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Beyond Fermats Last Theorem David Zureick-Brown Slides available at http://www.mathcs.emory.edu/~dzb/slides/ February 28, 2020 a 2 + b 2 = c 2 Parimala 1 Quadratic forms 2 Galois cohomology 3 Algebraic groups David Zureick-Brown (Emory

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Beyond Fermat’s Last Theorem

David Zureick-Brown

Slides available at http://www.mathcs.emory.edu/~dzb/slides/

February 28, 2020 a2 + b2 = c2

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Parimala

1 Quadratic forms 2 Galois cohomology 3 Algebraic groups David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 2 / 25

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Suresh Venapally

1 Quadratic forms 2 Galois cohomology David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 3 / 25

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Vicki Powers

1 positive polynomials 2 sums of squares 3 real algebraic geometry 4 mathematics of voting David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 4 / 25

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John Duncan

1 number theory 2 algebra 3 geometry 4 mathematical physics. 5 moonshine David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 5 / 25

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Brooke Ullery (new!)

1 classical algebraic geometry 2 commutative algebra 3 linear series 4 vector bundles David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 6 / 25

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David Zureick-Brown (DZB)

1 Number Theory 2 Arithmetic Geometry 3 Algebraic Geometry 1 p-adic Cohomology 2 Galois Representations 3 Arithmetic of Varieties 4 arithmetic statistics David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 7 / 25

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Basic Problem (Solving Diophantine Equations)

Setup

Let f1, . . . , fm ∈ Z[x1, ..., xn] be polynomials. Let R be a ring (e.g., R = Z, Q).

Problem

Describe the set

(a1, . . . , an) ∈ Rn : ∀i, fi(a1, . . . , an) = 0
.

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 8 / 25

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Basic Problem (Solving Diophantine Equations)

Setup

Let f1, . . . , fm ∈ Z[x1, ..., xn] be polynomials. Let R be a ring (e.g., R = Z, Q).

Problem

Describe the set

(a1, . . . , an) ∈ Rn : ∀i, fi(a1, . . . , an) = 0
.

Fact

Solving diophantine equations is hard.

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 8 / 25

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Hilbert’s Tenth Problem

The ring R = Z is especially hard.

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 9 / 25

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Hilbert’s Tenth Problem

The ring R = Z is especially hard.

Theorem (Davis-Putnam-Robinson 1961, Matijaseviˇ c 1970)

There does not exist an algorithm solving the following problem: input: f1, . . . , fm ∈ Z[x1, ..., xn];

utput: YES / NO according to whether the set
(a1, . . . , an) ∈ Zn : ∀i, fi(a1, . . . , an) = 0
is non-empty.

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 9 / 25

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Hilbert’s Tenth Problem

The ring R = Z is especially hard.

Theorem (Davis-Putnam-Robinson 1961, Matijaseviˇ c 1970)

There does not exist an algorithm solving the following problem: input: f1, . . . , fm ∈ Z[x1, ..., xn];

utput: YES / NO according to whether the set
(a1, . . . , an) ∈ Zn : ∀i, fi(a1, . . . , an) = 0
is non-empty.

This is also known for many rings (e.g., R = C, R, Fq, Qp, C(t)).

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 9 / 25

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Hilbert’s Tenth Problem

The ring R = Z is especially hard.

Theorem (Davis-Putnam-Robinson 1961, Matijaseviˇ c 1970)

There does not exist an algorithm solving the following problem: input: f1, . . . , fm ∈ Z[x1, ..., xn];

utput: YES / NO according to whether the set
(a1, . . . , an) ∈ Zn : ∀i, fi(a1, . . . , an) = 0
is non-empty.

This is also known for many rings (e.g., R = C, R, Fq, Qp, C(t)). This is still open for many other rings (e.g., R = Q).

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 9 / 25

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Fermat’s Last Theorem

Theorem (Wiles et. al)

The only solutions to the equation xn + yn = zn, n ≥ 3 are multiples of the triples (0, 0, 0), (±1, ∓1, 0), ±(1, 0, 1), (0, ±1, ±1).

