# Deformation Theory And The Computation Of Zeta Functions

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### There are many interactions between noncommutative algebra

theory on the one hand and classical algebraic geometry on the other, with important applications in both directions. The aim of this book is to provide a comprehensive introduction to some of the most signiﬁcant topics in this area, including noncom-mutative projective algebraic geometry, deformation theory, symplectic reﬂection

### Computing Zeta Functions of Nondegenerate Curves

An important research topic in computational number theory is the determination of the number of rational points on an algebraic curve C over a ﬁnite ﬁeld Fpn. More generally, one is interested in the computation of its Hasse-Weil zeta function ZC(t) = exp X∞ k=1 #C(Fpnk) tk k! ∈Q[[t]],

### Variation of the Unit Root along the Dwork Family of Calabi

the family in characteristic pvia -adic Fourier transforms. The moment zeta functions of the Dwork family are computed in [11]. In [10], the zeta functions for more general monomial deformations of Fermat type hypersurfaces in weighted projective spaces are studied via Dwork s deformation theory.

### Gross{Zagier reading seminar

A simple computation shows that L(E f=K;s) = X n 1 a nr(n)n s; where r(n) is the number of integral ideals in Kof norm n. In other words, nth coe cient in the above Dirichlet series is the product of the nth coe cient in L(E f=Q;s) and the nth coe cient of the Dedekind zeta function of K. The way to understand this type of product

### AN ALGORITHMIC APPROACH TO THE DWORK FAMILY

to compute Zeta functions using p-adic and ℓ-adic cohomology. In studying the Zeta function using ℓ-adic cohomology, Katz proved that there was a link between more general monomial deformations of Fermat hypersurfaces (of which the Dwork family is an example) and hypergeometric sheaves [20]. Rojas-Leon and Wan, inde-

### On the Zeta Function of a Hypersurface: IV. A Deformation

On the zeta function of a hypersurface: IV. A deformation theory for singular hypersurfaces By BERNARD M. DWORK Let f be a homogeneous form in n + 1 variables with coefficients in a field of characteristic zero. In previous articles we have considered the space a defined by equations similar to (1.1) below. We have shown that if f is the

### Doctoral Degrees Conferred - AMS

computation in the ﬁeld of numbers. Lim, Poon Chuan Adrian,Pathintegrals on a compact manifold with non-negative curvature. Miceli, Brian, A rook theory model for product formulas and poly-Stirling numbers. Newland, Derek, Kernels in the Selberg trace formula on Tk and distributions of the zeroes of the Ihara-zeta function.

### A celebration of the mathematical work of Glenn Stevens

Deformation, Overconvergence, and Euler Systems and their variants The concepts in the title above, and in the chart below, represent some intensely pursued programs in number theory, and will be themes well-covered in this conference They connect to basic constructions and arithmetic questions through the intermediary p-adically varying

### www.cosic.esat.kuleuven.be

Computing Zeta Functions of Nondegenerate Curves W. Castryck1?, J. Denef1 and F. Vercauteren2?? 1 Department of Mathematics, University of Leuven, Celestijnenlaan 200B, B-3001 Leu

### On the Zeta Function of a Hypersurface: II

Sep 27, 2020 On the Zeta Function of a Hypersurface: II* By BERNARD DWORK TABLE OF CONTENTS Introduction?1. Differential operators? 2. Geometrical theory? 3. Dual theory (a) General theory (b) Geometrical application (c) The parameter, s (d) The coefficients of f(X)? 4. Diagonal forms (a) Notation (b) One variable case (c) Several variable case? 5.

### Good Reductionof Affinoids on the Lubin-Tate Tower

Deligne-Carayolwouldthen be reduced tothe computation ofthe zeta functions associated to the components of the reduction of the semistable model. Before stating our main theorem, we introduce some notation. We write X(πn), n≥ 0, for the system of rigid-analytic spaces comprising the Lubin-Tate tower

### X arXiv:1906.06564v1 [math.NT] 15 Jun 2019

3.4. The computation of cohomology 15 3.5. A character sum and the Dwork splitting function 16 3.6. The Bell polynomials and a deformation formula for the Dwork operator ΨX 18 4. Appendix 19 References 22 1. Introduction The goal of this paper is to recapture the deformation theory of the zeta functions of an algebraic variety, which was

### ON ITERATED MAPS OF THE INTERVAL by Institute for Advanced

9. Computation of zeta functions I0. Periodic points of negative type Ii. Smooth deformations of f 12. An intermediate value theorem 13. More on quadratic maps 14. Constructing new functions out of old 469 474 477 482 488 493 497 507 515 521 529 539 545 553 References 560

