbols

2000-01-05

Cormac O’Sullivan

In this work a new kind of non-holomorphic Eisenstein series, first introduced by Goldfeld, is studied. For an arbitrary Fuchsian group of the first kind we fix a holomorphic cusp … In this work a new kind of non-holomorphic Eisenstein series, first introduced by Goldfeld, is studied. For an arbitrary Fuchsian group of the first kind we fix a holomorphic cusp form and consider Eisenstein series constructed with the modular symbol associated with this cusp form. We develop the theory analogously with that of the usual Eisenstein series starting with its meromorphic continuation to the entire complex plane. A functional equation is then obtained relating the values at s to those at 1− s. Introduction Let H = {z ∈ C : Im z > 0} be the upper half plane and let Γ ⊂ SL2(R) be a fixed non co-compact Fuchsian group of the first kind, (for example Γ(N), Γ0(N)), acting on H. For simplicity assume that Γ has a unique cusp at infinity with stability group Γ∞ = { ± ( 1 m 0 1 ) ,m ∈ Z } . For each γ in Γ we shall label its matrix elements ( γa γb γc γd ) . Let f(z) be an element of S2(Γ), the space holomorphic cusp forms of weight 2 for Γ. Following [Go] we define a modified Eisenstein series (0.1) E∗(z, s) = ∑ γ∈Γ∞\Γ 〈 γ, f 〉Im(γz)s, z ∈ H, where for γ ∈ Γ the modular symbol is given by 〈 γ, f 〉 = −2πi ∫ γw w f(τ) dτ, the definition being independent of w ∈ H. Note that since 〈 γ1γ2, f 〉 = 〈 γ1, f 〉+ 〈 γ2, f 〉 the series is not automorphic. The transformation rule is E∗(γz, s) = E∗(z, s)− 〈 γ, f 〉E(z, s), for all γ ∈ Γ where E(z, s) is the usual Eisenstein series for Γ. This new type of non-holomorphic Eisenstein series was introduced by Goldfeld in order to study the distribution properties of modular symbols 〈 γ, f 〉 as γ ranges over the group Γ. The series (0.1) converges for Re(s) > 2 and Goldfeld hypothesised that it should have an analytic continuation and a functional equation. In this paper Selberg’s method, (described in [Iw], [He]), is extended to establish the following results. Typeset by AMS-TEX 1

Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. II

1967-12-01

Walter Roelcke

Low lying zeros of families of L-functions

2000-12-01

Henryk Iwaniec , Wenzhi Luo , Peter Sarnak

Chat View PDF

Eisenstein series of 1/2-integral weight and the mean value of real DirichletL-series

1985-06-01

Dorian Goldfeld , Jeffrey Hoffstein

Fourier coefficients of cusp forms

1978-02-01

Roelof W. Bruggeman

A non-vanishing theorem for zeta functions ofGL n

1976-02-01

Herv� Jacquet , J. A. Shalika

�ber die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen

1967-12-01

Andr� Weil

A new method for lower bounds of L-functions

2004-06-11

Stephen Gelbart , Erez Lapid , Peter Sarnak

Chat View PDF

Real zeros of quadratic Dirichlet L-functions

2002-10-01

J. Brian Conrey , K. Soundararajan

Chat View PDF

On Some Exponential Sums

1948-05-01

André Weil

�ber eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichlet scher Reihen durch Funktionalgleichungen

1949-12-01

Hans Maaß

A property of L-functions on the real line

1980-02-01

H. Stärk , Don Zagier

On the determination of the Plancherel measure for Lebedev–Whittaker transforms on \mathop GL(n)

2012-01-01

Dorian Goldfeld , Alex Kontorovich

We give a simple derivation of the Plancherel measure for Lebedev-Whittaker transforms on GL(n). We give a simple derivation of the Plancherel measure for Lebedev-Whittaker transforms on GL(n).

Chat View PDF

Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces With Applications to Dirichlet Series

1956-09-01

Atle Selberg

In The following lectures we shall give a brief sketch of some representative parts of certain investigations that have been undertaken during the last five years. The center of these … In The following lectures we shall give a brief sketch of some representative parts of certain investigations that have been undertaken during the last five years. The center of these investigations is a general relation which can be considered as a generalization of the so-called Poisson summation formula (in one or more dimensions). This relation we here refer to as the trace-formula.

