Searcharxiv⌕ Search

SEARCH · Searcharxiv

Results for “math.NT”

Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 recordsLinked to original sources
arXiv2000

Invitation to higher local fields (Introduction)

The monograph "Invitation to higher local fields" is the result of the conference on higher local fields held in Muenster, August 29 to September 5, 1999. The aim is to provide an introduction to higher local fields (more generally complete discrete valuation fields with arbitrary residue field) and render the main ideas of this theory (Part I), as well as to discuss several applications and connections to other areas (Part II). The volume grew as an extended version of talks given at the conference. The two parts are separated by a paper of K. Kato, an IHES preprint from 1980 which has never been published. We hope that the volume will be a useful introduction and guide to the subject. The contributions to this volume were received over the period November 1999 to August 2000 and the electronic publication date is 10 December 2000. This is the introduction: math.NT/0012131. Other arXiv references are as follows: Part I: Sections 1: math.NT/0012132, 2: math.NT/0012133, A: math.NT/0012134, 4: math.NT/0012135, 5: math.NT/0012136, 6: math.NT/0012137, 7: math.NT/0012138, 8: math.NT/0012139, 9: math.NT/0012140, 10: math.NT/0012141, 11: math.NT/0012142, 12: math.NT/0012143, 13: math.NT/0012144, 14: math.NT/0012145, 15: math.NT/0012146, 16: math.NT/0012147, 17: math.NT/0012148, 18: math.NT/0012149 . Interlude: math.NT/0012150 . Part II: Sections 1: math.NT/0012151, 2: math.NT/0012152, 3: math.NT/0012153, 4: math.NT/0012154, 5: math.NT/0012155, 6: math.NT/0012156, 7: math.NT/0012157, 8: math.NT/0012158, 9: math.NT/0012159, 10: math.NT/0012160 .

math.NT↗
arXiv2016

A characteristic 2 recurrence related to $U_{5}$, with a Hecke algebra application

In arXiv:1603.03910 [math.NT] we introduced some $C_{n}$ in $Z/2[t]$ defined by a linear recurrence and showed that each $C_{n}$, $n\equiv 0 \bmod{4}$, is a sum of $C_{k}$, $k<n$. Combining this with results from arXiv:1508.07523 [math.NT] we proved that the space $K$, consisting of those odd mod~2 modular forms of level $Γ_{0}(3)$ that are annihilated by the operator $U_{3}+I$, has a basis $m_{i,j}$ "adapted to $T_{7}$ and $T_{13}$" in the sense of Nicolas and Serre. (And so the "completed shallow Hecke algebra" attached to $K$ is a power series ring in $T_{7}$ and $T_{13}$.) This note derives analogous results in level $Γ_{0}(5)$. Now $U_{3}+I$ is replaced by $U_{5}+I$, and the operators $T_{7}$ and $T_{13}$ by $T_{3}$ and $T_{7}$. In place of level $Γ_{0}(3)$ results from 1508.07523, we use level $Γ_{0}(5)$ results from arXiv:1603.07085 [math.NT]. A linear recurrence again plays the key role. Now $C_{n+6} = C_{n+5} + (t^{6}+t^{5}+t^{2}+t)C_{n}+t^{n}(t^{2}+t)$, $C_{0}=0$, $C_{1}=C_{2}=1$, $C_{3}=t$, $C_{4}=t^{2}$, $C_{5}=t^{4}+t^{2}+t$, and we prove that each $C_{n}$, $n\equiv 0$ or $2\bmod{6}$ is a sum of $C_{k}$, $k<n$.

math.NT↗
arXiv2001

The impact of the infinite primes on the Riemann hypothesis for characteristic p L-series

In math.NT/9907019 we proposed an analog of the classical Riemann hypothesis for characteristic p valued L-series based on the work of Wan, Diaz-Vargas, Thakur, Poonen, and Sheats for the zeta function $ζ_{\Fr[θ]}(s)$. During the writing of math.NT/9907019, we made two assumptions that have subsequently proved to be incorrect. The first assumption is that we can ignore the trivial zeroes of characteristic p L-series in formulating our conjectures. Instead, we show here how the trivial zeroes influence nearby zeroes and so lead to counter-examples of the original Riemann hypothesis analog. We then sketch an approach to handling such ``near-trivial'' zeroes via Hensel's and Krasner's Lemmas (whereas classically one uses Gamma-factors). Moreover, we show that $ζ_{\Fr[θ]}(s)$ is not representative of general L-series as, surprisingly, all its zeroes are near-trivial, much as the Artin-Weil zeta-function of $\mathbb{P}^1/\Fr$ is not representative of general complex L-functions of curves over finite fields. Consequently, the ``critical zeroes'' (= all zeroes not effected by the trivial zeroes) of characteristic p L-series now appear to be quite mysterious. The second assumption made while writing math.NT/9907019 is that certain Taylor expansions of classical L-series of number fields would exhibit complicated behavior with respect to their zeroes. We present a simple argument that this is not so, and, at the same time, give a characterization of functional equations.

