Gauss composition and integral arithmetic invariant theory
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1 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
2 Sums of Squares Recall (p prime) p = x 2 + y 2 if and only if p = 1 mod 4 or p = 2. For products (x 2 + y 2 )(z 2 + w 2 ) = (xz + yw) 2 + (xw yz) 2 David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
3 Sums of Squares Recall (p prime) p = x 2 + dy 2 if and only if [more complicated condition]. Example p = x 2 + 2y 2 for some x, y Z if and only if p = 2 or p = 1, 3 mod 8. Example p = x 2 + 3y 2 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, / 26
4 Sums of Squares Recall (p prime) p = x 2 + dy 2 if and only if [more complicated condition]. For products (x 2 + dy 2 )(z 2 + dw 2 ) = (xz + dyw) 2 + d(xw yz) 2 David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
5 Integers represented by a quadratic form General quadratic forms (initiated by Lagrange) Recall (p prime) Q(x, y) Z[x, y] 2 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, / 26
6 Sums of Squares (Euler s conjecture) Example p = x y 2 for some x, y Z if and only if ( ) 14 p = 1 and (z 2 + 1) 2 = 8 has a solution mod p. Example p = 2x 2 + 7y 2 for some x, y Z if and only if ( ) 14 p = 1 and (z 2 + 1) 2 8 factors into two irreducible quadratics mod p. David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
7 Integers represented by a quadratic form (equivalence) Equivalence of forms 1 Q(x, y) Z[x, y] 2 2 M SL 2 (Z), Q M (x, y) := Q(ax + by, cx + dy) 3 n Z is represented by Q iff it is represented by Q M. 4 Reduced forms: b a c and b 0 if a = c or a = b. Example 29x xy + 58y 2 x 2 + y 2. David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
8 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) x y 2, 2x 2 + 7y 2, 3x 2 ± 2xy + 5y 2 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, / 26
9 Bhargava cubes a e b f d h c g David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
10 Bhargava cubes a e b f d h c g 1 a, b, d, c, e, f, h, g Z, 2 Cube is really an element of Z 2 Z 2 Z 2, with a natural SL 2 (Z) 3 action David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
11 Gauss composition via Bhargava cubes a e b f h d c g Q 1 (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, / 26
12 Gauss composition via Bhargava cubes a e b f d h c g Q i (x, y) := Det (M i x N i y) David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
13 Gauss composition via Bhargava cubes a e b f d h c g Q i (x, y) := Det (M i x N i y) David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
14 Bhargava s theorem a e b f d h c g Theorem (Bhargava) Q i (x, y) := Det (M i x N i y) Q 1 (x, y) + Q 2 (x, y) + Q 3 (x, y) = 0 David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
15 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, / 26
16 Representation theoretic framework Space of forms 1 The space V of binary quadratic forms is 3-dimensional vector space (resp. R-module). 2 V = Sym 2 C 2 Representations SL 2 (C) Sym 2 C 2 SL 2 (R) Sym 2 R 2 SL 2 (Z) Sym 2 Z 2 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, / 26
17 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) x y 2 is not equivalent to 2x 2 + 7y 2. Fundamental object of study 1 SL 2 (Z)-orbits of an SL 2 (Q)-orbit David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
18 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, / 26
19 Arithmetic invariant theory Is every group a cohomology group H 1 ét (Spec Z, Res O/ZG m ) David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
20 Arithmetic invariant theory Is every group a cohomology group or a Manjul shaped asteroid that fell from the sky? Jordan Ellenberg H 1 ét (Spec Z, Res O/ZG m ) David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
21 Arithmetic invariant theory Is every group a cohomology group or a Manjul shaped asteroid that fell from the sky? Jordan Ellenberg H 1 ét (Spec Z, Res O/ZG m ) David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
22 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 = M σ f, so f = M 1 M σ f, i.e. M 1 M σ Stab f Cohomological framework The map Gal(Q/Q) Stab f ; σ M 1 M σ is a cocycle, and gives an element of H 1 (Gal(Q/Q), Stab f ). David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
23 Integral arithmetic invariant theory Remark 1 AIT only works for fields; can t recover Gauss composition 2 Analogue of Galois cohomology is étale cohomology. David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
24 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 = SL 2,Z, acting on X = Sym 2 A 2 Z David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
25 Main Theorem Theorem (Giraud; Geraschenko-ZB) Let v X (S). Then there is a functorial long exact sequence (of groups and pointed sets) 0 Stab v (S) G(S) g g v Orbit v (S) H 1 (S, Stab v ) H 1 (S, G). If Stab v is commutative, then is a group. Orbit v (S)/G(S) = ker ( H 1 (S, Stab v ) H 1 (S, G) ) Remark The image Orbit v (S)/G(S) of X (S) is the set of G(S) equivalence classes of v Orbit v (S) in the same local orbit as v. David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
26 Example: Gauss composition revisited Example ( Gauss ) G = SL 2,Z acts on X = Sym 2 A 2 Z ; Stab v is a non-split torus (thus commutative). Let f X (Z) be a primitive (non-zero mod all p) integral quadratic form. 0 Stab v (Z) SL 2 (Z) g g f Orbit f (Z) H 1 (Z, Stab v ) H 1 (Z, SL 2 ). Remark 1 H 1 (Z, SL 2 ) = 0 (this is Hilbert s theorem 90). 2 Orbit f (Z)/ SL 2 (Z) = integral equivalence classes of primitive forms with the same discriminantn. 3 H 1 (Z, Stab v ) = Orbit f (Z)/ SL 2 (Z). 4 H 1 (Z, Stab v ) = Pic Z[( f + f )/2] = Cl Q[ f ]. David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, / 26
27 Example: Gauss composition (non-primitive) Remark 1 If f Z 2 is not primitive, then Stab f is not flat over Spec Z. 2 (Easiest way to not be flat: dim Stab f,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, / 26
28 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, / 26
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