Introduction Modular forms Universality theorems Overview • “There are five fundamental operations of arithmetic: addition, subtraction, multiplication, division, and modular forms.” (Attributed to Martin Eichler.) Jeremy Rouse Quadratic forms 11/45
Introduction Modular forms Universality theorems Definitions • A modular form of weight k , level N and character χ is a holomorphic function f : H → C so that � az + b � = χ ( d )( cz + d ) k f ( z ) f cz + d � a � b for all ∈ Γ 0 ( N ). c d Jeremy Rouse Quadratic forms 12/45
Introduction Modular forms Universality theorems Definitions • A modular form of weight k , level N and character χ is a holomorphic function f : H → C so that � az + b � = χ ( d )( cz + d ) k f ( z ) f cz + d � a � b for all ∈ Γ 0 ( N ). c d • Let M k (Γ 0 ( N ) , χ ) denote the C -vector space of such modular forms, and S k (Γ 0 ( N ) , χ ) the subspace of cusp forms. Jeremy Rouse Quadratic forms 12/45
Introduction Modular forms Universality theorems Definitions • A modular form of weight k , level N and character χ is a holomorphic function f : H → C so that � az + b � = χ ( d )( cz + d ) k f ( z ) f cz + d � a � b for all ∈ Γ 0 ( N ). c d • Let M k (Γ 0 ( N ) , χ ) denote the C -vector space of such modular forms, and S k (Γ 0 ( N ) , χ ) the subspace of cusp forms. • These vector spaces are finite-dimensional! Jeremy Rouse Quadratic forms 12/45
Introduction Modular forms Universality theorems Theta series • Let Q be a quaternary quadratic form and let x ∈ Z 4 : Q ( � r Q ( n ) = # { � x ) = n } be the number of representations of n by Q . Jeremy Rouse Quadratic forms 13/45
Introduction Modular forms Universality theorems Theta series • Let Q be a quaternary quadratic form and let x ∈ Z 4 : Q ( � r Q ( n ) = # { � x ) = n } be the number of representations of n by Q . • Define ∞ � r Q ( n ) q n , q = e 2 π iz . θ Q ( z ) = n =0 Jeremy Rouse Quadratic forms 13/45
Introduction Modular forms Universality theorems Theta series • Let Q be a quaternary quadratic form and let x ∈ Z 4 : Q ( � r Q ( n ) = # { � x ) = n } be the number of representations of n by Q . • Define ∞ � r Q ( n ) q n , q = e 2 π iz . θ Q ( z ) = n =0 • The generating function θ Q ( z ) is a modular form of weight 2 on Γ 0 ( D ( Q )) with character χ D ( Q ) . Jeremy Rouse Quadratic forms 13/45
Introduction Modular forms Universality theorems Decomposition • We can decompose θ Q ( z ) as the sum of an Eisenstein series E ( z ) and a cusp form C ( z ). Jeremy Rouse Quadratic forms 14/45
Introduction Modular forms Universality theorems Decomposition • We can decompose θ Q ( z ) as the sum of an Eisenstein series E ( z ) and a cusp form C ( z ). • The coefficients a E ( n ) of E ( z ) are large and predictable ( a E ( n ) ≫ n 1 − ǫ if n is locally represented and coprime to D ( Q )). Jeremy Rouse Quadratic forms 14/45
Introduction Modular forms Universality theorems Decomposition • We can decompose θ Q ( z ) as the sum of an Eisenstein series E ( z ) and a cusp form C ( z ). • The coefficients a E ( n ) of E ( z ) are large and predictable ( a E ( n ) ≫ n 1 − ǫ if n is locally represented and coprime to D ( Q )). • The coefficients of a C ( n ) are small and mysterious ( | a C ( n ) | ≪ d ( n ) √ n ). Jeremy Rouse Quadratic forms 14/45
