## Abelprisen.no

**Abel Prize 2010**

**Number Theory; the **

**ground of John Tate**

**Algebraic integers, finite fields, p-adic numbers. Class field theory, rigid analytic spaces, **

**elliptic curves. These are some of the concepts you must know if your task is to describe **

**the mathematical achievements of John Torrence Tate, the Abel Laureate 2010. **

**If you don' t know anything about any of the listed words, you can still share something **

**with one of the most brilliant scientist of our time; the fascination for the natural num-**

**bers. At the very first glance, they look innocent and easily accessible. Counting, 1, 2, 3, **

**., or computing, 1+2=3, 3+5=8, it's a childs game. But as you learn more about them, **

**you realize that the world you are diving into is huge, mysteriuos and unpredictable.**

**"Mathematics is the queen of the sciences and **

**number theory is the queen of mathematics." **

*Carl Friedrich Gauss*
**"Number is the **

**within of all **

*Pythagoras of Samos*
**"God invented the integers; all **

**else is the work of man."**

*Leopold Kronecker*
**Abel Prize 2010**

**Number theory is the study of the proper-**

**ties of numbers in general, and integers in **

**particular. Number theory may be subdi-**

**vided into several fields, according to the **

**methods used and the type of questions **

**Elementary number theory**

In elementary number theory, integers are studied
without use of techniques from other mathematical
fields. Important discoveries of this field are Fermat's
little theorem, Euler's theorem, the Chinese remain-
**Algebraic number theory**

der theorem and the law of quadratic reciprocity, to
In algebraic number theory, the concept of a number mention a few.

is expanded to algebraic numbers, i.e. roots of pol-
ynomials with rational coefficients. These domains **Analytic number theory**

contain elements analogous to the integers, the so-
Analytic number theory employs the machinery of
called algebraic integers, which is a main subject of calculus and complex analysis to tackle questions
study in this field.
about integers. Examples include the prime number
Many number theoretic questions are attacked by theorem concerning the asymptotic behavior of the
reduction modulo p for various primes p. This lo-
primes and the Riemann hypothesis, but also proofs
calization procedure leads to the construction of the of the transcendence of π or e, are classified as ana-
p-adic numbers, another main subject of study in lytical number theory.
the field of algebraic number theory.

**Arithmetic algebraic geometry**

Arithmetic (algebraic) geometry is the study of
schemes of finite type over the spectrum of the ring
of integers **Z**.

Diophantine geometry is the study of algebraic vari-
eties over number fields.

John Tates investigations
**Combinatorial number theory**

mainly belong to the subfield
Combinatorial number theory deals with number the-
**Algebraic number theory. **

oretic problems which involve combinatorial ideas
in their formulations or solutions. Paul Erdös is the
main founder of this branch of number theory. Exam-
ples are the problems of finding arithmetic progres-
sions in a set of integers.

Other subfields of number
Modular forms are analytic functions on the upper
half-plane satisfying a certain kind of functional
equation and a growth condition. The theory of mod-
ular forms therefore belongs to complex analysis but
the main importance of the theory has traditionally
been in its connections with number theory.
**Abel Prize 2010**

**John Tate´s influence in modern number **

**theory can be read out of the numerous **

**results and concepts named after him. **

**Here are some of them:**

* Hodge-Tate theory;* p-adic analogue of the Hodge

decomposition for complex cohomology.

*; a Galois module constructed*

**Tate module**from an abelian variety over a field The

*is the*

**Lubin–Tate formal group law**unique (1-dimensional) formal group law F such that e(x) = px + xp is an endomorphism of F, i.e. such that e(F(x,y)) = F(e(x), e(y)) The

*says that abelian varie-*

**Tate Isogeny theorem**ties with isomorphic Tate modules are isogenous.

The

*is a statistical state-*

**Sato–Tate conjecture**ment about the family of elliptic curves E over the finite field with p elements, with p a prime number, obtained from an elliptic curve E over the rational

*a particular abelian group with an*

**Tate twist;**number field, by the process of reduction modulo a action of a Galois group constructed from a prime for almost all p.