math.NT↗
arXiv2007

Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)

This paper is third in a series of three, following "Summation Formulas, from Poisson and Voronoi to the Present" (math.NT/0304187) and "Distributions and Analytic Continuation of Dirichlet Series" (math.FA/0403030). The first is primarily an expository paper explaining the present one, whereas the second contains some distributional machinery used here as well. These papers concern the boundary distributions of automorphic forms, and how they can be applied to study questions about cusp forms on semisimple Lie groups. The main result of this paper is a Voronoi-style summation formula for the Fourier coefficients of a cusp form on GL(3,Z)\GL(3,R). We also give a treatment of the standard L-function on GL(3), focusing on the archimedean analysis as performed using distributions. Finally a new proof is given of the GL(3)xGL(1) converse theorem of Jacquet, Piatetski-Shapiro, and Shalika. This paper is also related to the later papers math.NT/0402382 and math.NT/0404521.

math.NT↗
arXiv2006

Twisted character of a small representation of GL(4)

We compute by a purely local method the (elliptic) twisted by transpose-inverse character χ_{π_Y} of the representation π_Y=I_{(3,1)}(1_3xχ_Y) of G=GL(4,F), where F is a p-adic field, p not 2, and Y is an unramified quadratic extension of F; χ_Y is the nontrivial character of F^\x/N_{Y/F}Y^x. The representation π_Y is normalizedly induced from \pmatrix m_3&\ast 0&m_1\endpmatrix \mapstoχ_Y(m_1), m_i in GL(i,F), on the maximal parabolic subgroup of type (3,1). We show that the twisted character χ_{π_Y} of π_Y is an unstable function: its value at a twisted regular elliptic conjugacy class with norm in C_Y=``GL(2,Y)/F^x'' is minus its value at the other class within the twisted stable conjugacy class. It is zero at the classes without norm in C_Y. Moreover π_Y is the endoscopic lift of the trivial representation of C_Y. We deal only with unramified Y/F, as globally this case occurs almost everywhere. Naturally this computation plays a role in the theory of lifting of C_Y and GSp(2) to GL(4) using the trace formula. Our work extends -- to the context of nontrivial central characters -- the work of math.NT/0606262, where representations of PGL(4,F) are studied. In math.NT/0606262 a 4-dimensional analogue of the model of the small representation of PGL(3,F) introduced with Kazhdan in a 3-dimensional case is developed, and the local method of computation introduced by us in the 3-dimensional case is extended. As in math.NT/0606262 we use here the classification of twisted (stable) regular conjugacy classes in GL(4,F).

math.NT↗
arXiv2011

Boundary behavior of special cohomology classes arising from the Weil representation

In our previous paper [math.NT/0408050], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p,q)$. This correspondence is realized using theta functions associated to explicitly constructed "special" Schwartz forms. Furthermore, the theta functions give rise to generating series of certain "special cycles" in $X$ with coefficients. In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel-Sere compactification $\bar{X}$ of $X$. However, for the $\Q$-split case for signature $(p,p)$, we have to construct and consider a slightly larger compactification, the "big" Borel-Serre compactification. The restriction to each face of $\bar{X}$ is again a theta series as in [math.NT/0408050], now for a smaller orthogonal group and a larger coefficient system. As application we establish the cohomological nonvanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish.

math.NT↗
arXiv2013

Apéry's theorem and problems for the values of Riemann's zeta function and their $q$-analogues

