Geometric Approaches to Solving Diophantine Equations

Home , Geometry of numbers

Geometric approaches to solving Diophantine equations

Alex J. Best

Introduction Geometric approaches to solving Geometry of Numbers

Diophantine equations Rational points on surfaces

Alex J. Best

Tomorrows Mathematicians Today 2013

16-2-2013 I x2 + y 2 = z2 (Pythagorean triples)

I x2 − ny 2 = 1 (Pell’s equation)

I x3 + 48 = y 4

I 9x − 8y = 1

Geometric Introduction approaches to solving Diophantine equations Named for Diophantus of Alexandria (≈ 250AD) Alex J. Best Some Diophantine equations: Introduction

Geometry of Numbers

Rational points on surfaces I x2 − ny 2 = 1 (Pell’s equation)

I x3 + 48 = y 4

I 9x − 8y = 1

Geometric Introduction approaches to solving Diophantine equations Named for Diophantus of Alexandria (≈ 250AD) Alex J. Best Some Diophantine equations: Introduction I 2 2 2 Geometry of x + y = z Numbers Rational points on (Pythagorean triples) surfaces I x3 + 48 = y 4

I 9x − 8y = 1

I x2 − ny 2 = 1 (Pell’s equation) I 9x − 8y = 1

I x2 − ny 2 = 1 (Pell’s equation)

I x3 + 48 = y 4 Geometric Introduction approaches to solving Diophantine equations Named for Diophantus of Alexandria (≈ 250AD) Alex J. Best Some Diophantine equations: Introduction I 2 2 2 Geometry of x + y = z Numbers Rational points on (Pythagorean triples) surfaces

I x2 − ny 2 = 1 (Pell’s equation)

I x3 + 48 = y 4

I 9x − 8y = 1 I Are there any solutions?

I How many are there?

I Can we classify them?

I How can we compute them?

Geometric What do we want to know? approaches to solving Diophantine equations

Alex J. Best

Introduction

Geometry of Numbers

Rational points on surfaces I How many are there?

I Can we classify them?

I How can we compute them?

Geometric What do we want to know? approaches to solving Diophantine equations

Alex J. Best

Introduction

Geometry of Numbers

Rational points on I Are there any solutions? surfaces I Can we classify them?

I How can we compute them?

Geometric What do we want to know? approaches to solving Diophantine equations

Alex J. Best

Introduction

Geometry of Numbers

Rational points on I Are there any solutions? surfaces

I How many are there? I How can we compute them?

Geometric What do we want to know? approaches to solving Diophantine equations

Alex J. Best

Introduction

Geometry of Numbers

Rational points on I Are there any solutions? surfaces

I How many are there?

I Can we classify them? Geometric What do we want to know? approaches to solving Diophantine equations

Alex J. Best

Introduction

Geometry of Numbers

Rational points on I Are there any solutions? surfaces

I How many are there?

I Can we classify them?

I How can we compute them? I Geometry of numbers

I Rational points on surfaces

Geometric In this talk approaches to solving Diophantine equations

Alex J. Best

Introduction

Geometry of Numbers

Rational points on Two general ideas: surfaces I Rational points on surfaces

Geometric In this talk approaches to solving Diophantine equations

Alex J. Best

Introduction

Geometry of Numbers

Rational points on Two general ideas: surfaces

I Geometry of numbers Geometric In this talk approaches to solving Diophantine equations

Alex J. Best

Introduction

Geometry of Numbers

Rational points on Two general ideas: surfaces

I Geometry of numbers

I Rational points on surfaces Geometric Minkowski and the Geometry of Numbers approaches to solving Diophantine equations 1910 - Hermann Minkowski publishes his paper “Geometrie Alex J. Best der Zahlen” and sparks a new ﬁeld called the Geometry of Numbers. Introduction Geometry of Numbers

Rational points on surfaces We can try a few primes, and notice that 22 + 12 = 5, 32 + 22 = 13, 42 + 12 = 17, ... all work. But 7, 11, 19, ... do not. So maybe x2 + y 2 = p when p ≡ 1 (mod 4). Theorem (Fermat’s Christmas Theorem) An odd prime p can be written as p = x2 + y 2 if and only if p ≡ 1 (mod 4).

