Half Integral Weight Modular Forms

Deﬁnitions Application Quaternions HMF Application II

Half integral weight modular forms

Ariel Pacetti

Universidad de Buenos Aires

Explicit Methods for Modular Forms March 20, 2013

Ariel Pacetti Half integral weight modular forms The Dedekind eta function ∞ πiz Y 2πinz η(z) = e 12 (1 − e ). n=1

It is well know that η(z)24 = ∆(z) a weight 12 cusp form, so η “should be” of weight 1/2. Actually η turns out to be weight 1/2 but with a character of order 24.

We can consider two classical examples:

Deﬁnitions Application Quaternions HMF Application II

Motivation

What is a half integral modular form?

Ariel Pacetti Half integral weight modular forms The Dedekind eta function ∞ πiz Y 2πinz η(z) = e 12 (1 − e ). n=1

It is well know that η(z)24 = ∆(z) a weight 12 cusp form, so η “should be” of weight 1/2. Actually η turns out to be weight 1/2 but with a character of order 24.

Deﬁnitions Application Quaternions HMF Application II

Motivation

What is a half integral modular form? We can consider two classical examples:

Ariel Pacetti Half integral weight modular forms It is well know that η(z)24 = ∆(z) a weight 12 cusp form, so η “should be” of weight 1/2. Actually η turns out to be weight 1/2 but with a character of order 24.

Deﬁnitions Application Quaternions HMF Application II

Motivation

What is a half integral modular form? We can consider two classical examples: The Dedekind eta function ∞ πiz Y 2πinz η(z) = e 12 (1 − e ). n=1

Ariel Pacetti Half integral weight modular forms Actually η turns out to be weight 1/2 but with a character of order 24.

Deﬁnitions Application Quaternions HMF Application II

Motivation

What is a half integral modular form? We can consider two classical examples: The Dedekind eta function ∞ πiz Y 2πinz η(z) = e 12 (1 − e ). n=1

It is well know that η(z)24 = ∆(z) a weight 12 cusp form, so η “should be” of weight 1/2.

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Motivation

What is a half integral modular form? We can consider two classical examples: The Dedekind eta function ∞ πiz Y 2πinz η(z) = e 12 (1 − e ). n=1

It is well know that η(z)24 = ∆(z) a weight 12 cusp form, so η “should be” of weight 1/2. Actually η turns out to be weight 1/2 but with a character of order 24.

Ariel Pacetti Half integral weight modular forms a b It is not hard to see that if γ = c d ∈ Γ0(4), then

θ(γz)2 −1 = (cz + d). θ(z) d

2 So θ(z) ∈ M1(Γ0(4), χ−1).

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Motivation

The classical theta function ∞ X 2 θ(z) = e2πin z . n=−∞

Ariel Pacetti Half integral weight modular forms 2 So θ(z) ∈ M1(Γ0(4), χ−1).

Deﬁnitions Application Quaternions HMF Application II

Motivation

The classical theta function ∞ X 2 θ(z) = e2πin z . n=−∞

a b It is not hard to see that if γ = c d ∈ Γ0(4), then

θ(γz)2 −1 = (cz + d). θ(z) d

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Motivation

The classical theta function ∞ X 2 θ(z) = e2πin z . n=−∞

a b It is not hard to see that if γ = c d ∈ Γ0(4), then

θ(γz)2 −1 = (cz + d). θ(z) d

2 So θ(z) ∈ M1(Γ0(4), χ−1).

Ariel Pacetti Half integral weight modular forms k a b f (γz) = J(γ, z) ψ(d)f (z) ∀γ = c d ∈ Γ0(4N) f (z) is holomorphic at the cusps.

We denote by Mk/2(4N, ψ) the space of such forms and Sk/2(4N, ψ) the subspace of cuspidal ones.

Let k be an odd positive integer, N a positive integer and ψ a character modulo N. Deﬁnition A modular form of weight k/2, level 4N and character ψ is an holomorphic function f : H → C such that

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Deﬁnition

θ(γz) We consider the factor of automorphy J(γ, z) = θ(z) .

Ariel Pacetti Half integral weight modular forms k a b f (γz) = J(γ, z) ψ(d)f (z) ∀γ = c d ∈ Γ0(4N) f (z) is holomorphic at the cusps.

