Gauss composition and integral arithmetic invariant theorydzb/slides/ZBGer-UNC-integral-AIT.pdf ·...

Gauss composition and integral arithmetic invarianttheory

David Zureick-Brown (Emory University)Anton Gerschenko (Google)

Connections in Number TheoryFall 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.


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


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 complicatedcondition].

Composition law?

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


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.


Integers represented by a quadratic form (equivalence)

Equivalence of forms

1 Q(x , y) ∈ Z[x , y ]22 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.


Gauss composition

Theorem (Gauss composition)

The reduced, non-degenerate positive definite forms of discriminant −Dform a finite abelian group, isomorphic to the class group of Q(

√−D).

Example (D = −56)x2 + 14y2, 2x2 + 7y2, 3x2 ± 2xy + 5y2

Remark1 Gauss’s proof was long and complicated; difficult to compute with.

2 Later reformulated by Dirichlet.

3 Much later reformulated by Bhargava.


Bhargava cubes

a b

d c

e f

h g

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Bhargava cubes

a b

d c

e f

h g

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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


Gauss composition via Bhargava cubes

a b

d c

e f

h g

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Q1(x , y) := −Det((

a bd c

)x −

(e fh g

)y)



a b

d c

e f

h g

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Qi (x , y) := −Det (Mix − Niy)



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d c

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Bhargava’s theorem

a b

d c

e f

h g

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Theorem (Bhargava)

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


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

Remark1 14 more (Bhargava)

2 many more (Bhargava-Ho)


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..

Invariants1 C-Invariants: two non-zero forms f , g are C equivalent iff ∆(f ) = ∆(g).

2 Z-Invariants: ∆(f ) = ∆(g) 6⇒ Z equivalence.


Representation theoretic framework

Invariants1 C-Invariants: two non-zero forms f , g are C equivalent iff ∆(f ) = ∆(g).

2 Z-Invariants: ∆(f ) = ∆(g) 6⇒ 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


General representation theoretic framework

Framework1 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


Arithmetic invariant theory

“Is every group a cohomology group

or a Manjul shapedasteroid that fell from the sky?” – Jordan Ellenberg

H1et(SpecZ,ResO/ZGm)



“Is every group a cohomology group or a Manjul shapedasteroid that fell from the sky?” – Jordan Ellenberg




“Is every group a cohomology group or a Manjul shapedasteroid that fell from the sky?” – Jordan Ellenberg



Bhargava–Gross–Wang

Setup

1 f , g ∈ V (Q)

2 M ∈ G (Q) s.t. g = M · f3 σ ∈ Gal(Q/Q)

4 Then g = Mσ · f , so f = M−1Mσ · f , i.e. M−1Mσ ∈ Stabf

Cohomological framework

The mapGal(Q/Q)→ Stabf ; σ 7→ M−1Mσ

is a cocycle, and gives an element of H1(Gal(Q/Q),Stabf ).


Integral arithmetic invariant theory

Remark1 AIT only works for fields; can’t recover Gauss composition

2 Analogue of Galois cohomology is etale cohomology.


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 A2Z


Main Theorem

Theorem (Giraud; Geraschenko-ZB)

Let v ∈ X (S). Then there is a functorial long exact sequence (of groupsand pointed sets)

0→ Stabv (S)→ G (S)g 7→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 classesof v ′ ∈ Orbitv (S) in the same local orbit as v .


Example: Gauss composition revisited

Example (“Gauss”)

G = SL2,Z acts on X = Sym2 A2Z; 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 7→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 ) ∼= PicZ[(∆f +√

∆f )/2] = Cl Q[√

∆f ].


Example: Gauss composition (non-primitive)

Remark

1 If f ∈ Z2 is not primitive, then Stabf is not flat over SpecZ.

2 (Easiest way to not be flat: dim Stabf ,Fp is not constant.)

3 Our machinery does not care; and recovers Gauss composition fornon-primitive forms.


Still to come

More applications wanted.

1 We’re currently iterating through the known literature, derivingparamaterizations where possible.

2 E.g. Delone–Faddeev (ternary cubic forms vs cubic rings): stabilizer isa finite flat group scheme.

3 Future predictive power, especially of degenerate objects/orbits.


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