Introduction Modular forms Universality theorems Example (1/2) • If Q = x 2 + y 2 + 3 z 2 + 3 w 2 + xz + yw , then θ Q ( z ) = 1 + 4 q + 4 q 2 + 8 q 3 + 20 q 4 + 16 q 5 + · · · ∈ M 2 (Γ 0 (11) , χ 1 ) . Jeremy Rouse Quadratic forms 15/45
Introduction Modular forms Universality theorems Example (1/2) • If Q = x 2 + y 2 + 3 z 2 + 3 w 2 + xz + yw , then θ Q ( z ) = 1 + 4 q + 4 q 2 + 8 q 3 + 20 q 4 + 16 q 5 + · · · ∈ M 2 (Γ 0 (11) , χ 1 ) . • We have ∞ E ( z ) = 1 + 12 � ( σ ( n ) − 11 σ ( n / 11)) q n . 5 n =1 Jeremy Rouse Quadratic forms 15/45
Introduction Modular forms Universality theorems Example (1/2) • If Q = x 2 + y 2 + 3 z 2 + 3 w 2 + xz + yw , then θ Q ( z ) = 1 + 4 q + 4 q 2 + 8 q 3 + 20 q 4 + 16 q 5 + · · · ∈ M 2 (Γ 0 (11) , χ 1 ) . • We have ∞ E ( z ) = 1 + 12 � ( σ ( n ) − 11 σ ( n / 11)) q n . 5 n =1 • If ∞ ∞ (1 − q n ) 2 (1 − q 11 n ) 2 = � � a ( n ) q n , f ( z ) = q n =1 n =1 then C ( z ) = 8 5 f ( z ). Jeremy Rouse Quadratic forms 15/45
Introduction Modular forms Universality theorems Example (2/2) • The Hasse bound gives that | a ( n ) | ≤ d ( n ) √ n and so 5 d ( n ) √ n . r Q ( n ) ≥ 12 d − 8 � 5 d | n 11 ∤ d Jeremy Rouse Quadratic forms 16/45
Introduction Modular forms Universality theorems Example (2/2) • The Hasse bound gives that | a ( n ) | ≤ d ( n ) √ n and so 5 d ( n ) √ n . r Q ( n ) ≥ 12 d − 8 � 5 d | n 11 ∤ d • There are 110 squarefree integers for which the right hand side is negative. Jeremy Rouse Quadratic forms 16/45
Introduction Modular forms Universality theorems Example (2/2) • The Hasse bound gives that | a ( n ) | ≤ d ( n ) √ n and so 5 d ( n ) √ n . r Q ( n ) ≥ 12 d − 8 � 5 d | n 11 ∤ d • There are 110 squarefree integers for which the right hand side is negative. • One can check that Q represents all of these. It follows that Q represents all positive integers. Jeremy Rouse Quadratic forms 16/45
Introduction Modular forms Universality theorems Eisenstein part • The coefficient a E ( n ) of the Eisenstein series can be written � a E ( n ) = β p ( Q , n ) p ≤∞ as a product of local densities. Jeremy Rouse Quadratic forms 17/45
Introduction Modular forms Universality theorems Eisenstein part • The coefficient a E ( n ) of the Eisenstein series can be written � a E ( n ) = β p ( Q , n ) p ≤∞ as a product of local densities. • Here x ∈ ( Z / p k Z ) 4 : Q ( � (mod p k ) } # { � x ) ≡ n β p ( Q , n ) = lim . p 3 k k →∞ Jeremy Rouse Quadratic forms 17/45
Introduction Modular forms Universality theorems Eisenstein part • The coefficient a E ( n ) of the Eisenstein series can be written � a E ( n ) = β p ( Q , n ) p ≤∞ as a product of local densities. • Here x ∈ ( Z / p k Z ) 4 : Q ( � (mod p k ) } # { � x ) ≡ n β p ( Q , n ) = lim . p 3 k k →∞ π 2 n √ • We have β ∞ ( n ) = D ( Q ) . If p ∤ nD ( Q ), then β p ( Q , n ) = 1 + O (1 / p 2 ). If p | n but p ∤ D ( Q ), then β p ( Q ) = 1 + O (1 / p ). Jeremy Rouse Quadratic forms 17/45
Introduction Modular forms Universality theorems Bounds on β p ( n ) • Let p be a prime and decompose Q over Z p as p a 1 Q 1 ⊥ p a 2 Q 2 ⊥ · · · ⊥ p a k Q k . x ∈ Z 4 For � p , decompose � x = � x 1 ⊥ · · · ⊥ � x k . Jeremy Rouse Quadratic forms 18/45