We denote by Mk/2(4N, ψ) the space of such forms and Sk/2(4N, ψ) the subspace of cuspidal ones.

Deﬁnition A modular form of weight k/2, level 4N and character ψ is an holomorphic function f : H → C such that

Deﬁnitions Application Quaternions HMF Application II

Deﬁnition

θ(γz) We consider the factor of automorphy J(γ, z) = θ(z) . Let k be an odd positive integer, N a positive integer and ψ a character modulo N.

Ariel Pacetti Half integral weight modular forms k a b f (γz) = J(γ, z) ψ(d)f (z) ∀γ = c d ∈ Γ0(4N) f (z) is holomorphic at the cusps.

We denote by Mk/2(4N, ψ) the space of such forms and Sk/2(4N, ψ) the subspace of cuspidal ones.

Deﬁnitions Application Quaternions HMF Application II

Deﬁnition

θ(γz) We consider the factor of automorphy J(γ, z) = θ(z) . Let k be an odd positive integer, N a positive integer and ψ a character modulo N. Deﬁnition A modular form of weight k/2, level 4N and character ψ is an holomorphic function f : H → C such that

Ariel Pacetti Half integral weight modular forms f (z) is holomorphic at the cusps.

We denote by Mk/2(4N, ψ) the space of such forms and Sk/2(4N, ψ) the subspace of cuspidal ones.

Deﬁnitions Application Quaternions HMF Application II

Deﬁnition

Ariel Pacetti Half integral weight modular forms We denote by Mk/2(4N, ψ) the space of such forms and Sk/2(4N, ψ) the subspace of cuspidal ones.

Deﬁnitions Application Quaternions HMF Application II

Deﬁnition

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Deﬁnition

We denote by Mk/2(4N, ψ) the space of such forms and Sk/2(4N, ψ) the subspace of cuspidal ones.

Ariel Pacetti Half integral weight modular forms 1 Tn = 0 if n is not a square.

2 If (n : 4N) = 1, Tn2 is self adjoint for an inner product.

3 Tn2 Tm2 = Tm2 Tn2 . 4 k−1 If terms of q-expansion, let ω = 2 , then Tp2 acts like ω −1 n ω−1 2 k−1 a 2 + ψ(n) p a + ψ(p )p a 2 . p n p p n n/p

Hence there exists a basis of eigenforms for the Hecke operators prime to 4N.

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Hecke operators

Via a double coset action, one can deﬁne Hecke operators {Tn}n≥1 acting on Sk/2(4N, ψ). They satisfy the properties:

Ariel Pacetti Half integral weight modular forms 2 If (n : 4N) = 1, Tn2 is self adjoint for an inner product.

3 Tn2 Tm2 = Tm2 Tn2 . 4 k−1 If terms of q-expansion, let ω = 2 , then Tp2 acts like ω −1 n ω−1 2 k−1 a 2 + ψ(n) p a + ψ(p )p a 2 . p n p p n n/p

Hence there exists a basis of eigenforms for the Hecke operators prime to 4N.

Deﬁnitions Application Quaternions HMF Application II

Hecke operators

Via a double coset action, one can deﬁne Hecke operators {Tn}n≥1 acting on Sk/2(4N, ψ). They satisfy the properties:

1 Tn = 0 if n is not a square.

Ariel Pacetti Half integral weight modular forms 3 Tn2 Tm2 = Tm2 Tn2 . 4 k−1 If terms of q-expansion, let ω = 2 , then Tp2 acts like ω −1 n ω−1 2 k−1 a 2 + ψ(n) p a + ψ(p )p a 2 . p n p p n n/p

Hence there exists a basis of eigenforms for the Hecke operators prime to 4N.

Deﬁnitions Application Quaternions HMF Application II

Hecke operators

Via a double coset action, one can deﬁne Hecke operators {Tn}n≥1 acting on Sk/2(4N, ψ). They satisfy the properties:

1 Tn = 0 if n is not a square.

2 If (n : 4N) = 1, Tn2 is self adjoint for an inner product.

Ariel Pacetti Half integral weight modular forms 4 k−1 If terms of q-expansion, let ω = 2 , then Tp2 acts like ω −1 n ω−1 2 k−1 a 2 + ψ(n) p a + ψ(p )p a 2 . p n p p n n/p

Hence there exists a basis of eigenforms for the Hecke operators prime to 4N.