On the estimation of Fourier coefficients of modular forms

1965-01-01

Atle Selberg

Coefficients of Maass Forms and the Siegel Zero

1994-07-01

Jeffrey Hoffstein , Paul Lockhart

Perspectives on the Analytic Theory of L-Functions

2000-01-01

Henryk Iwaniec , Peter Sarnak

To the general mathematician L-functions might appear to be an esoteric and special topic in number theory. We hope that the discussion below will convince the reader otherwise. Time and … To the general mathematician L-functions might appear to be an esoteric and special topic in number theory. We hope that the discussion below will convince the reader otherwise. Time and again it has turned out that the crux of a problem lies in the theory of these functions. At some level it is not entirely clear to us why L-functions should enter decisively, though in hindsight one can give reasons. Our plan is to introduce L-functions and describe the central problems connected with them. We give a sample (this is certainly not meant to be a survey) of results towards these conjectures as well as some problems that can be resolved by finessing these conjectures. We also mention briefly some of the successful present-day tools and the role they might play in the big picture.

Analytic Proof of the Strong Multiplicity One Theorem

1985-02-01

Carlos Moreno

The Multiplicity One Theorem for GL n

1974-09-01

J. A. Shalika

The purpose of this paper is to establish preliminary results in the theory of representations in the direction of a systematic treatment of Hecke-theory for the group GLn. In particular, … The purpose of this paper is to establish preliminary results in the theory of representations in the direction of a systematic treatment of Hecke-theory for the group GLn. In particular, special functions, which may be called Whittaker functions, play a central role. Whether or not the further study of Whittaker functions will lead to definitive results concerning cusp forms for general groups is very much an open question at present. Needless to say, a development of Hecke-theory would necessarily follow the logical outline of the work of H. Jacquet and R. P. Langlands [161. It now seems possible that many of their results can be extended to GLU. This is also indicated by the recent work of I. M. Gelfand and D. A. Kajdan [7]. In the present paper we will be primarily concerned with the study of certain induced representations of the group G = GLn(k) where k is a local field. These representation are of the form

Modular Elliptic Curves and Diophantine Problems

1990-12-31

Dorian Goldfeld

Poincaré series and Kloosterman sums for SL(3, Z)

1988-01-01

Daniel Bump , Solomon Friedberg , Dorian Goldfeld

Chat View PDF

Modular curves and the eisenstein ideal

1977-12-01

Barry Mazur

Chat View PDF

Hecke operators on ?0(m)

1970-06-01

A. O. L. Atkin , Joseph Lehner

Random Matrices, Frobenius Eigenvalues, and Monodromy

1998-11-24

Nicholas M. Katz , Peter Sarnak

Statements of the main results Reformulation of the main results Reduction steps in proving the main theorems Test functions Haar measure Tail estimates Large $N$ limits and Fredholm determinants Several … Statements of the main results Reformulation of the main results Reduction steps in proving the main theorems Test functions Haar measure Tail estimates Large $N$ limits and Fredholm determinants Several variables Equidistribution Monodromy of families of curves Monodromy of some other families GUE discrepancies in various families Distribution of low-lying Frobenius eigenvalues in various families Appendix AD: Densities Appendix AG: Graphs References.

Functional Equations With Multiple Gamma Factors and the Average Order of Arithmetical Functions

1962-07-01

K. Chandrasekharan , Raghavan Narasimhan

Dirichlet Series for the Group GL(N)

1981-01-01

Hervé Jacquet

On the large sieve

1965-12-01

Enrico Bombieri

Effective multiplicity one on GL n and narrow zero-free regions for Rankin-Selberg L -functions

2006-12-01

Farrell Brumley

We establish zero-free regions tapering as an inverse power of the analytic conductor for Rankin-Selberg L -functions on GL n ×GL n' . Such zero-free regions are equivalent to commensurate … We establish zero-free regions tapering as an inverse power of the analytic conductor for Rankin-Selberg L -functions on GL n ×GL n' . Such zero-free regions are equivalent to commensurate lower bounds on the edge of the critical strip, and in the case of L ( s , π X), on the residue at s = 1. As an application we show that a cuspidal automorphic representation on GL n is determined by a finite number of its Dirichlet series coefficients, and that this number grows at most polynomially in the analytic conductor.