Introduction Modular forms Universality theorems Bounds on β p ( n ) • Let p be a prime and decompose Q over Z p as p a 1 Q 1 ⊥ p a 2 Q 2 ⊥ · · · ⊥ p a k Q k . x ∈ Z 4 For � p , decompose � x = � x 1 ⊥ · · · ⊥ � x k . • Define ord p ( a i ) + ord p ( � r p ( Q ) = min inf x i ) . x ∈ Z r 1 ≤ i ≤ k � p Q ( � x )=0 Jeremy Rouse Quadratic forms 18/45
Introduction Modular forms Universality theorems Bounds on β p ( n ) • Let p be a prime and decompose Q over Z p as p a 1 Q 1 ⊥ p a 2 Q 2 ⊥ · · · ⊥ p a k Q k . x ∈ Z 4 For � p , decompose � x = � x 1 ⊥ · · · ⊥ � x k . • Define ord p ( a i ) + ord p ( � r p ( Q ) = min inf x i ) . x ∈ Z r 1 ≤ i ≤ k � p Q ( � x )=0 • The r p ( Q ) is a measure of how anisotropic Q is. If Q is anisotropic, then r p ( Q ) = ∞ . Jeremy Rouse Quadratic forms 18/45
Introduction Modular forms Universality theorems Local density bounds Lemma Suppose that Q is primitive and n is locally represented by Q and p > 2 . Jeremy Rouse Quadratic forms 19/45
Introduction Modular forms Universality theorems Local density bounds Lemma Suppose that Q is primitive and n is locally represented by Q and p > 2 . If n satisfies the strong local solubility condition, then β p ( n ) ≥ 1 − 1 / p. Jeremy Rouse Quadratic forms 19/45
Introduction Modular forms Universality theorems Local density bounds Lemma Suppose that Q is primitive and n is locally represented by Q and p > 2 . If n satisfies the strong local solubility condition, then β p ( n ) ≥ 1 − 1 / p. If n is primitively locally represented by Q, then β p ( n ) ≥ (1 − 1 / p ) p −⌊ ord p ( D ( Q )) / 2 ⌋ . Jeremy Rouse Quadratic forms 19/45
Introduction Modular forms Universality theorems Local density bounds Lemma Suppose that Q is primitive and n is locally represented by Q and p > 2 . If n satisfies the strong local solubility condition, then β p ( n ) ≥ 1 − 1 / p. If n is primitively locally represented by Q, then β p ( n ) ≥ (1 − 1 / p ) p −⌊ ord p ( D ( Q )) / 2 ⌋ . In general, β p ( n ) ≥ (1 − 1 / p ) p − min { r p ( Q ) , ord p ( n ) } . Jeremy Rouse Quadratic forms 19/45
Introduction Modular forms Universality theorems Local density bounds Lemma Suppose that Q is primitive and n is locally represented by Q and p > 2 . If n satisfies the strong local solubility condition, then β p ( n ) ≥ 1 − 1 / p. If n is primitively locally represented by Q, then β p ( n ) ≥ (1 − 1 / p ) p −⌊ ord p ( D ( Q )) / 2 ⌋ . In general, β p ( n ) ≥ (1 − 1 / p ) p − min { r p ( Q ) , ord p ( n ) } . • We have similar results if p = 2. Jeremy Rouse Quadratic forms 19/45
Introduction Modular forms Universality theorems Proof of lemma (1/2) • Following Jonathan Hanke, we divide the solutions to Q ( � x ) ≡ 0 (mod p k ) into four classes: Jeremy Rouse Quadratic forms 20/45
Introduction Modular forms Universality theorems Proof of lemma (1/2) • Following Jonathan Hanke, we divide the solutions to Q ( � x ) ≡ 0 (mod p k ) into four classes: Good type: These are solutions where p a i � x i �≡ 0 (mod p ) for some i . Jeremy Rouse Quadratic forms 20/45