Deﬁnitions Application Quaternions HMF Application II

Hecke operators

Via a double coset action, one can deﬁne Hecke operators {Tn}n≥1 acting on Sk/2(4N, ψ). They satisfy the properties:

1 Tn = 0 if n is not a square.

2 If (n : 4N) = 1, Tn2 is self adjoint for an inner product.

3 Tn2 Tm2 = Tm2 Tn2 .

Ariel Pacetti Half integral weight modular forms Hence there exists a basis of eigenforms for the Hecke operators prime to 4N.

Deﬁnitions Application Quaternions HMF Application II

Hecke operators

Via a double coset action, one can deﬁne Hecke operators {Tn}n≥1 acting on Sk/2(4N, ψ). They satisfy the properties:

1 Tn = 0 if n is not a square.

2 If (n : 4N) = 1, Tn2 is self adjoint for an inner product.

3 Tn2 Tm2 = Tm2 Tn2 . 4 k−1 If terms of q-expansion, let ω = 2 , then Tp2 acts like ω −1 n ω−1 2 k−1 a 2 + ψ(n) p a + ψ(p )p a 2 . p n p p n n/p

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Hecke operators

Via a double coset action, one can deﬁne Hecke operators {Tn}n≥1 acting on Sk/2(4N, ψ). They satisfy the properties:

1 Tn = 0 if n is not a square.

2 If (n : 4N) = 1, Tn2 is self adjoint for an inner product.

3 Tn2 Tm2 = Tm2 Tn2 . 4 k−1 If terms of q-expansion, let ω = 2 , then Tp2 acts like ω −1 n ω−1 2 k−1 a 2 + ψ(n) p a + ψ(p )p a 2 . p n p p n n/p

Hence there exists a basis of eigenforms for the Hecke operators prime to 4N.

Ariel Pacetti Half integral weight modular forms What information encode the non-square Fourier coeﬃcients?

Furthermore, if f ∈ Sk/2(4N, ψ) is an eigenform for all the Hecke operators with eigenvalues λn, then Shimn(f ) is (up to a constant) given by Y −s 2 k−2−2s −1 (1 − λpp + ψ(p )p ) . p

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Shimura’s Theorem

Theorem (Shimura)

For each square-free positive integer n, there exists a T0-linear map

2 1 Shimn : Sk/2(4N, ψ) → Mk−1(2N, ψ ).

1The actual level may be smaller Ariel Pacetti Half integral weight modular forms What information encode the non-square Fourier coeﬃcients?

Deﬁnitions Application Quaternions HMF Application II

Shimura’s Theorem

Theorem (Shimura)

For each square-free positive integer n, there exists a T0-linear map

2 1 Shimn : Sk/2(4N, ψ) → Mk−1(2N, ψ ).

Furthermore, if f ∈ Sk/2(4N, ψ) is an eigenform for all the Hecke operators with eigenvalues λn, then Shimn(f ) is (up to a constant) given by Y −s 2 k−2−2s −1 (1 − λpp + ψ(p )p ) . p

1The actual level may be smaller Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Shimura’s Theorem

Theorem (Shimura)

For each square-free positive integer n, there exists a T0-linear map

2 1 Shimn : Sk/2(4N, ψ) → Mk−1(2N, ψ ).

Furthermore, if f ∈ Sk/2(4N, ψ) is an eigenform for all the Hecke operators with eigenvalues λn, then Shimn(f ) is (up to a constant) given by Y −s 2 k−2−2s −1 (1 − λpp + ψ(p )p ) . p What information encode the non-square Fourier coeﬃcients?

1The actual level may be smaller Ariel Pacetti Half integral weight modular forms If we ﬁxed n1, for all n as above

2 −1 k−1 k/2−1 an = κL(F , ψ0 χn, 2 )ψ(n)n where κ is a global constant.

