Gauss composition and integral arithmetic invariant theory David - - PowerPoint PPT Presentation

Gauss composition and integral arithmetic invariant theory

David Zureick-Brown (Emory University) Anton Gerschenko (Google) Connections in Number Theory Fall Southeastern Sectional Meeting University of North Carolina at Greensboro, Greensboro, NC Nov 8, 2014

Sums of Squares

Recall (p prime)

p = x2 + y2 if and only if p = 1 mod 4 or p = 2.

For products

(x2 + y2)(z2 + w2) = (xz + yw)2 + (xw − yz)2

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 2 / 26

Sums of Squares

Recall (p prime)

p = x2 + dy2 if and only if [more complicated condition].

Example

p = x2 + 2y2 for some x, y ∈ Z if and only if p = 2 or p = 1, 3 mod 8.

Example

p = x2 + 3y2 for some x, y ∈ Z if and only if p = 3 or p = 1 mod 3.

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 3 / 26

Sums of Squares

Recall (p prime)

p = x2 + dy2 if and only if [more complicated condition].

For products

(x2 + dy2)(z2 + dw2) = (xz + dyw)2 + d(xw − yz)2

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 4 / 26

Integers represented by a quadratic form

General quadratic forms (initiated by Lagrange)

Q(x, y) ∈ Z[x, y]2

Recall (p prime)

p = Q(x, y) for some x, y ∈ Z if and only if [more complicated condition].

Composition law?

Q(x, y)Q(z, w) = Q(a, b)

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 5 / 26

Sums of Squares (Euler’s conjecture)

Example

p = x2 + 14y2 for some x, y ∈ Z if and only if −14

p

• = −1 and

(z2 + 1)2 = 8 has a solution mod p.

Example

p = 2x2 + 7y2 for some x, y ∈ Z if and only if −14

p

• = −1 and

(z2 + 1)2 − 8 factors into two irreducible quadratics mod p.

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 6 / 26

Integers represented by a quadratic form (equivalence)

Equivalence of forms

1 Q(x, y) ∈ Z[x, y]2 2 M ∈ SL2(Z), QM(x, y) := Q(ax + by, cx + dy) 3 n ∈ Z is represented by Q iff it is represented by QM. 4 Reduced forms: |b| ≤ a ≤ c and b ≥ 0 if a = c or a = |b|.

Example

29x2 + 82xy + 58y2 ∼ x2 + y2.

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 7 / 26

Gauss composition

Theorem (Gauss composition)

The reduced, non-degenerate positive definite forms of discriminant −D form a finite abelian group, isomorphic to the class group of Q( √ −D).

Example (D = −56)

x2 + 14y2, 2x2 + 7y2, 3x2 ± 2xy + 5y2

Remark

1 Gauss’s proof was long and complicated; difficult to compute with. 2 Later reformulated by Dirichlet. 3 Much later reformulated by Bhargava. David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 8 / 26

Bhargava cubes

a b d c e f h g

• David Zureick-Brown (Emory University)

Gauss composition and integral AIT Nov 8, 2014 9 / 26

Bhargava cubes

a b d c e f h g

• 1 a, b, d, c, e, f , h, g ∈ Z,

2 Cube is really an element of Z2 ⊗ Z2 ⊗ Z2, with a natural SL2(Z)3

action

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 10 / 26

Gauss composition via Bhargava cubes

a b d c e f h g

• Q1(x, y) := −Det

a b

d c

• x −

e f

h g

• y
• David Zureick-Brown (Emory University)

Gauss composition and integral AIT Nov 8, 2014 11 / 26

Gauss composition via Bhargava cubes

a b d c e f h g

• Qi(x, y) := −Det (Mix − Niy)

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 12 / 26

Gauss composition via Bhargava cubes

a b d c e f h g

• Qi(x, y) := −Det (Mix − Niy)

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 13 / 26

Bhargava’s theorem

a b d c e f h g

• Qi(x, y) := −Det (Mix − Niy)

Theorem (Bhargava)

Q1(x, y) + Q2(x, y) + Q3(x, y) = 0

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 14 / 26

Lots of parameterizations

Example

binary cubic forms ↔ cubic fields pairs (ternary, quadratic) forms ↔ quartic fields quadruples of quinary ↔ quintic fields alternating bilinear forms binary quartic forms ↔

2-Selmer elements of Elliptic curves

Remark

1 14 more (Bhargava) 2 many more (Bhargava-Ho) David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 15 / 26

Representation theoretic framework

Space of forms

1 The space V of binary quadratic forms is 3-dimensional vector space

(resp. R-module).

2 V = Sym2 C2

Representations

SL2(C) Sym2 C2 SL2(R) Sym2 R2 SL2(Z) Sym2 Z2 etc..