### PAPER Related content 7ZLVWHGHOOLSWLFPXOWLSOH]HWDYDOXHVDQGQRQ

to classical multiple zeta values. We investigate properties of twisted elliptic multiple zeta values and utilize them in the evaluation of the non-planar part of the four-point one-loop open-superstring amplitude. Keywords: elliptic multiple zeta values, string theory, cyclotomic multiple zeta values, open-string one-loop amplitude

### Introduction

j is a zero of the Selberg zeta function, and the question of how certain L-functions behave at these points appears in the work of Phillips and Sarnak [23] on deformation of cusp forms. These points are also distinguished in the ana-lytic theory of L-functions for leading to conductor dropping. The latter phenomenon

### Low dimensional topology and number theory IX

Deformation varieties of hyperbolic two-bridge link complements and their zeta functions After a brief survey on SL2-character varieties and their zeta functions of hyperbolic 3-manifolds, I will talk on a work in progress about the deformation varieties attached to the canonical decompositions of cer-

### Zeta function for the Laplace operator acting on forms in a

In this paper we obtain the zeta function of the Laplace operator acting on antisymmetric tensor fields defined in a D-dimensional ball with gauge-invariant boundary conditions. Mathematically this computation is quite an imposing challenge, as is proven by the number of erroneous results reported in the literature on this and related computations

### Poles of the complex zeta fu nction of a plane cu rve

zeta function introduced by Denef and Loeser [29], for which more cases are known. The reader is referred to the classical reports of Denef [28] and Igusa [40], and the references therein for the concrete deﬁnitions, results and conjectures in the theory of Igusa s zeta functions. A survey of Meuser [54] includes the more recent developments.

### An open letter concerning Mass matrix transforms in qubit eld

n basis states. For example, an N qubit computation uses n = 2N basis states. The transform is unitary and it may be built from unitary gates, namely the Hadamard gate H = p1 2 ( x + z) and the series Bk = 0 @ 1 0 0 e 2 i 2k 1 A By analogy, a mass computation with 3N basis states uses ternary digits, so the gates Bk would be replaced by gates

### Introduction

zeta functions of a Borcea-Voisin threefold over Q may be obtained in certain cases by computing those of a simpler Calabi-Yau threefold constructed using the twist map, which is amenable to explicit counting and toric methods as shown by by Goto-Kloosterman-Yui [7]. 1. Introduction

### Lectures on the Mass of Topological Solitons Heat kernel/Zeta

of the partition function provides the residua at the poles of the generalized zeta function in terms of the Seeley coeﬃcients of the asymptotic approximation. In this way a formula is derived that allows computation of one-loop mass shifts for kinks, multi-component kinks, and self-dual Abrikosov-Nielsen-Olesen vortices.

### Геометрия и Анализ на Комплексных Алгебраических Многообразиях

Monodromy zeta functions over general toric occur naturally in the context of the deformation theory. Thus in order to classify non- computation of free

### AN INTEGRABLE EVOLUTION EQUATION IN GEOMETRY

deformation always leads to an expansion of the original space, featuring a fast in ationary start. The nonlinear evolution leaves the Laplacian L= D2 invariant so that linear Schr odinger or wave dynamics is not a ected. The expansion has the following e ects: a complex struc-ture can develop and the nonlinear quantum mechanics asymptotically

### Best number theory books pdf - uploads.strikinglycdn.com

Best number theory books pdf P-adic Numbers, p-adic Analysis and Zeta-Functions, (2nd edn.) N. Koblitz, Graduate Text 54, Springer 1996. Algorithmic Number Theory, Vol. 1, E. Bach and J. Shallit , MIT Press, August 1996 Automorphic Forms and Representations, D. Bump, CUP 1996 Notes on Fermat's Last Theorem, A.J. van der Poorten,

### Higher Hasse{Witt matrices Masha Vlasenko

B. Dwork Deformation theory for zeta functions (1962): can one use solutions to the Picard{Fuchs di erential equations to give a p-adic analytic formula for Z(X

### Computing zeta functions in families of curves using deformation

Abstract. We apply deformation theory to compute zeta functions in a family of C a;b curves over a nite eld of small characteristic. The method combines Denef and Vercauteren s extension of Kedlaya s algo-rithm to C a;b curves with Hubrechts recent work on point counting on hyperelliptic curves using deformation. As a result, it is now

### LARGE DEFLECTION OF CANTILEVER BEAMS*

desired accuracy but for practical values the amount of computation required is not excessive. 2 When n = 0, the result reduces to / e'2t tanh tdt = log 2 J. 3 A four-figure table is given in E. Jahnke and F. Emde, Tables oj functions, Dover Publications, New York, 1943, p. 273. LARGE DEFLECTION OF CANTILEVER BEAMS*

### Hartshorne Solutions Chapter 1

Algebraic Curves and Riemann Surfaces Develops the theory of algebraic curves over finite fields, their zeta and L-functions and the theory of algebraic geometric Goppa codes. Algebraic Curves Over Finite Fields First Published in 2018. Routledge is an imprint of Taylor & Francis, an Informa company.