Theorem (Waldspurger) × 2 Let n1, n2 be square free positive integers such that n1/n2 ∈ (Qp ) for all p | 4N. Then 2 −1 n2 k/2−1 2 −1 k/2−1 an1 L(F , ψ0 χn2 , ω)ψ n2 = an2 L(F , ψ0 χn1 , ω)n1 n1

ω where ψ (n) = ψ(n) −1 , χ is the quadratic character 0 n √n corresponding to the ﬁeld Q[ n]

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Waldspurger’s theorem

2 Let f ∈ Sk/2(4N, ψ), F = Shim(f ) ∈ Sk−1(2N, ψ ) eigenforms.

Ariel Pacetti Half integral weight modular forms If we ﬁxed n1, for all n as above

2 −1 k−1 k/2−1 an = κL(F , ψ0 χn, 2 )ψ(n)n where κ is a global constant.

Then 2 −1 n2 k/2−1 2 −1 k/2−1 an1 L(F , ψ0 χn2 , ω)ψ n2 = an2 L(F , ψ0 χn1 , ω)n1 n1

ω where ψ (n) = ψ(n) −1 , χ is the quadratic character 0 n √n corresponding to the ﬁeld Q[ n]

Deﬁnitions Application Quaternions HMF Application II

Waldspurger’s theorem

2 Let f ∈ Sk/2(4N, ψ), F = Shim(f ) ∈ Sk−1(2N, ψ ) eigenforms. Theorem (Waldspurger) × 2 Let n1, n2 be square free positive integers such that n1/n2 ∈ (Qp ) for all p | 4N.

Ariel Pacetti Half integral weight modular forms If we ﬁxed n1, for all n as above

2 −1 k−1 k/2−1 an = κL(F , ψ0 χn, 2 )ψ(n)n where κ is a global constant.

Deﬁnitions Application Quaternions HMF Application II

Waldspurger’s theorem

2 Let f ∈ Sk/2(4N, ψ), F = Shim(f ) ∈ Sk−1(2N, ψ ) eigenforms. Theorem (Waldspurger) × 2 Let n1, n2 be square free positive integers such that n1/n2 ∈ (Qp ) for all p | 4N. Then 2 −1 n2 k/2−1 2 −1 k/2−1 an1 L(F , ψ0 χn2 , ω)ψ n2 = an2 L(F , ψ0 χn1 , ω)n1 n1

ω where ψ (n) = ψ(n) −1 , χ is the quadratic character 0 n √n corresponding to the ﬁeld Q[ n]

Ariel Pacetti Half integral weight modular forms 2 −1 k−1 k/2−1 an = κL(F , ψ0 χn, 2 )ψ(n)n where κ is a global constant.

Deﬁnitions Application Quaternions HMF Application II

Waldspurger’s theorem

ω where ψ (n) = ψ(n) −1 , χ is the quadratic character 0 n √n corresponding to the ﬁeld Q[ n]

If we ﬁxed n1, for all n as above

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Waldspurger’s theorem

ω where ψ (n) = ψ(n) −1 , χ is the quadratic character 0 n √n corresponding to the ﬁeld Q[ n]

If we ﬁxed n1, for all n as above

2 −1 k−1 k/2−1 an = κL(F , ψ0 χn, 2 )ψ(n)n where κ is a global constant.

Ariel Pacetti Half integral weight modular forms Theorem (Tunnell) If n ∈ N is odd, (assuming BSD) it is a congruent number iﬀ

3 2 2 2 # (x, y, z) ∈ Z : n = 2x + y + 32z = 1 #(x, y, z) ∈ 3 : n = 2x2 + y 2 + 8z2 2 Z For even n, iﬀ

3 2 2 2 # (x, y, z) ∈ Z : n/2 = 4x + y + 32z = 1 #(x, y, z) ∈ 3 : n/2 = 4x2 + y 2 + 8z2 . 2 Z

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Congruent Number Problem

Deﬁnition: n ∈ N is called a congruent number if it is the area of a right triangle with rational sides.