Invariants

1

C-Invariants: two non-zero forms f , g are C equivalent iff ∆(f ) = ∆(g).

2

Z-Invariants: ∆(f ) = ∆(g) ⇒ Z equivalence.

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 16 / 26

Representation theoretic framework

Invariants

1

C-Invariants: two non-zero forms f , g are C equivalent iff ∆(f ) = ∆(g).

2

Z-Invariants: ∆(f ) = ∆(g) ⇒ Z equivalence.

Example (D = −14 · 4)

x2 + 14y2 is not equivalent to 2x2 + 7y2.

Fundamental object of study

1 SL2(Z)-orbits of an SL2(Q)-orbit David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 17 / 26

General representation theoretic framework

Framework

1 V = free R module 2 G V 3 R → R′ ring extension 4 v ∈ V (R)

Goal

Understand the G(R)-orbits of the G(R′)-orbit of v

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 18 / 26

Arithmetic invariant theory

“Is every group a cohomology group H1

´ et(Spec Z, ResO/ZGm)

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 19 / 26

Arithmetic invariant theory

“Is every group a cohomology group

• r a Manjul shaped

asteroid that fell from the sky?” – Jordan Ellenberg H1

´ et(Spec Z, ResO/ZGm)

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 19 / 26

Arithmetic invariant theory

“Is every group a cohomology group

• r a Manjul shaped

asteroid that fell from the sky?” – Jordan Ellenberg H1

´ et(Spec Z, ResO/ZGm)

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 19 / 26

Bhargava–Gross–Wang

Setup

1 f , g ∈ V (Q) 2 M ∈ G(Q) s.t. g = M · f 3 σ ∈ Gal(Q/Q) 4 Then g = Mσ · f , so f = M−1Mσ · f , i.e. M−1Mσ ∈ Stabf

Cohomological framework

The map Gal(Q/Q) → Stabf ; σ → M−1Mσ is a cocycle, and gives an element of H1(Gal(Q/Q), Stabf ).

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 20 / 26

Integral arithmetic invariant theory

Remark

1 AIT only works for fields; can’t recover Gauss composition 2 Analogue of Galois cohomology is ´

etale cohomology.

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 21 / 26

Integral arithmetic invariant theory – setup

Setup

1 S any base (e.g. Z); 2 G/S any group scheme (not necessarily smooth, or even flat); 3 X (usually a vector space); 4 G X an action.

Example (“Gauss”)

G = SL2,Z, acting on X = Sym2 A2

Z

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 22 / 26

Main Theorem

Theorem (Giraud; Geraschenko-ZB)

Let v ∈ X(S). Then there is a functorial long exact sequence (of groups and pointed sets) 0 → Stabv(S) → G(S)

g→g·v

− − − − → Orbitv(S) → H1(S, Stabv) → H1(S, G). If Stabv is commutative, then Orbitv(S)/G(S) ∼ = ker

• H1(S, Stabv) → H1(S, G)
• is a group.

Remark

The image Orbitv(S)/G(S) of X(S) is the set of G(S) equivalence classes

• f v′ ∈ Orbitv(S) in the same local orbit as v.

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 23 / 26

Example: Gauss composition revisited

Example (“Gauss”)

G = SL2,Z acts on X = Sym2 A2

Z; Stabv is a non-split torus (thus commutative).

Let f ∈ X(Z) be a primitive (non-zero mod all p) integral quadratic form. 0 → Stabv(Z) → SL2(Z)

g→g·f

− − − − → Orbitf (Z) → H1(Z, Stabv) → H1(Z, SL2).

Remark

1 H1(Z, SL2) = 0 (this is Hilbert’s theorem 90). 2 Orbitf (Z)/ SL2(Z) = integral equivalence classes of primitive forms with

the same discriminantn.

3

H1(Z, Stabv) ∼ = Orbitf (Z)/ SL2(Z).

4

H1(Z, Stabv) ∼ = Pic Z[(∆f + √∆f )/2] = Cl Q[√∆f ].

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 24 / 26

Example: Gauss composition (non-primitive)

Remark

1 If f ∈ Z2 is not primitive, then Stabf is not flat over Spec Z. 2 (Easiest way to not be flat: dim Stabf ,Fp is not constant.) 3 Our machinery does not care; and recovers Gauss composition for

non-primitive forms.

David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 25 / 26

Still to come

More applications wanted.

1 We’re currently iterating through the known literature, deriving

paramaterizations where possible.

2 E.g. Delone–Faddeev (ternary cubic forms vs cubic rings): stabilizer is

a finite flat group scheme.

3 Future predictive power, especially of degenerate objects/orbits. David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 26 / 26