Introduction Modular forms Universality theorems Proof of lemma (1/2) • Following Jonathan Hanke, we divide the solutions to Q ( � x ) ≡ 0 (mod p k ) into four classes: Good type: These are solutions where p a i � x i �≡ 0 (mod p ) for some i . Zero type: These are solutions where � x ≡ 0 (mod p ). Jeremy Rouse Quadratic forms 20/45
Introduction Modular forms Universality theorems Proof of lemma (1/2) • Following Jonathan Hanke, we divide the solutions to Q ( � x ) ≡ 0 (mod p k ) into four classes: Good type: These are solutions where p a i � x i �≡ 0 (mod p ) for some i . Zero type: These are solutions where � x ≡ 0 (mod p ). Bad type I: There is some i with a i = 1 and � x i �≡ 0 (mod p ) Jeremy Rouse Quadratic forms 20/45
Introduction Modular forms Universality theorems Proof of lemma (1/2) • Following Jonathan Hanke, we divide the solutions to Q ( � x ) ≡ 0 (mod p k ) into four classes: Good type: These are solutions where p a i � x i �≡ 0 (mod p ) for some i . Zero type: These are solutions where � x ≡ 0 (mod p ). Bad type I: There is some i with a i = 1 and � x i �≡ 0 (mod p ) Bad type II: All i with � x i �≡ 0 (mod p ) have a i ≥ 2. Jeremy Rouse Quadratic forms 20/45
Introduction Modular forms Universality theorems Proof of lemma (2/2) • Hensel’s lemma makes it easy to count good type solutions. Jeremy Rouse Quadratic forms 21/45
Introduction Modular forms Universality theorems Proof of lemma (2/2) • Hensel’s lemma makes it easy to count good type solutions. • Zero type solutions have a contribution ( Q , n ) = β p ( Q , n / p 2 ). β Zero p Jeremy Rouse Quadratic forms 21/45
Introduction Modular forms Universality theorems Proof of lemma (2/2) • Hensel’s lemma makes it easy to count good type solutions. • Zero type solutions have a contribution ( Q , n ) = β p ( Q , n / p 2 ). β Zero p • There are reduction maps that relate β Bad ( Q , n ) to β p ( Q ′ , n / p ) p and β p ( Q ′′ , n / p 2 ) for other quadratic forms Q ′ and Q ′′ . Jeremy Rouse Quadratic forms 21/45
Introduction Modular forms Universality theorems The cusp form piece • To bound a C ( n ), we use the same approach as the work of Fomenko and Schulze-Pillot. This is to bound 3 �� | C ( z ) | 2 dx dy . � C , C � = π [ SL 2 ( Z ) : Γ 0 ( N ( Q ))] H / Γ 0 ( N ) Jeremy Rouse Quadratic forms 22/45
Introduction Modular forms Universality theorems The cusp form piece • To bound a C ( n ), we use the same approach as the work of Fomenko and Schulze-Pillot. This is to bound 3 �� | C ( z ) | 2 dx dy . � C , C � = π [ SL 2 ( Z ) : Γ 0 ( N ( Q ))] H / Γ 0 ( N ) • Blomer and Mili´ cevi´ c show that there is an orthonormal basis h i = � a i ( n ) q n for S 2 (Γ 0 ( N ( Q )) , χ ) so that a i ( n ) ≪ N ( Q ) 1 / 2+ ǫ d ( n ) √ n provided gcd( n , D ( Q )) = 1. Jeremy Rouse Quadratic forms 22/45
Introduction Modular forms Universality theorems Lattice • Goal: Bound � C , C � . Jeremy Rouse Quadratic forms 23/45
Introduction Modular forms Universality theorems Lattice • Goal: Bound � C , C � . • To do so, we use an explicit formula for the Weil representation due to Scheithauer to compute how θ Q transforms under any matrix in SL 2 ( Z ). Jeremy Rouse Quadratic forms 23/45