In the theory of elliptic curves,

**Tate's algo-***is the tensor inverse of the Lefschetz*

**Tate motive***takes as input an integral model of an*

**rithm,**elliptic curve E over

**Q**and a prime p. It returns

the exponent f of p in the conductor of E, the type of reduction at p, and the local index c .p The

*, named for Tate*

**Tate-Shafarevich group**and Igor Shafarevich, of an abelian variety de- Via the

*one can control (part*

**Serre-Tate theorem**fined over a number field K consists of the ele- of) the char p deformations of an abelian scheme ments of the Weil-Châtelet group that become coming from the local part of the Barsotti-Tate trivial in all of the completions of K

*arise in "nature"*

**Barsotti-Tate groups;**when one consider the sequence of kernels

*(or canonical height) is a quad-*

**Néron–Tate height**of multiplication by successive powers of p ratic form on the Mordell-Weil group of rational on an abelian variety.

points of an abelian variety defined over a global

*are a slightly modified*

**Tate cohomology groups**form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence.

*; i.e. classification of abelian*

**Honda-Tate theory**varieties over finite fields up to isogeny.

**Abel Prize 2010**

**A fundamental result in number theory**

**is the Unique-Prime-Factorization Theo-**

**rem for integers. In 1847, in an attempt**

**to prove Fermats Last Theorem, Gabriel**

**Lamé incorrectly assumed that this prop-**

**erty holds in general. He was immediate-**

**ly corrected by Joseph Liouville who re-**

**The Fundamental Theorem of Arithmetic****ferred to results of Ernst Kummer about**

The Fundamental Theorem of Arithmetic (or

**failure of unique prime factorization in**

Unique-Prime-Factorization Theorem) states

**certain rings of algebraic integers, pub-**

that any integer greater than 1 can be written as a

**lished in 1843.**

unique product (up to ordering of the factors) of

**This innocent little dispute became the**

prime numbers. Intuitively, this theorem character-

**origin of a branch of number theory, in**

izes prime numbers uniquely in the sense that they are the "fundamental numbers." The theorem was

**which John Tate has been a main figure**

practically proved by Euclid but the first full and

**during the last 50 years.**

correct proof is found in the Disquisitiones Arith- meticae by Carl Friedrich Gauss.

**Prime factorization in**

**algebraic number fields**A fundamental property of the integers is the unique In number theory the starting points is the set of in- factorization property. There is only on way of writ- tegers, ., -3, -2, -1, 0, 1, 2, 3, ., denoted by

**Z**. The

ing 105 as a product of primes (105=3·5·7) when we integers are included in the rational numbers

**Q**, i.e.

do not bother about the order of the factors. But in all fractions of integers, where the denominator is a general algebraic number field this is no longer different from 0. Unfortunately the number �2, de- true. The favourite example for (nearly) all math- fined as the root of the polynomial equation x2-2=0 is ematicians is the extension of

**Q**by the square root

not included in

**Q**, as observed by the Pythagoreans

of -5. (If you have bad feelings for the square root around 400 BC. Nevertheless, we are interested in of a negative number, just close your eyes and keep studying the properties of �2, so we extend

**Q**by �2

walking. You will get used to it.) In this extension, to obtain our first homemade algebraic number field, or rather the integral part of it, the number 6 has two denoted

**Q**(�2), consisting of all number which can

different prime factorizations.

be written as a+b�2, for rational numbers a and b.

6=2·3=(1+ �-5)·(1- �-5) The number �2 is defined as the solution of certain All factors involved are prime numbers, i.e. only di- polynomial equation. One can in fact show that all visible by 1 and itself. numbers in

**Q**(�2) satisfy some polynomial equation.

The extent to which unique factorization fails in the Not the same one for all, but at least one for each. ring of integers of an algebraic number field can be Some rational numbers are integers, and some of the described by a certain group known as a class group. algebraic numbers are algebraic integers. The way If the class group is finite, then the order of the group we decide if a number should be called an integer, is is called the class number. The class group of the as follows; we look at the monic polynomial equa- algebraic integers of an algebraic number field is tion for the number (monic means that the coefficient trivial if and only if the ring of algebraic integers of the highest degree term is 1). If all the coefficients has the unique factorization property. The size of the are integers, all roots of the equation are algebraic ideal class group can thus be considered as a meas- integers, If not, they are not! Examples of algebraic ure for the deviation from being a unique factoriza- integers in

**Q**(�2) are �2 (root of the polynomial x2-

tion domain.

2) and 1+�2 (root of x2-2x-1).

Source: http://www.abelprisen.no/nedlastning/2010/tate_en.pdf

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