Geometric A motivating example approaches to solving Diophantine equations

Alex J. Best

Introduction We want to ﬁnd when integers x, y such that x2 + y 2 = p Geometry of where p is prime. Numbers Rational points on surfaces But 7, 11, 19, ... do not. So maybe x2 + y 2 = p when p ≡ 1 (mod 4). Theorem (Fermat’s Christmas Theorem) An odd prime p can be written as p = x2 + y 2 if and only if p ≡ 1 (mod 4).

Geometric A motivating example approaches to solving Diophantine equations

Alex J. Best

Introduction We want to ﬁnd when integers x, y such that x2 + y 2 = p Geometry of where p is prime. Numbers We can try a few primes, and notice that 22 + 12 = 5, Rational points on surfaces 32 + 22 = 13, 42 + 12 = 17, ... all work. So maybe x2 + y 2 = p when p ≡ 1 (mod 4). Theorem (Fermat’s Christmas Theorem) An odd prime p can be written as p = x2 + y 2 if and only if p ≡ 1 (mod 4).

Geometric A motivating example approaches to solving Diophantine equations

Alex J. Best

Introduction We want to ﬁnd when integers x, y such that x2 + y 2 = p Geometry of where p is prime. Numbers We can try a few primes, and notice that 22 + 12 = 5, Rational points on surfaces 32 + 22 = 13, 42 + 12 = 17, ... all work. But 7, 11, 19, ... do not. So maybe x2 + y 2 = p when p ≡ 1 (mod 4). Theorem (Fermat’s Christmas Theorem) An odd prime p can be written as p = x2 + y 2 if and only if p ≡ 1 (mod 4). Geometric What can geometry do for us? approaches to solving Diophantine When p = 5 we can look at points satisfying our criteria: equations

Alex J. Best

Introduction

Geometry of Numbers

Rational points on surfaces Geometric What can geometry do for us? approaches to solving Diophantine When p = 5 we can look at points satisfying our criteria: equations

Alex J. Best

Introduction

Geometry of Numbers

Rational points on surfaces Geometric What can geometry do for us? approaches to solving Diophantine When p = 5 we can look at points satisfying our criteria: equations

Alex J. Best

Introduction

Geometry of Numbers

Rational points on surfaces Geometric What can geometry do for us? approaches to solving Diophantine When p = 5 we can look at points satisfying our criteria: equations

Alex J. Best

Introduction

Geometry of Numbers

Rational points on surfaces Theorem (Minkowski) n If B is a convex symmetric body and Λ a lattice in R then B contains a non-zero point of the lattice if:

Vol(B) > 2d i

Where i is the area of a single cell of the lattice Λ. We can ﬁnd it using determinants, or the order of our lattice as a n subgroup of Z for the more algebraically inclined.

Geometric Minkowski’s ﬁrst theorem approaches to solving Diophantine equations

Alex J. Best n We say a set B in R is symmetric if x ∈ B =⇒ −x ∈ B. Introduction We say a set B is n is convex if R Geometry of x, y ∈ B =⇒ x + λ(y − x) ∈ B for 0 ≤ λ ≤ 1. Numbers Rational points on surfaces Geometric Minkowski’s ﬁrst theorem approaches to solving Diophantine equations

Alex J. Best n We say a set B in R is symmetric if x ∈ B =⇒ −x ∈ B. Introduction We say a set B is n is convex if R Geometry of x, y ∈ B =⇒ x + λ(y − x) ∈ B for 0 ≤ λ ≤ 1. Numbers Rational points on Theorem (Minkowski) surfaces n If B is a convex symmetric body and Λ a lattice in R then B contains a non-zero point of the lattice if:

Vol(B) > 2d i

Where i is the area of a single cell of the lattice Λ. We can ﬁnd it using determinants, or the order of our lattice as a n subgroup of Z for the more algebraically inclined. The area of our lattice cell is p. The area of our circle is π2p So Minkowski tells us that as:

π2p = Vol(B) > 2d i = 22p = 4p

We have a point which satisﬁes x2 + y 2 = kp for some k ≥ 1, and also x2 + y 2 < 2p. So x2 + y 2 = p and we are done.

