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 = 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, 2014 2 / 26
Sums of Squares Recall ( p prime) p = x 2 + dy 2 if and only if [more complicated condition] . Example p = x 2 + 2 y 2 for some x , y ∈ Z if and only if p = 2 or p = 1 , 3 mod 8. Example p = x 2 + 3 y 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, 2014 3 / 26
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, 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 = x 2 + 14 y 2 for some x , y ∈ Z if and only if � − 14 � = − 1 and p ( z 2 + 1) 2 = 8 has a solution mod p . Example p = 2 x 2 + 7 y 2 for some x , y ∈ Z if and only if � − 14 � = − 1 and p ( z 2 + 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 ∈ 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 29 x 2 + 82 xy + 58 y 2 ∼ x 2 + y 2 . 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) x 2 + 14 y 2 , 2 x 2 + 7 y 2 , 3 x 2 ± 2 xy + 5 y 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, 2014 8 / 26
Bhargava cubes e f � � � � a b g h � � � � c d David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 9 / 26
Bhargava cubes e f � � � � a b g h � � � � c d 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, 2014 10 / 26
Gauss composition via Bhargava cubes e f � � � � a b g h � � � � c d �� a b � e f � � � Q 1 ( x , y ) := − Det x − y d c h g David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 11 / 26
Gauss composition via Bhargava cubes e f � � � � a b g h � � � � c d Q i ( x , y ) := − Det ( M i x − N i y ) David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 12 / 26
Gauss composition via Bhargava cubes e f � � � � a b g h � � � � c d Q i ( x , y ) := − Det ( M i x − N i y ) David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 13 / 26
Bhargava’s theorem e f � � � � a b g h � � � � c d Q i ( x , y ) := − Det ( M i x − N i y ) Theorem (Bhargava) 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, 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 = 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 C -Invariants : two non-zero forms f , g are C equivalent iff ∆( f ) = ∆( g ). 1 Z -Invariants : ∆( f ) = ∆( g ) �⇒ Z equivalence. 2 David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 16 / 26
Representation theoretic framework Invariants C -Invariants : two non-zero forms f , g are C equivalent iff ∆( f ) = ∆( g ). 1 Z -Invariants : ∆( f ) = ∆( g ) �⇒ Z equivalence. 2 Example ( D = − 14 · 4) x 2 + 14 y 2 is not equivalent to 2 x 2 + 7 y 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, 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 H 1 et (Spec Z , Res O / Z G m ) ´ 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 shaped asteroid that fell from the sky?” – Jordan Ellenberg H 1 et (Spec Z , Res O / Z G m ) ´ 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 shaped asteroid that fell from the sky?” – Jordan Ellenberg H 1 et (Spec Z , Res O / Z G m ) ´ 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 = 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, 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 = SL 2 , Z , acting on X = Sym 2 A 2 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) g �→ g · v → Orbit v ( S ) → H 1 ( S , Stab v ) → H 1 ( S , G ) . 0 → Stab v ( S ) → G ( S ) − − − − If Stab v is commutative, then Orbit v ( S ) / G ( S ) ∼ H 1 ( S , Stab v ) → H 1 ( S , G ) � � = ker is a group. 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, 2014 23 / 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. g �→ g · f → Orbit f ( Z ) → H 1 ( Z , Stab v ) → H 1 ( Z , SL 2 ) . 0 → Stab v ( Z ) → SL 2 ( Z ) − − − − 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. H 1 ( Z , Stab v ) ∼ = Orbit f ( Z ) / SL 2 ( Z ). 3 = Pic Z [(∆ f + √ ∆ f ) / 2] = Cl Q [ √ ∆ f ]. H 1 ( Z , Stab v ) ∼ 4 David Zureick-Brown (Emory University) Gauss composition and integral AIT Nov 8, 2014 24 / 26
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