### University of Oxford

A recursive method for computing zeta functions of varieties Alan G.B. Lauder∗ February 16, 2006 1 Introduction We present a method for calculating the zeta function of a smooth

### Symplectic and Poisson Geometry in Interaction with Analysis

theory has contributed new insights to quantum ﬁeld theory. In a related direction, noncommutative geometry has developed an alternative mathematical quantization scheme, based on a geometric approach to operator algebras. Deformation quanti-zation, a blend of symplectic methods and noncommutative geometry, approaches

### A RECURSIVE METHOD FOR COMPUTING ZETA FUNCTIONS OF VARIETIES

yields a method for computing zeta functions which proceeds by induction on the dimension The inductive step com-bines previous work of the author on the deformation of Frobe-nius with a higher rank generalisation of Kedlaya s algorithm. The analysis of the loss of precision during the algorithm uses

### Introduction

tion to the Witten deformation of classical Hodge theory. Let us simply recall that if f : X → R is a smooth function, the associated Witten Laplacian is a one parameter deformation X T of the classical Laplacian X, which coincides with X for T = 0, which also consists of elliptic self-adjoint op

### Description Real Analysis Measure Theory Description

Its focus is the theory of the Rie-mann zeta-function and Dirichlet L-functions, the distribution of prime numbers, and Dirichlet s theorem on primes in arithmetic progressions. The development of the theory and application of Dirichlet series in number theory leads to surprisingly powerful results on the distribution of prime

### AcelebrationofthemathematicalworkofGlennStevens

Deformation, Overconvergence, and Euler Systems and their variants The concepts in the title above, and in the chart below, represent some intensely pursued programs in number theory, and will be themes well-covered in this conference They connect to basic constructions and arithmetic questions through the intermediary p-adically varying

### Symmetric Square L-Functions and Shafarevich-Tate Groups

. , functions attached to motives. Previous conjectures of Deligne, Beilinson and Bloch had predicted these values only up to rational multiples. Bloch and Kato provided strong evidence for their conjecture in the case of the Riemann zeta function, and for the L-functions of complex-multiplication elliptic curves at 5 = 2 (for s > 2 see

### DEFORMATION THEORY AND THE COMPUTATION OF ZETA FUNCTIONS

DEFORMATION THEORY AND THE COMPUTATION OF ZETA FUNCTIONS ALAN G. B. LAUDER 1. Introduction An attractive and challenging problem in computational number theory is to

### Correlation functions of the CFTs on torus with deformation

so on. One example is the four-point functions which are related to out of time order correlator (OTOC), a quantity that can be used to diagnose the chaotic behavior in ﬁeld theory with/without the TT¯ deformation [50 53]. To measure the quantum entan-glement, the computation of entanglement (or R´enyi) entropies involves the correlation

### ANTS XIII Proceedings of the Thirteenth Algorithmic Number

p1=2Co.1/algorithm for computing zeta functions of generic projective hypersurfaces of higher dimension. Tuitman s algorithm has a similar theoretical dependence on the degree of the curve and the degree of the ﬁeld (over Fp) as our algorithm. Throughout, we use a bit complexity model for computation and the notation Oe.x/D S k O.x logk.x//.

### Characteristic 1, entropy and the absolute point

Finally, we test the computation of the zeta function on elliptic curves over the rational numbers. Contents 1. Introduction 2. Working in characteristic 1: Entropy and Witt construction 2.1 Additive structure 2.2 Characteristic p= 2 2.3 Characteristic p= 1 and idempotent analysis 2.4 Witt ring in characteristic p= 1 and entropy

### PHYSICAL REVIEW D 102, 026023 (2020)

field theories with TT¯ deformation. A. Correlation functions in the TT¯-deformed CFTs To obtain the correlation functions of the CFTs with TT¯ deformation on a torus, the procedure is similar to the case in which there is only a single T insertion [78,79], where the correlation functions were derived in the operator formalism.