Introduction Modular forms Universality theorems Lattice • Goal: Bound � C , C � . • To do so, we use an explicit formula for the Weil representation due to Scheithauer to compute how θ Q transforms under any matrix in SL 2 ( Z ). • Let L be the lattice attached to Q . This is the set Z 4 with the inner product y � = 1 � � x , � 2 ( Q ( � x + � y ) − Q ( � x ) − Q ( � y )) . Jeremy Rouse Quadratic forms 23/45
Introduction Modular forms Universality theorems Lattice • Goal: Bound � C , C � . • To do so, we use an explicit formula for the Weil representation due to Scheithauer to compute how θ Q transforms under any matrix in SL 2 ( Z ). • Let L be the lattice attached to Q . This is the set Z 4 with the inner product y � = 1 � � x , � 2 ( Q ( � x + � y ) − Q ( � x ) − Q ( � y )) . • The dual lattice L ′ of L is L ′ = { � x ∈ R 4 : � � x , � y � ∈ Z for all � y ∈ L } . Jeremy Rouse Quadratic forms 23/45
Introduction Modular forms Universality theorems Notation • Define D = L ′ / L to be the discriminant group. It’s order is D ( Q ). Jeremy Rouse Quadratic forms 24/45
Introduction Modular forms Universality theorems Notation • Define D = L ′ / L to be the discriminant group. It’s order is D ( Q ). • For a number c | N ( Q ), define D c to be the kernel of the map [ c ] : D → D and D c to be the image. Define D c ∗ = { α ∈ D : 1 2 c � γ, γ � + � α, γ � ≡ 0 (mod 1) for all γ ∈ D c } . Jeremy Rouse Quadratic forms 24/45
Introduction Modular forms Universality theorems Notation • Define D = L ′ / L to be the discriminant group. It’s order is D ( Q ). • For a number c | N ( Q ), define D c to be the kernel of the map [ c ] : D → D and D c to be the image. Define D c ∗ = { α ∈ D : 1 2 c � γ, γ � + � α, γ � ≡ 0 (mod 1) for all γ ∈ D c } . c • Let w = gcd( N ( Q ) , c ) . Jeremy Rouse Quadratic forms 24/45
Introduction Modular forms Universality theorems Formula • With all the notation on the previous slide, the coefficient of q n / w in the Fourier expansion of ( cz + d ) − 2 θ Q (( az + b ) / ( cz + d )) is a root of unity times 1 � � e π ia � β,β � / 2 # { � L + β : β ∈ D c ∗ , Q ( � v ∈ v ) = n / w } . � | D c ∗ | β ∈ D c ∗ Jeremy Rouse Quadratic forms 25/45
Introduction Modular forms Universality theorems Formula • With all the notation on the previous slide, the coefficient of q n / w in the Fourier expansion of ( cz + d ) − 2 θ Q (( az + b ) / ( cz + d )) is a root of unity times 1 � � e π ia � β,β � / 2 # { � L + β : β ∈ D c ∗ , Q ( � v ∈ v ) = n / w } . � | D c ∗ | β ∈ D c ∗ x ∈ L ′ : � x mod L ⊆ D c ∪ D c ∗ } . • Let T = { � Jeremy Rouse Quadratic forms 25/45
Introduction Modular forms Universality theorems Formula • With all the notation on the previous slide, the coefficient of q n / w in the Fourier expansion of ( cz + d ) − 2 θ Q (( az + b ) / ( cz + d )) is a root of unity times 1 � � e π ia � β,β � / 2 # { � L + β : β ∈ D c ∗ , Q ( � v ∈ v ) = n / w } . � | D c ∗ | β ∈ D c ∗ x ∈ L ′ : � x mod L ⊆ D c ∪ D c ∗ } . • Let T = { � • If we define R : T → Q by R ( � x ) = 4 w � � x , � x � , then R is an integral quadratic form with discriminant ≤ (4 w ) 4 D ( Q ) . | D c | 2 Jeremy Rouse Quadratic forms 25/45
