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.
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 complicatedcondition].
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 ]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.
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 −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.
David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 8 / 26
Bhargava cubes
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h g
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David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 9 / 26
Bhargava cubes
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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
David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 10 / 26
Gauss composition via Bhargava cubes
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d c
e f
h g
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Q1(x , y) := −Det((
a bd c
)x −
(e fh g
)y)
David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 11 / 26
Gauss composition via Bhargava cubes
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d c
e f
h g
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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
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d c
e f
h g
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Qi (x , y) := −Det (Mix − Niy)
David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 13 / 26
Bhargava’s theorem
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d c
e f
h g
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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
Remark1 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..
Invariants1 C-Invariants: two non-zero forms f , g are C equivalent iff ∆(f ) = ∆(g).
2 Z-Invariants: ∆(f ) = ∆(g) 6⇒ Z equivalence.
David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 16 / 26
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
David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 17 / 26
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
David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 18 / 26
Arithmetic invariant theory
“Is every group a cohomology group
or a Manjul shapedasteroid that fell from the sky?” – Jordan Ellenberg
H1et(SpecZ,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 or a Manjul shapedasteroid that fell from the sky?” – Jordan Ellenberg
H1et(SpecZ,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 or a Manjul shapedasteroid that fell from the sky?” – Jordan Ellenberg
H1et(SpecZ,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 · 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 ).
David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 20 / 26
Integral arithmetic invariant theory
Remark1 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 A2Z
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 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 .
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 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 ].
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 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.
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, 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.
David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 26 / 26