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Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Nicholas M. Katz

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.001.0001
Subject:
Mathematics, Number Theory

This book explores an important aspect of number theory—the theory of exponential sums over finite fields and their Mellin transforms—from a new, categorical point of view. The book presents ... More


The Situation over a Finite Field

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0005
Subject:
Mathematics, Number Theory

This chapter deals with the case of a finite field k, and a group scheme 𝓖/k. which is a form of 𝔾ₘ.


SPIN 1/2 CHAINS

Thierry Giamarchi

in Quantum Physics in One Dimension

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198525004
eISBN:
9780191711909
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198525004.003.0006
Subject:
Physics, Condensed Matter Physics / Materials

This chapter examines the question of spin systems, the simplest realization of Luttinger liquids. The peculiarities of one dimension make the solution of this problem quite surprising and different ... More


Convolution of Perverse Sheaves

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0003
Subject:
Mathematics, Number Theory

Let k be a finite field, q its cardinality, p its characteristic, 𝓁 ≠ p a prime number, and 𝓖/k a smooth commutative group scheme which over k¯ becomes isomorphic to 𝔾ₘ/k¯. This chapter is concerned ... More


An <i>O</i>(2<i>n</i>) Example

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0025
Subject:
Mathematics, Number Theory

This chapter works over a finite field k of odd characteristic. Fix an even integer 2n ≤ 4 and a monic polynomial f(x) ɛ k[x] of degree 2n, f(x)=∑i=02nAixi, A2n=1. The following three assumptions are ... More


<i>G</i><sub>2</sub> Examples: Construction in Characteristic Two

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0027
Subject:
Mathematics, Number Theory

This chapter treats the case of characteristic two separately because it is somewhat simpler than the case of odd characteristic. Recall from the first paragraph of Chapter 25 that for k a finite ... More


Introduction

Nicholas M. Katz

in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153308
eISBN:
9781400842704
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153308.003.0001
Subject:
Mathematics, Number Theory

This introductory chapter sets out the book's focus, namely equidistribution results over larger and larger finite extensions of a given finite field. Emanuel Kowalski drew attention to the interest ... More


Modular curves, modular forms, lattices, Galois representations

Bas Edixhoven

in Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691142012
eISBN:
9781400839001
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142012.003.0002
Subject:
Mathematics, Number Theory

This chapter provides the necessary background concerning modular curves and modular forms. It covers modular curves, modular forms, lattices and modular forms, Galois representations attached to ... More


Derivative Methods

Stephan P. A. Sauer

in Molecular Electromagnetism: A Computational Chemistry Approach

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199575398
eISBN:
9780191775109
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199575398.003.0012
Subject:
Physics, Atomic, Laser, and Optical Physics

This chapter presents computational chemistry methods derived by taking derivatives of the approximate expressions for the energy of a molecule in the presence of electric or magnetic fields. It ... More


First description of the algorithms

Jean-Marc Couveignes and Bas Edixhoven

in Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691142012
eISBN:
9781400839001
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142012.003.0003
Subject:
Mathematics, Number Theory

This chapter provides the first, informal description of the algorithms. It explains how the computation of the Galois representations V attached to modular forms over finite fields should proceed. ... More


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