Chat View PDF

Some new results on the Eichler cohomology of automorphic forms

1974-01-01

Marvin I. Knopp

We here formulate and prove several new results concerning the Eichler cohomology of automorphic forms on finitely generated Fuchsian groups of the first kind and, in particular, on //-groups.The Eichler … We here formulate and prove several new results concerning the Eichler cohomology of automorphic forms on finitely generated Fuchsian groups of the first kind and, in particular, on //-groups.The Eichler cohomology we introduce is connected with automorphic forms of arbitrary real degree (as opposed to integral degree), with a suitably chosen underlying space of functions analytic in the upper half-plane.We obtain structure theorems for the Eichler cohomology groups which are analogous to earlier results of Eichler and Gunning.

Chat View PDF

On some applications of automorphic forms to number theory

1996-01-01

Daniel Bump , Solomon Friedberg , Jeffrey Hoffstein

A basic idea of Dirichlet is to study a collection of interesting quantities <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace a Subscript n Baseline right-brace Subscript n greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mo … A basic idea of Dirichlet is to study a collection of interesting quantities <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace a Subscript n Baseline right-brace Subscript n greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\{a_n\}_{n\geq 1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by means of its Dirichlet series in a complex variable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w"> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding="application/x-tex">w</mml:annotation> </mml:semantics> </mml:math> </inline-formula>: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n greater-than-or-equal-to 1 Endscripts a Subscript n Baseline n Superscript negative w"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo>∑</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:munder> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mi>w</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\sum _{n\geq 1}a_nn^{-w}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper we examine this construction when the quantities <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript n"> <mml:semantics> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">a_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are themselves infinite series in a second complex variable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, arising from number theory or representation theory. We survey a body of recent work on such series and present a new conjecture concerning them.

Chat View PDF

Lectures on Modular Functions of One Complex Variable

1983-01-01

Hans Maaß

A complete determination of the complex quadratic fields of class-number one.

1967-04-01

H. Stärk

Chat View PDF

A unimodality result in the enumeration of subgroups of a finite abelian group

1987-01-01

Lynne M. Butler

The number of subgroups of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Superscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{p^k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in an abelian group … The number of subgroups of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Superscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{p^k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in an abelian group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{p^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a polynomial in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p comma alpha Subscript left-semidirect-product Baseline left-parenthesis k semicolon p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>α</mml:mi> <mml:mo>⋋</mml:mo> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>;</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p,{\alpha _ \leftthreetimes }(k;p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, determined by the type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda"> <mml:semantics> <mml:mi>λ</mml:mi> <mml:annotation encoding="application/x-tex">\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is well known that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha Subscript left-semidirect-product Baseline left-parenthesis k semicolon p right-parenthesis equals alpha Subscript left-semidirect-product Baseline left-parenthesis n minus k semicolon p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>α</mml:mi> <mml:mo>⋋</mml:mo> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>;</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>α</mml:mi> <mml:mo>⋋</mml:mo> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mi>k</mml:mi> <mml:mo>;</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\alpha _ \leftthreetimes }(k;p) = {\alpha _ \leftthreetimes }(n - k;p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Using a recent result from the theory of Hall-Littlewood symmetric functions, we prove that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha Subscript left-semidirect-product Baseline left-parenthesis k semicolon p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>α</mml:mi> <mml:mo>⋋</mml:mo> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>;</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\alpha _ \leftthreetimes }(k;p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is a unimodal sequence of polynomials. That is, for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-equal-to k less-than-or-equal-to n slash 2 comma alpha Subscript lamda Baseline left-parenthesis k semicolon p right-parenthesis minus alpha Subscript lamda Baseline left-parenthesis k minus 1 semicolon p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>k</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>;</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>−</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>;</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">1 \leq k \leq n/2,{\alpha _\lambda }(k;p) - {\alpha _\lambda }(k - 1;p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a polynomial in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with nonnegative coefficients.