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 10 / 25

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Fermat’s Last Theorem

Theorem (Wiles et. al)

The only solutions to the equation xn + yn = zn, n ≥ 3 are multiples of the triples (0, 0, 0), (±1, ∓1, 0), ±(1, 0, 1), (0, ±1, ±1). This took 300 years to prove!

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 10 / 25

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Fermat’s Last Theorem

Theorem (Wiles et. al)

The only solutions to the equation xn + yn = zn, n ≥ 3 are multiples of the triples (0, 0, 0), (±1, ∓1, 0), ±(1, 0, 1), (0, ±1, ±1). This took 300 years to prove!

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 10 / 25

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Basic Problem: f1, . . . , fm ∈ Z[x1, ..., xn]

Qualitative:

Does there exist a solution? Do there exist infinitely many solutions? Does the set of solutions have some extra structure (e.g., geometric structure, group structure).

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 11 / 25

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Basic Problem: f1, . . . , fm ∈ Z[x1, ..., xn]

Qualitative:

Does there exist a solution? Do there exist infinitely many solutions? Does the set of solutions have some extra structure (e.g., geometric structure, group structure).

Quantitative

How many solutions are there? How large is the smallest solution? How can we explicitly find all solutions? (With proof?)

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 11 / 25

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Basic Problem: f1, . . . , fm ∈ Z[x1, ..., xn]

Qualitative:

Does there exist a solution? Do there exist infinitely many solutions? Does the set of solutions have some extra structure (e.g., geometric structure, group structure).

Quantitative

How many solutions are there? How large is the smallest solution? How can we explicitly find all solutions? (With proof?)

Implicit question

Why do equations have (or fail to have) solutions? Why do some have many and some have none? What underlying mathematical structures control this?

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 11 / 25

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The Mordell Conjecture

Example

The equation y2 + x2 = 1 has infinitely many solutions.

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 12 / 25

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The Mordell Conjecture

Example

The equation y2 + x2 = 1 has infinitely many solutions.

Theorem (Faltings)

For n ≥ 5, the equation y2 + xn = 1 has only finitely many solutions.

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 12 / 25

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The Mordell Conjecture

Example

The equation y2 + x2 = 1 has infinitely many solutions.

Theorem (Faltings)

For n ≥ 5, the equation y2 + xn = 1 has only finitely many solutions.

Theorem (Faltings)

For n ≥ 5, the equation y2 = f (x) has only finitely many solutions if f (x) is squarefree, with degree > 4.

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 12 / 25

SLIDE 23

Fermat Curves

Question

Why is Fermat’s last theorem believable?

1 xn + yn − zn = 0 looks like a surface (3 variables) 2 xn + yn − 1 = 0 looks like a curve (2 variables) David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 13 / 25

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Mordell Conjecture

Example

y2 = (x2 − 1)(x2 − 2)(x2 − 3) This is a cross section of a two holed torus. The genus is the number of holes.

Conjecture (Mordell)

A curve of genus g ≥ 2 has only finitely many rational solutions.

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 14 / 25

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Fermat Curves

Question

Why is Fermat’s last theorem believable?

1 xn + yn − 1 = 0 is a curve of genus (n − 1)(n − 2)/2. 2 Mordell implies that for fixed n > 3, the nth Fermat equation has

nly finitely many solutions.

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 15 / 25

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Fermat Curves

Question

What if n = 3?

1 x3 + y3 − 1 = 0 is a curve of genus (3 − 1)(3 − 2)/2 = 1. 2 We were lucky; Ax3 + By3 = Cz3 can have infinitely many solutions. David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 16 / 25

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Fermat Surfaces

Conjecture

The only solutions to the equation xn + yn = zn + wn, n ≥ 5 satisfy xyzw = 0 or lie on the lines ‘lines’ x = ±y, z = ±w (and permutations).