Geometric Let’s apply it approaches to solving Diophantine equations

Alex J. Best

Introduction 2 A circle in R is symmetric and convex, great! Geometry of Numbers

Rational points on surfaces The area of our circle is π2p So Minkowski tells us that as:

π2p = Vol(B) > 2d i = 22p = 4p

We have a point which satisﬁes x2 + y 2 = kp for some k ≥ 1, and also x2 + y 2 < 2p. So x2 + y 2 = p and we are done.

Geometric Let’s apply it approaches to solving Diophantine equations

Alex J. Best

Introduction 2 A circle in R is symmetric and convex, great! Geometry of The area of our lattice cell is p. Numbers Rational points on surfaces So Minkowski tells us that as:

π2p = Vol(B) > 2d i = 22p = 4p

We have a point which satisﬁes x2 + y 2 = kp for some k ≥ 1, and also x2 + y 2 < 2p. So x2 + y 2 = p and we are done.

Geometric Let’s apply it approaches to solving Diophantine equations

Alex J. Best

Introduction 2 A circle in R is symmetric and convex, great! Geometry of The area of our lattice cell is p. Numbers Rational points on The area of our circle is π2p surfaces Geometric Let’s apply it approaches to solving Diophantine equations

Alex J. Best

Introduction 2 A circle in R is symmetric and convex, great! Geometry of The area of our lattice cell is p. Numbers Rational points on The area of our circle is π2p surfaces So Minkowski tells us that as:

π2p = Vol(B) > 2d i = 22p = 4p

We have a point which satisﬁes x2 + y 2 = kp for some k ≥ 1, and also x2 + y 2 < 2p. So x2 + y 2 = p and we are done. One parametrisation of solutions[1]:

a(s, t) = s7 + s5t2 − 2s3t4 + 3s2t5 + st6 b(s, t) = s6t − 3s5t2 − 2s4t3 + s2t5 + t7 c(s, t) = s7 + s5t2 − 2s3t4 − 3s2t5 + st6 d(s, t) = s6t + 3s5t2 − 2s4t3 + s2t5 + t7

Unfortunately this does not give us all solutions.

Geometric Rational points on surfaces approaches to solving Diophantine equations

Alex J. Best

Euler looked at Introduction 4 4 4 4 A + B = C + D Geometry of Numbers with A, B, C, D ∈ Q Rational points on 4 surfaces This deﬁnes a surface in R . Unfortunately this does not give us all solutions.

Geometric Rational points on surfaces approaches to solving Diophantine equations

Alex J. Best

Euler looked at Introduction 4 4 4 4 A + B = C + D Geometry of Numbers with A, B, C, D ∈ Q Rational points on 4 surfaces This deﬁnes a surface in R . One parametrisation of solutions[1]:

a(s, t) = s7 + s5t2 − 2s3t4 + 3s2t5 + st6 b(s, t) = s6t − 3s5t2 − 2s4t3 + s2t5 + t7 c(s, t) = s7 + s5t2 − 2s3t4 − 3s2t5 + st6 d(s, t) = s6t + 3s5t2 − 2s4t3 + s2t5 + t7 Geometric Rational points on surfaces approaches to solving Diophantine equations

Alex J. Best

Euler looked at Introduction 4 4 4 4 A + B = C + D Geometry of Numbers with A, B, C, D ∈ Q Rational points on 4 surfaces This deﬁnes a surface in R . One parametrisation of solutions[1]:

a(s, t) = s7 + s5t2 − 2s3t4 + 3s2t5 + st6 b(s, t) = s6t − 3s5t2 − 2s4t3 + s2t5 + t7 c(s, t) = s7 + s5t2 − 2s3t4 − 3s2t5 + st6 d(s, t) = s6t + 3s5t2 − 2s4t3 + s2t5 + t7

Unfortunately this does not give us all solutions. A: ∞ Can we ﬁnd a better way of counting them? We want to estimate the density of our solutions.