Chat View PDF

On the factors of the jacobian variety of a modular function field

1973-07-01

Goro Shimura

Chat View PDF

Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes

1916-01-01

G. H. Hardy , J. E. Littlewood

solution of I~ ' The idea which dominates the critical stage of the argument is also LANvxu's, but is to be found in another of his papers ('(~ber die Anzahl … solution of I~ ' The idea which dominates the critical stage of the argument is also LANvxu's, but is to be found in another of his papers ('(~ber die Anzahl der Gitterpunkte in gewissen Bereichen',

Chat View PDF

Mean-Value Theorems in the Theory of the Riemann Zeta-Function

1928-01-01

A. E. Ingham

Kloosterman sums and Fourier coefficients of cusp forms

1982-06-01

Jean-Marc Deshouillérs , Henryk Iwaniec

Mellin transforms of GL ( n , [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]) whittaker functions

2001-02-01

Eric Stade

Using a known recursive formula for the class one principal series GL(n, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]) Whittaker function, we deduce a recursive formula for the multiple Mellin transform of this … Using a known recursive formula for the class one principal series GL(n, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]) Whittaker function, we deduce a recursive formula for the multiple Mellin transform of this function. From the latter formula, we verify a conjecture of Goldfeld regarding the location of poles of our Mellin transform. We further express the residues at these poles in terms of Mellin transforms of lower-rank Whittaker functions. Our next result concerns the simplification of our Mellin transform under a certain restriction on the transform parameter. We show, by applying a change of variable to our above result on poles of the Mellin transform, that the GL(n, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]) transform reduces essentially to a GL(n - 1, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]) transform under this restriction. We then demonstrate that, under further restriction of the Mellin transform parameter, this Mellin transform in fact reduces to a ratio of products of gamma functions. Our result proves a conjecture of Bump and Friedberg that is motivated by the theory of exterior square automorphic L-functions. Finally, we show that a certain Mellin transform of a product of two Whittaker functions (one on GL(n - 1, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /]), and the other on GL(n, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /])) reduces to a product of gamma functions. This last result verifies a conjecture of Bump regarding archimedean Euler factors of automorphic L-functions on GL(n - 1, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="07i" /]) x GL(n, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="08i" /]).

The highest lowest zero and other applications of positivity

2002-03-15

Stephen D. Miller

The first nontrivial zeros of the Riemann $zeta$-function are $\approx1/2\pm 14:13472i$. We investigate the question of whether or not any other $L$-function has a higher lowest zero. To do so, … The first nontrivial zeros of the Riemann $zeta$-function are $\approx1/2\pm 14:13472i$. 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 zeros, and vice-versa. We prove that many types of automorphic $L$-functions have a lower first zero than $\zetas$'s (see Theorems 1.1 and 1.2). This is done using Weil's explicit formula with carefully chosen test functions. When this method does not immediately show that $L$-functions of a certain type have low zeros, we then attempt to turn the tables and show that no $L$-functions of that type exist. Thus, the argument is a combination of proving that low zeros exist and that certain cusp forms do not. Consequently, we are able to prove vanishing theorems and improve upon existing bounds on the Laplace spectrum on $L\sp 2({\rm SL}\sb n(\mathbb {Z})\backslash{\rm SL}\sb n(\mathbb {R})/{\rm SO}\sb n(\mathbb {R}))$. These in turn can be used to show that ${\rm SL}\sb 68(\mathbb {Z})\backslash{\rm SL}\sb 68(\mathbb {R})/{\rm SO}\sb 68(\mathbb {R})$ has a discrete, nonconstant, noncuspidal eigenvalue outside the range of the continuous spectrum on $L\sp 2({\rm SL}\sb 68(\mathbb {R})/{\rm SO}\sb 68(\mathbb {R}))$, but that this never happens for ${\rm SL}\sb n(\mathbb {Z})\backslash/{\rm SL}\sb n(\mathbb {R})/{\rm SO}\sb n(\mathbb {R})$ in lower rank. Another application is to cuspidal cohomology: we show there are no cuspidal harmonic forms on ${\rm SL}\sb n(\mathbb {Z})\backslash{\rm SL}\sb n(\mathbb {R})/{\rm SO}\sb n(\mathbb {R})$ for $n<27$.

Chat View PDF