Introduction Modular forms Universality theorems Bound on � C , C � • Putting this together, we get ∞ r R (4 n ) 2 √ 1 � � | D c ∗ | ( n / w ) e − 2 π 3 n / w . � C , C � ≪ w [ SL 2 ( Z ) : Γ 0 ( N )] n =1 a / c Jeremy Rouse Quadratic forms 26/45
Introduction Modular forms Universality theorems Bound on � C , C � • Putting this together, we get ∞ r R (4 n ) 2 √ 1 � � | D c ∗ | ( n / w ) e − 2 π 3 n / w . � C , C � ≪ w [ SL 2 ( Z ) : Γ 0 ( N )] n =1 a / c n ≤ x r R ( n ) 2 . • To estimate the sum, we need to bound � Jeremy Rouse Quadratic forms 26/45
Introduction Modular forms Universality theorems Analyzing this sum • The easiest way to do this is to use � � r R ( n ) 2 ≤ � � · r R ( n ) max n ≤ x r R ( n ) . n ≤ x n ≤ x Jeremy Rouse Quadratic forms 27/45
Introduction Modular forms Universality theorems Analyzing this sum • The easiest way to do this is to use � � r R ( n ) 2 ≤ � � · r R ( n ) max n ≤ x r R ( n ) . n ≤ x n ≤ x • The first term is straightforward to analyze. We get x 2 � D ( R ) 1 / 2 + x 3 / 2 . r R ( n ) ≪ n ≤ x Jeremy Rouse Quadratic forms 27/45
Introduction Modular forms Universality theorems Analyzing this sum • The easiest way to do this is to use � � r R ( n ) 2 ≤ � � · r R ( n ) max n ≤ x r R ( n ) . n ≤ x n ≤ x • The first term is straightforward to analyze. We get x 2 � D ( R ) 1 / 2 + x 3 / 2 . r R ( n ) ≪ n ≤ x • There’s a clever argument I learned from MathOverflow that gives n ≤ x r R ( n ) ≪ x 1+ ǫ D ( R ) − 1 / 4+ ǫ + x 1 / 2 . max Jeremy Rouse Quadratic forms 27/45
Introduction Modular forms Universality theorems Cusp form bound • From this we get that w 3 1 � � C , C � ≪ | D c ∗ | . [ SL 2 ( Z ) : Γ 0 ( N ( Q ))] a / c Jeremy Rouse Quadratic forms 28/45
Introduction Modular forms Universality theorems Cusp form bound • From this we get that w 3 1 � � C , C � ≪ | D c ∗ | . [ SL 2 ( Z ) : Γ 0 ( N ( Q ))] a / c • This is ≪ E ( Q ) = max { N ( Q ) 1 / 2+ ǫ D ( Q ) 1 / 4+ ǫ , N ( Q ) 1+ ǫ } . Jeremy Rouse Quadratic forms 28/45
Introduction Modular forms Universality theorems Cusp form bound • From this we get that w 3 1 � � C , C � ≪ | D c ∗ | . [ SL 2 ( Z ) : Γ 0 ( N ( Q ))] a / c • This is ≪ E ( Q ) = max { N ( Q ) 1 / 2+ ǫ D ( Q ) 1 / 4+ ǫ , N ( Q ) 1+ ǫ } . • It follows from this that | a C ( n ) | ≪ E ( Q ) d ( n ) √ n if gcd( n , D ( Q )) = 1. Jeremy Rouse Quadratic forms 28/45
Introduction Modular forms Universality theorems Conclusion • We have that r Q ( n ) = a E ( n ) + a C ( n ). Jeremy Rouse Quadratic forms 29/45
Introduction Modular forms Universality theorems Conclusion • We have that r Q ( n ) = a E ( n ) + a C ( n ). • If gcd( n , D ( Q )) = 1 then a E ( n ) ≫ n 1 − ǫ D ( Q ) − 1 / 2 . Jeremy Rouse Quadratic forms 29/45
Introduction Modular forms Universality theorems Conclusion • We have that r Q ( n ) = a E ( n ) + a C ( n ). • If gcd( n , D ( Q )) = 1 then a E ( n ) ≫ n 1 − ǫ D ( Q ) − 1 / 2 . • If gcd( n , D ( Q )) = 1, then | a C ( n ) | ≪ E ( Q ) d ( n ) √ n . Jeremy Rouse Quadratic forms 29/45
Introduction Modular forms Universality theorems Conclusion • We have that r Q ( n ) = a E ( n ) + a C ( n ). • If gcd( n , D ( Q )) = 1 then a E ( n ) ≫ n 1 − ǫ D ( Q ) − 1 / 2 . • If gcd( n , D ( Q )) = 1, then | a C ( n ) | ≪ E ( Q ) d ( n ) √ n . • It follows that r Q ( n ) > 0 if n ≫ D ( Q ) E ( Q ) 2+ ǫ . Jeremy Rouse Quadratic forms 29/45