This monograph is intended to be considered as my habilitation (D.Sc.) thesis; because of that and as everything has already appeared in English, it is performed exclusively in Russian. The monograph comprises a detailed introduction and seven chapters that represent part of my work influenced by Apéry's proof from 1978 of the irrationality of $ζ(2)$ and $ζ(3)$, the values of Riemann's zeta function. Chapter 1 is about "at least one of the four numbers $ζ(5)$, $ζ(7)$, $ζ(9)$ and $ζ(11)$ is irrational" (based in part on arXiv:math.NT/0206176). Chapter 2 explains a connection between the generalized multiple integrals introduced by Beukers in his proof of Apéry's result and the very-well-poised hypergeometric series; it is based on arXiv:math.CA/0206177. Chapter 3 surveys some arithmetic and hypergeometric $q$-analogies and establishes the irrationality measure $μ(ζ_q(2))<3.518876$ for a $q$-analogue of $ζ(2)$; it closely follows the text in Sb. Math. 193 (2002), 1151--1172, but also incorporates the sharper analysis of the hypergeometric construction by Smet and Van Assche (arXiv:0809.2501 [math.CA]) to produce the improvement upon the 2002 result. Chapter 4 is devoted to the measure $μ(ζ(2))<5.095412$ and is based on arXiv:1310.1526 [math.NT]; Chapter 5 is establishing the estimate $||(3/2)^k||>0.5803^k$ for the distance from $(3/2)^k$ to the nearest integer, with the English version published in J. Théor. Nombres Bordeaux 19 (2007), 313--325. Chapter 6 reproduces the solution (from arXiv:math.CA/0311195) to the problem of Asmus Schmidt about generalized Apéry's numbers. Finally, Chapter 7 is about expressing the special $L$-values as periods (in the sense of Kontsevich and Zagier), in particular, as values of hypergeometric functions; it is based on the publication in Springer Proc. Math. Stat. 43 (2013), 381--395.

math.NT↗
arXiv2014

On the distribution of gaps between consecutive primes

Erdös conjectured that the set J of limit points of d_n/logn contains all nonnegative numbers, where d_n denotes the nth primegap. The author proved a year ago (arXiv: 1305.6289) that J contains an interval of type [0,c] with a positive ineffective value c. In the present work we extend this result for a large class of normalizing functions. The only essential requirement is that the function f(n) replacing logn should satisfy f(n)<<lognloglognloglogloglogn/(logloglogn)^2 (with a small implied constant), the well-known Erdös-Rankin bound for the largest known gaps between consecutive primes. The work also proves that apart from a thin set of exceptional functions the original Erdös conjecture holds if logn is replaced by a non-exceptional function f(n). The paper also gives a new proof for a result of Helmut Maier which generalized the Erdös-Rankin bound for an arbitrarily long finite chain of consecutive primegaps. The proof uses a combination of methods of Erdös-Rankin,Maynard-Tao and Banks-Freiberg-Maynard. Since the submission of the present work the very important recent simultaneous and independent works of Ford-Green-Konjagin-Tao (arXiv:1408.4505 [math.NT] and Maynard (aerXiv:1408.5110 [math.NT]) appeared on arXiv and they proved the old conjecture of Erdös which asserts that the lower bound for large gaps exceeds Clognloglognloglogloglogn/(logloglogn)^2 with an arbitrarily large constant C. In this new version we prove the same assertions as in the original work for the case when f(n)<<Clognloglognloglogloglogn/(logloglogn)^2 with an arbi8trarily large constant C, in particular we show that there are blocks of m primes for any m such that all gaps between these primes simultaneously satisfy the lower estimate Clognloglognloglogloglogn/(logloglogn)^2 with an arbitrarily large constant C. The proof uses the method of Maynard.

math.NT↗
arXiv2016

Sparse generalised polynomials

We investigate generalised polynomials (i.e. polynomial-like expressions involving the use of the floor function) which take the value $0$ on all integers except for a set of density $0$. Our main result is that the set of integers where a sparse generalised polynomial takes non-zero value cannot contain a translate of an IP set. We also study some explicit constructions, and show that the characteristic functions of the Fibonacci and Tribonacci numbers are given by generalised polynomails. Finally, we show that any sufficiently sparse $\{0,1\}$-valued sequence is given by a generalised polynomial. (This paper is essentially the first half of our earlier submission arXiv:1610.03900 [math.NT]. Because the material in arXiv:1610.03900 [math.NT] touches upon many different subjects, we believe it is preferable to split it into two independent papers.)

math.NT↗
arXiv2004

Theory of The Generalized Bernoulli-Hurwitz Numbers for The Algebraic Functions of Cyclotomic Type and The Universal Bernoulli Numbers

Hurwitz numbers are the Laurent coefficients of an elliptic function $\wp(u)$ of cyclotomic type, and they are natural generalization of the Bernoulli numbers. This paper gives new generalization of Bernoulli and Hurwitz numbers for higher genus cases. They satisfy completely von Staudt-Clausen type theorem, an extension of von Staudt second theorem, and Kummer type congruence relation. The present paper is revised and combined version of math.NT/0304377 and math.NT/0312178 containing many numerical examples.