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 17 / 25

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Fermat-like equations

Theorem (Poonen, Schaefer, Stoll)

The coprime integer solutions to x2 + y3 = z7 are the 16 triples (±1, −1, 0), (±1, 0, 1), ±(0, 1, 1),

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 18 / 25

SLIDE 29

Fermat-like equations

Theorem (Poonen, Schaefer, Stoll)

The coprime integer solutions to x2 + y3 = z7 are the 16 triples (±1, −1, 0), (±1, 0, 1), ±(0, 1, 1), (±3, −2, 1),

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 18 / 25

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Fermat-like equations

Theorem (Poonen, Schaefer, Stoll)

The coprime integer solutions to x2 + y3 = z7 are the 16 triples (±1, −1, 0), (±1, 0, 1), ±(0, 1, 1), (±3, −2, 1), (±71, −17, 2),

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 18 / 25

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Fermat-like equations

Theorem (Poonen, Schaefer, Stoll)

The coprime integer solutions to x2 + y3 = z7 are the 16 triples (±1, −1, 0), (±1, 0, 1), ±(0, 1, 1), (±3, −2, 1), (±71, −17, 2), (±2213459, 1414, 65), (±15312283, 9262, 113), (±21063928, −76271, 17) .

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 18 / 25

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Generalized Fermat Equations

Problem

What are the solutions to the equation xa + yb = zc?

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 19 / 25

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Generalized Fermat Equations

Problem

What are the solutions to the equation xa + yb = zc?

Theorem (Darmon and Granville)

Fix a, b, c ≥ 2. Then the equation xa + yb = zc has only finitely many coprime integer solutions iff χ = 1

a + 1 b + 1 c − 1 ≤ 0.

µa µb µc

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 19 / 25

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Known Solutions to xa + y b = zc

The ‘known’ solutions with 1 a + 1 b + 1 c < 1 are the following: 1p + 23 = 32 25 + 72 = 34, 73 + 132 = 29, 27 + 173 = 712, 35 + 114 = 1222 177 + 762713 = 210639282, 14143 + 22134592 = 657 92623 + 1531228322 = 1137 438 + 962223 = 300429072, 338 + 15490342 = 156133

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 20 / 25

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Known Solutions to xa + y b = zc

The ‘known’ solutions with 1 a + 1 b + 1 c < 1 are the following: 1p + 23 = 32 25 + 72 = 34, 73 + 132 = 29, 27 + 173 = 712, 35 + 114 = 1222 177 + 762713 = 210639282, 14143 + 22134592 = 657 92623 + 1531228322 = 1137 438 + 962223 = 300429072, 338 + 15490342 = 156133

Problem (Beal’s conjecture)

These are all solutions with 1

a + 1 b + 1 c − 1 < 0.

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 20 / 25

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Generalized Fermat Equations – Known Solutions

Conjecture (Beal, Granville, Tijdeman-Zagier)

This is a complete list of coprime non-zero solutions such that

1 p + 1 q + 1 r − 1 < 0.

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 21 / 25

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Generalized Fermat Equations – Known Solutions

Conjecture (Beal, Granville, Tijdeman-Zagier)

This is a complete list of coprime non-zero solutions such that

1 p + 1 q + 1 r − 1 < 0.

$1,000,000 prize for proof of conjecture...

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 21 / 25

SLIDE 38

Generalized Fermat Equations – Known Solutions

Conjecture (Beal, Granville, Tijdeman-Zagier)

This is a complete list of coprime non-zero solutions such that

1 p + 1 q + 1 r − 1 < 0.

$1,000,000 prize for proof of conjecture... ...or even for a counterexample.

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 21 / 25

SLIDE 39

Examples of Generalized Fermat Equations

Theorem (Poonen, Schaefer, Stoll)

The coprime integer solutions to x2 + y3 = z7 are the 16 triples (±1, −1, 0), (±1, 0, 1), ±(0, 1, 1), (±3, −2, 1), (±71, −17, 2), (±2213459, 1414, 65), (±15312283, 9262, 113), (±21063928, −76271, 17) . 1 2 + 1 3 + 1 7 − 1 = − 1 42 < 0

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 22 / 25

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Examples of Generalized Fermat Equations

Theorem (Poonen, Schaefer, Stoll)