Introduction Modular forms Universality theorems Motivation Theorem (Lagrange, 1770) Every positive integer can be written as a sum of four squares. Jeremy Rouse Quadratic forms 30/45
Introduction Modular forms Universality theorems Motivation Theorem (Lagrange, 1770) Every positive integer can be written as a sum of four squares. • What other expressions represent all positive integers? Jeremy Rouse Quadratic forms 30/45
Introduction Modular forms Universality theorems Motivation Theorem (Lagrange, 1770) Every positive integer can be written as a sum of four squares. • What other expressions represent all positive integers? x ) = 1 x T A � • Write Q ( � 2 � x . We say that Q is integer-matrix if all the entries of A are even. Jeremy Rouse Quadratic forms 30/45
Introduction Modular forms Universality theorems 15 • We say a quadratic form is integer-valued if the diagonal entries of A are even. Jeremy Rouse Quadratic forms 31/45
Introduction Modular forms Universality theorems 15 • We say a quadratic form is integer-valued if the diagonal entries of A are even. Theorem (Conway-Schneeberger-Bhargava) A positive-definite integer matrix form Q represents every positive integer if and only if it represents 1 , 2 , 3 , 5 , 6 , 7 , 10 , 14 , and 15 . Jeremy Rouse Quadratic forms 31/45
Introduction Modular forms Universality theorems 290 Theorem (Bhargava-Hanke) A positive-definite, integer-valued form Q represents every positive integer if and only if it represents 1 , 2 , 3 , 5 , 6 , 7 , 10 , 13 , 14 , 15 , 17 , 19 , 21 , 22 , 23 , 26 , 29 , 30 , 31 , 34 , 35 , 37 , 42 , 58 , 93 , 110 , 145 , 203 , and 290 . Jeremy Rouse Quadratic forms 32/45
Introduction Modular forms Universality theorems Consequences • Each of these results is sharp. The form x 2 + 2 y 2 + 4 z 2 + 29 w 2 + 145 v 2 − xz − yz represents every positive integer except 290. Jeremy Rouse Quadratic forms 33/45
Introduction Modular forms Universality theorems Consequences • Each of these results is sharp. The form x 2 + 2 y 2 + 4 z 2 + 29 w 2 + 145 v 2 − xz − yz represents every positive integer except 290. • If a form represents every positive integer less than 290, it represents every integer greater than 290. Jeremy Rouse Quadratic forms 33/45
Introduction Modular forms Universality theorems Consequences • Each of these results is sharp. The form x 2 + 2 y 2 + 4 z 2 + 29 w 2 + 145 v 2 − xz − yz represents every positive integer except 290. • If a form represents every positive integer less than 290, it represents every integer greater than 290. • There are 6436 integer-valued quaternary forms that represent all positive integers. Jeremy Rouse Quadratic forms 33/45
Introduction Modular forms Universality theorems Later results Theorem (R, 2014) Assume GRH. Then a positive-definite, integer-valued form Q represents all positive odds if and only if it represents 1 , 3 , 5 , 7 , 11 , 13 , 15 , 17 , 19 , 21 , 23 , 29 , 31 , 33 , 35 , 37 , 39 , 41 , 47 , 51 , 53 , 57 , 59 , 77 , 83 , 85 , 87 , 89 , 91 , 93 , 105 , 119 , 123 , 133 , 137 , 143 , 145 , 187 , 195 , 203 , 205 , 209 , 231 , 319 , 385 , and 451 . Jeremy Rouse Quadratic forms 34/45