math.NT↗
arXiv2004

Galois representations modulo $p$ and cohomology of Hilbert modular varieties

The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let's mention : (1) the control of the image of the Galois representation modulo $p$, (2) Hida's congruence criterion outside an explicit set of primes $p$, and (3) the freeness of the integral cohomology of the Hilbert modular variety over certain local components of the Hecke algebra and the Gorenstein property of these local algebras. We study the arithmetic of the Hilbert modular forms by studying their modulo $p$ Galois representations and our main tool is the action of the inertia groups at the primes above $p$. In order to determine this action, we compute the Hodge-Tate (resp. the Fontaine-Laffaille) weights of the $p$-adic (resp. the modulo $p$) etale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine on the cohomology of the Siegel modular varieties and builds upon the geometric constructions of math.NT/0212071 and math.NT/0212072.

math.NT↗
arXiv2007

On an Argument of Shkredov on Two-Dimensional Corners

Let $\mathbb F_2^n$ be the finite field of cardinality $2 ^{n}$. For all large $n$, any subset $A\subset \mathbb F_2^n\times \mathbb F_2 ^n$ of cardinality \begin{equation*} \abs{A} \gtrsim 4^n \log\log n (\log n) ^{-1} \end{equation*} must contain three points $ \{(x,y) ,(x+d,y) ,(x,y+d)\}$ for $x,y,d\in \mathbb F_2^n$ and $d\neq0$. Our argument is an elaboration of an argument of Shkredov \cite {math.NT/0405406}, building upon the finite field analog of Ben Green \cite {math.NT/0409420}. The interest in our result is in the exponent on $ \log n$, which is larger than has been obtained previously.

math.CO↗
arXiv2007

Modular shadows and the Levy-Mellin infinity-adic transform

This paper continues the study of the structures induced on the ``invisible boundary'' of the modular tower and extends some results of math.NT/0102006. We start with a systematic formalism of pseudo-measures generalizing the well-known theory of modular symbols for SL(2). These pseudo-measures, and the related integral formula which we call the Levy-Mellin transform, can be considered as an ``infinity-adic'' version of Mazur's p-adic measures introduced in the seventies in the theory of p-adic interpolation of Mellin transforms of cusp forms. A formalism of iterated Levy-Mellin transform in the style of math.NT/0502576 is sketched. Finally, we discuss the invisible boundary from the perspective of non-commutative geometry.

math.NT↗
arXiv2023

Spectrum of p-adic linear differential equations I: The shape of the spectrum

This paper extends our previous works arXiv:1802.07306 [math.NT], arXiv:1808.02382 [math.NT] on determining the spectrum, in the Berkovich sense, of ultrametric linear differential equations. Our previous works focused on equations with constant coefficients or over a field of formal power series. In this paper, we investigate the spectrum of $p$-adic differential equations at a generic point on a quasi-smooth curve. This analysis allows us to establish a significant connection between the spectrum and the spectral radii of convergence of a differential equation when considering the affine line. Furthermore, the spectrum offers a more detailed decomposition compared to Robba's decomposition based on spectral radii.

math.NT↗
arXiv2003

Binomial sums related to rational approximations to $ζ(4)$

For the solution $\{u_n\}_{n=0}^\infty$ to the polynomial recursion $(n+1)^5u_{n+1}-3(2n+1)(3n^2+3n+1)(15n^2+15n+4)u_n -3n^3(3n-1)(3n+1)u_{n-1}=0$, where $n=1,2,...$, with the initial data $u_0=1$, $u_1=12$, we prove that all $u_n$ are integers. The numbers $u_n$, $n=0,1,2,...$, are denominators of rational approximations to $ζ(4)$ (see math.NT/0201024). We use Andrews's generalization of Whipple's transformation of a terminating ${}_7F_6(1)$-series and the method from math.NT/0311114.

math.CA↗
arXiv2001

Irrationality of values of zeta-function

We present several results on the number of irrational and linear independent values among $ζ(s),ζ(s+2),...,ζ(s+2n)$, where $s>2$ is an odd integer and $n>0$ is an integer. The main tool in our proofs is a certain generalization of Rivoal's construction (math.NT/0008051, math.NT/0104221).

math.NT↗
arXiv2024

Exact formulae for ranks of partitions

In 2009, Bringmann arXiv:0708.0691 [math.NT] used the circle method to prove an asymptotic formula for the Fourier coefficients of rank generating functions. In this paper, we prove that Bringmann's formula, when summing up to infinity and in the case of prime modulus, gives a Rademacher-type exact formula involving sums of vector-valued Kloosterman sums. As a corollary, in another paper arXiv:2406.07469 [math.NT], we will provide a new proof of Dyson's conjectures by showing that the certain Kloosterman sums vanish.

math.NT↗