The coprime integer solutions to x2 + y3 = z7 are the 16 triples (±1, −1, 0), (±1, 0, 1), ±(0, 1, 1), (±3, −2, 1), (±71, −17, 2), (±2213459, 1414, 65), (±15312283, 9262, 113), (±21063928, −76271, 17) . 1 2 + 1 3 + 1 7 − 1 = − 1 42 < 0 1 2 + 1 3 + 1 6 − 1 = 0

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 22 / 25

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Examples of Generalized Fermat Equations

Theorem (Darmon, Merel)

Any pairwise coprime solution to the equation xn + yn = z2, n > 4 satisfies xyz = 0. 1 n + 1 n + 1 2 − 1 = 2 n − 1 2 < 0

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 23 / 25

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Examples of Generalized Fermat Equations

Theorem (Klein, Zagier, Beukers, Edwards, others)

The equation x2 + y3 = z5

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 24 / 25

SLIDE 43

Examples of Generalized Fermat Equations

Theorem (Klein, Zagier, Beukers, Edwards, others)

The equation x2 + y3 = z5 1 2 + 1 3 + 1 5 − 1 = 1 30 > 0

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 24 / 25

SLIDE 44

Examples of Generalized Fermat Equations

Theorem (Klein, Zagier, Beukers, Edwards, others)

The equation x2 + y3 = z5 has infinitely many coprime solutions 1 2 + 1 3 + 1 5 − 1 = 1 30 > 0

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 24 / 25

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Examples of Generalized Fermat Equations

Theorem (Klein, Zagier, Beukers, Edwards, others)

The equation x2 + y3 = z5 has infinitely many coprime solutions 1 2 + 1 3 + 1 5 − 1 = 1 30 > 0 (T/2)2 + H3 + (f /123)5

1 f = st(t10 − 11t5s5 − s10), 2 H = Hessian of f , 3 T = a degree 3 covariant of the dodecahedron. David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 24 / 25

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(p, q, r) such that χ < 0 and the solutions to xp + y q = zr have been determined.

{n, n, n} Wiles,Taylor-Wiles, building on work of many others {2, n, n} Darmon-Merel, others for small n {3, n, n} Darmon-Merel, others for small n {5, 2n, 2n} Bennett (2, 4, n) Ellenberg, Bruin, Ghioca n ≥ 4 (2, n, 4) Bennett-Skinner; n ≥ 4 {2, 3, n} Poonen-Shaefer-Stoll, Bruin. 6 ≤ n ≤ 9 {2, 2ℓ, 3} Chen, Dahmen, Siksek; primes 7 < ℓ < 1000 with ℓ = 31 {3, 3, n} Bruin; n = 4, 5 {3, 3, ℓ} Kraus; primes 17 ≤ ℓ ≤ 10000 (2, 2n, 5) Chen n ≥ 3∗ (4, 2n, 3) Bennett-Chen n ≥ 3 (6, 2n, 2) Bennett-Chen n ≥ 3 (2, 6, n) Bennett-Chen n ≥ 3

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 25 / 25

SLIDE 47

(p, q, r) such that χ < 0 and the solutions to xp + y q = zr have been determined.

{n, n, n} Wiles,Taylor-Wiles, building on work of many others {2, n, n} Darmon-Merel, others for small n {3, n, n} Darmon-Merel, others for small n {5, 2n, 2n} Bennett (2, 4, n) Ellenberg, Bruin, Ghioca n ≥ 4 (2, n, 4) Bennett-Skinner; n ≥ 4 {2, 3, n} Poonen-Shaefer-Stoll, Bruin. 6 ≤ n ≤ 9 {2, 2ℓ, 3} Chen, Dahmen, Siksek; primes 7 < ℓ < 1000 with ℓ = 31 {3, 3, n} Bruin; n = 4, 5 {3, 3, ℓ} Kraus; primes 17 ≤ ℓ ≤ 10000 (2, 2n, 5) Chen n ≥ 3∗ (4, 2n, 3) Bennett-Chen n ≥ 3 (6, 2n, 2) Bennett-Chen n ≥ 3 (2, 6, n) Bennett-Chen n ≥ 3 (2, 3, 10) ZB

David Zureick-Brown (Emory University) Beyond Fermat’s Last Theorem February 28, 2020 25 / 25