Introduction Modular forms Universality theorems Later results Theorem (R, 2014) Assume GRH. Then a positive-definite, integer-valued form Q represents all positive odds if and only if it represents 1 , 3 , 5 , 7 , 11 , 13 , 15 , 17 , 19 , 21 , 23 , 29 , 31 , 33 , 35 , 37 , 39 , 41 , 47 , 51 , 53 , 57 , 59 , 77 , 83 , 85 , 87 , 89 , 91 , 93 , 105 , 119 , 123 , 133 , 137 , 143 , 145 , 187 , 195 , 203 , 205 , 209 , 231 , 319 , 385 , and 451 . Theorem (DeBenedetto-R, 2016) A positive-definite, integer-valued form Q represents every positive integer coprime to 3 if and only if it represents 1 , 2 , 5 , 7 , 10 , 11 , 13 , 14 , 17 , 19 , 22 , 23 , 26 , 29 , 31 , 34 , 35 37 , 38 , 46 , 47 , 55 , 58 , 62 , 70 , 94 , 110 , 119 , 145 , 203 , and 290 . Jeremy Rouse Quadratic forms 34/45
Introduction Modular forms Universality theorems Two exceptions • It follows from the proof of the 15-theorem that if an integer-valued form Q represents all positive integers with one exception, then that exception must be 1, 2, 3, 5, 6, 7, 10, 14, or 15. Jeremy Rouse Quadratic forms 35/45
Introduction Modular forms Universality theorems Two exceptions • It follows from the proof of the 15-theorem that if an integer-valued form Q represents all positive integers with one exception, then that exception must be 1, 2, 3, 5, 6, 7, 10, 14, or 15. Theorem (BDMSST, 2017) If a positive-definite integer-matrix form Q represents all positive integers with two exceptions, the pair of exceptions { m , n } must be one of the following: { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 1 , 5 } , { 1 , 6 } , { 1 , 7 } , { 1 , 9 } , { 1 , 10 } , { 1 , 11 } , { 1 , 13 } , { 1 , 14 } , { 1 , 15 } , { 1 , 17 } , { 1 , 19 } , { 1 , 21 } , { 1 , 23 } , { 1 , 25 } , { 1 , 30 } , { 1 , 41 } , { 1 , 55 } , { 2 , 3 } , { 2 , 5 } , { 2 , 6 } , { 2 , 8 } , { 2 , 10 } , { 2 , 11 } , { 2 , 14 } , { 2 , 15 } , { 2 , 18 } , { 2 , 22 } , { 2 , 30 } , { 2 , 38 } , { 2 , 50 } , { 3 , 6 } , { 3 , 7 } , { 3 , 11 } , { 3 , 12 } , { 3 , 19 } , { 3 , 21 } , { 3 , 27 } , { 3 , 30 } , { 3 , 35 } , { 3 , 39 } , { 5 , 7 } , { 5 , 10 } , { 5 , 13 } , { 5 , 14 } , { 5 , 20 } , { 5 , 21 } , { 5 , 29 } , { 5 , 30 } , { 5 , 35 } , { 5 , 37 } , { 5 , 42 } , { 5 , 125 } , { 6 , 15 } , { 6 , 54 } , { 7 , 10 } , { 7 , 15 } , { 7 , 23 } , { 7 , 28 } , { 7 , 31 } , { 7 , 39 } , { 7 , 55 } , { 10 , 15 } , { 10 , 26 } , { 10 , 40 } , { 10 , 58 } , { 10 , 250 } , { 14 , 30 } , { 14 , 56 } , { 14 , 78 } . Jeremy Rouse Quadratic forms 35/45
Introduction Modular forms Universality theorems Overview • Bhargava’s escalator method is used to reduce problems like those above to a finite calculation involving specific quaternary quadratic forms. Jeremy Rouse Quadratic forms 36/45
Introduction Modular forms Universality theorems Overview • Bhargava’s escalator method is used to reduce problems like those above to a finite calculation involving specific quaternary quadratic forms. • The modular symbols algorithm can be used to decompose C ( z ) into newforms and to derive an explicit bound on a C ( n ). Jeremy Rouse Quadratic forms 36/45
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