Geometric Counting solutions approaches to solving Diophantine equations

Alex J. Best

Introduction

Geometry of Numbers

Rational points on Q: How many solutions are there? surfaces Geometric Counting solutions approaches to solving Diophantine equations

Alex J. Best

Introduction

Geometry of Numbers

Rational points on Q: How many solutions are there? surfaces A: ∞ Can we ﬁnd a better way of counting them? We want to estimate the density of our solutions. Then we say height of x is:

H(x) := max{log(|a|), log(|b|)}

Hurrah! There are only ﬁnitely many points with height less than a given value.

Geometric Heights approaches to solving Diophantine equations

Alex J. Best

We want a diﬀerent way of describing the size of a rational Introduction

number. Geometry of Take Numbers a x = ∈ , a, b ∈ Rational points on b Q Z surfaces gcd(a, b) = 1 Hurrah! There are only ﬁnitely many points with height less than a given value.

Geometric Heights approaches to solving Diophantine equations

Alex J. Best

We want a diﬀerent way of describing the size of a rational Introduction

number. Geometry of Take Numbers a x = ∈ , a, b ∈ Rational points on b Q Z surfaces gcd(a, b) = 1 Then we say height of x is:

H(x) := max{log(|a|), log(|b|)} Geometric Heights approaches to solving Diophantine equations

Alex J. Best

We want a diﬀerent way of describing the size of a rational Introduction

number. Geometry of Take Numbers a x = ∈ , a, b ∈ Rational points on b Q Z surfaces gcd(a, b) = 1 Then we say height of x is:

H(x) := max{log(|a|), log(|b|)}

Hurrah! There are only ﬁnitely many points with height less than a given value. Manin and others have conjectured that the growth of N(X , B) is asymptotically governed by geometric properties of the surface for many problems[2].

Geometric Counting points approaches to solving Diophantine equations

Alex J. Best

Introduction We now want to count points in a set X with bounded Geometry of Numbers height. We do this in a simple way and deﬁne: Rational points on surfaces N(X , B) := #{x ∈ X |H(x) ≤ B}

We can analyse the growth of this function as B grows. Geometric Counting points approaches to solving Diophantine equations

Alex J. Best

Introduction We now want to count points in a set X with bounded Geometry of Numbers height. We do this in a simple way and deﬁne: Rational points on surfaces N(X , B) := #{x ∈ X |H(x) ≤ B}

We can analyse the growth of this function as B grows. Manin and others have conjectured that the growth of N(X , B) is asymptotically governed by geometric properties of the surface for many problems[2]. And there is a lot more interplay between these two areas.

Geometric Morals approaches to solving Diophantine equations

Alex J. Best

Introduction

Geometry of Numbers Geometry can help us show solutions exist to Diophantine Rational points on surfaces equations. Geometric properties govern the density of the solutions to some problems. Geometric Morals approaches to solving Diophantine equations

Alex J. Best

Introduction

Geometry of Numbers Geometry can help us show solutions exist to Diophantine Rational points on surfaces equations. Geometric properties govern the density of the solutions to some problems. And there is a lot more interplay between these two areas. Geometric References approaches to solving Diophantine equations

Alex J. Best

Introduction

Geometry of Numbers

Hardy, G.H. and Wright, E.M. and Heath-Brown, D.R. Rational points on and Silverman, J.H., An Introduction to the Theory of surfaces Numbers, Oxford University Press, 2008. Heath-Brown, D.R., Diophantine equations, Algebra, Geometry, Analysis & Logic, Talk at Warwick Mathematics Institute.