Number Theory - an introduction - NYU Couranttschinke/teaching/Fall05/intro.pdf · 2-adic numbers...

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Number Theory - an introduction September 2005 Number Theory - an introduction

Transcript of Number Theory - an introduction - NYU Couranttschinke/teaching/Fall05/intro.pdf · 2-adic numbers...

Number Theory - an introduction

September 2005

Number Theory - an introduction

Numbers

Integers: Z: ...,−2,−1, 0, 1, 2...Primes: 2, 3, 5, 7, 11, ...Rationals: Q: a/b with a, b ∈ Z and b 6= 0Irrationals: 3

√2, e, π...

Number Theory - an introduction

The queen of mathematics

Number Theory - an introduction

Equations

3x + 5 = 0x2 − Dy2 = 1x2 + y2 = z2

3x3 + 4y3 = 5z3

x3 + 4y3 = 25z3 + 10t3

x4 + 2y4 = z4 + 4t4

Number Theory - an introduction

Questions

I Existence of solutions in Z or Q?

I Qualitative description of the set of solutions: finite, dense?

I Quantitative description: how many solutions?

Number Theory - an introduction

Diophantus of Alexandria

Solutions in Z of x2 + y2 = z2 are given by

x = 2mny = m2 − n2

z = m2 + n2,

with m, n ∈ Z.

Number Theory - an introduction

Pell’s equation: x2 − Dy 2 = 1, for D ∈ Z, D > 0

D = 61 : x = 1766319049 y = 226153980

D = 63 : x = 8 y = 1

D = 73 : x = 2281249 y = 267000

D = 97 : x = 62809633 y = 6377352

D = 99 : x = 10 y = 1

Number Theory - an introduction

Cubic equations

y2 = x3 + Ax + B, A,B ∈ Z

If (x1, y1), (x2, y2) are solutions, with x1 6= x2, then also (x3, y3)where

x3 = −x1 − x2 +(

y1−y2x1−x2

)2

y3 = y1−y2x1−x2

(x1 − x3)− y1.

In particular, if x1, x2, y1, y2 ∈ Q, then also x3, y3.

Number Theory - an introduction

More equations

I Euler (1769):x4 + y4 + z4 = t4 has no nontrivial solutions.(Elkies, 1998):26824404 + 153656394 + 187967604 = 206156734

I Swinnerton-Dyer (2001):x4 + 2y4 = z4 + 4t4 has no nontrivial solutions.(Elsenhans/Jahnel, 2004):4848014 + 2 · 12031204 = 11694074 + 4 · 11575204

Number Theory - an introduction

Hilbert’s problems, Paris 1900

10.“Given a diophantine equation with any number of unknownquantities and with rational integral numerical coefficients: Todevise a process according to which it can be determined by afinite number of operations whether the equation is solvable inrational integers.”

Number Theory - an introduction

Matiyasevich’s theorem (1970)/M. Davis, H. Putnam, J.Robinson

“The solubility of diophantine equations is not decidedable.”

“There is a single equation

F (t, x1, . . . , xn) = 0

with coefficients in Z, which is equivalent to all of (formalmathematics): the statement #t is provable if and only if theabove equation is solvable in x1, . . . , xn ∈ Z.”

Number Theory - an introduction

Dimension 1

axn + byn = czn

with a, b, c ∈ Z, abc 6= 0, and n ≥ 2.

I n = 2 - no solutions or infinitely many solutions

I n = 3 - none, finitely many or infinitely many solutions

I n ≥ 4 - at most finitely many solutions

Number Theory - an introduction

Diophantine geometry: curves

most important geometric invariant: genus g

I g = 0: Gauss, Minkowski, Hurwitz, Hilbert

I g = 1: Poincare, Mordell, Weil, Birch/Swinnerton-Dyer, Wiles

I g ≥ 2: Klein, ..., Faltings

Number Theory - an introduction

Cubic equations - geometry

PQ

P ⊕Q

This is how one “adds” rational points.

Number Theory - an introduction

Bending curves

Number Theory - an introduction

Lattices and curve families

Number Theory - an introduction

Dimension 2

axn + byn = czn + dtn

with a, b, c , d ∈ Z, abcd 6= 0, and n ≥ 2.

I n = 2 - no solutions or a dense set of solutions

I n = 3 - no solutions or a dense set of solutions

I n ≥ 4 - ???

Number Theory - an introduction

Cubic surface

Number Theory - an introduction

Cubic surfaces II

Number Theory - an introduction

E6-lattice

Number Theory - an introduction

E6-cubic I

Number Theory - an introduction

E6-cubic II

Number Theory - an introduction

Quartic surface

Number Theory - an introduction

Quartic surface - sliced

Number Theory - an introduction

Nano-Lattices

Number Theory - an introduction

2-adic numbers

Number Theory - an introduction

Spectral sequence

Number Theory - an introduction

Deformation of a Riemann surface

Number Theory - an introduction

π and e

Number Theory - an introduction

“Imagine this surface is a building. Isn’t one of the lines a goodplace for a balcony and the other one a great place to i nstall an

escalator straight from the bottom to the top?”

Number Theory - an introduction

Plan of the course

1. Primes, primality testing, factorization and applications

2. Reciprocity laws

3. Algebraic number theory

4. Geometry of numbers

5. Diophantine equations of small degrees

6. Diophantine approximations

7. Analytic tools

Number Theory - an introduction