Ariel Pacetti Half integral weight modular forms For even n, iﬀ

3 2 2 2 # (x, y, z) ∈ Z : n/2 = 4x + y + 32z = 1 #(x, y, z) ∈ 3 : n/2 = 4x2 + y 2 + 8z2 . 2 Z

Deﬁnitions Application Quaternions HMF Application II

Congruent Number Problem

Deﬁnition: n ∈ N is called a congruent number if it is the area of a right triangle with rational sides. Theorem (Tunnell) If n ∈ N is odd, (assuming BSD) it is a congruent number iﬀ

3 2 2 2 # (x, y, z) ∈ Z : n = 2x + y + 32z = 1 #(x, y, z) ∈ 3 : n = 2x2 + y 2 + 8z2 2 Z

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Congruent Number Problem

Deﬁnition: n ∈ N is called a congruent number if it is the area of a right triangle with rational sides. Theorem (Tunnell) If n ∈ N is odd, (assuming BSD) it is a congruent number iﬀ

3 2 2 2 # (x, y, z) ∈ Z : n = 2x + y + 32z = 1 #(x, y, z) ∈ 3 : n = 2x2 + y 2 + 8z2 2 Z For even n, iﬀ

3 2 2 2 # (x, y, z) ∈ Z : n/2 = 4x + y + 32z = 1 #(x, y, z) ∈ 3 : n/2 = 4x2 + y 2 + 8z2 . 2 Z

Ariel Pacetti Half integral weight modular forms 1 Given F ∈ S2k (N, 1), construct preimages under Shim. 2 Give an explicit constant in Waldspurger Theorem. 3 Generalize this to Hilbert modular forms.

For simplicity we will consider the case of weight k = 2 (where modular forms correspond with elliptic curves).

Deﬁnitions Application Quaternions HMF Application II

Preimages

What we would like to do:

Ariel Pacetti Half integral weight modular forms 2 Give an explicit constant in Waldspurger Theorem. 3 Generalize this to Hilbert modular forms.

For simplicity we will consider the case of weight k = 2 (where modular forms correspond with elliptic curves).

Deﬁnitions Application Quaternions HMF Application II

Preimages

What we would like to do:

1 Given F ∈ S2k (N, 1), construct preimages under Shim.

Ariel Pacetti Half integral weight modular forms 3 Generalize this to Hilbert modular forms.

For simplicity we will consider the case of weight k = 2 (where modular forms correspond with elliptic curves).

Deﬁnitions Application Quaternions HMF Application II

Preimages

What we would like to do:

1 Given F ∈ S2k (N, 1), construct preimages under Shim. 2 Give an explicit constant in Waldspurger Theorem.

Ariel Pacetti Half integral weight modular forms For simplicity we will consider the case of weight k = 2 (where modular forms correspond with elliptic curves).

Deﬁnitions Application Quaternions HMF Application II

Preimages

What we would like to do:

1 Given F ∈ S2k (N, 1), construct preimages under Shim. 2 Give an explicit constant in Waldspurger Theorem. 3 Generalize this to Hilbert modular forms.

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Preimages

What we would like to do:

1 Given F ∈ S2k (N, 1), construct preimages under Shim. 2 Give an explicit constant in Waldspurger Theorem. 3 Generalize this to Hilbert modular forms.

For simplicity we will consider the case of weight k = 2 (where modular forms correspond with elliptic curves).

Ariel Pacetti Half integral weight modular forms Let R ⊂ B be an Eichler order of level N.

Let J (R) be the set of left R-ideals and let {[a1],..., [an]} be ideal classes representatives. Let M(R) be the C-v.s. spanned by these representatives. Consider the inner product given by ( 0 if i 6= j, h[a ], [a ]i = i j 1 × 2 #Rr (ai ) if i = j.

Given m ∈ N and a ∈ J (R), let

2 tm(a) = {b ∈ J (R): b ⊂ a, [a : b] = m }.

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Quaternionic modular forms

Let B be a quaternion algebra over Q ramiﬁed at ∞.

Ariel Pacetti Half integral weight modular forms Let M(R) be the C-v.s. spanned by these representatives. Consider the inner product given by ( 0 if i 6= j, h[a ], [a ]i = i j 1 × 2 #Rr (ai ) if i = j.

Given m ∈ N and a ∈ J (R), let

2 tm(a) = {b ∈ J (R): b ⊂ a, [a : b] = m }.

Deﬁnitions Application Quaternions HMF Application II

Quaternionic modular forms

Let B be a quaternion algebra over Q ramiﬁed at ∞. Let R ⊂ B be an Eichler order of level N.

Let J (R) be the set of left R-ideals and let {[a1],..., [an]} be ideal classes representatives.

Ariel Pacetti Half integral weight modular forms Consider the inner product given by ( 0 if i 6= j, h[a ], [a ]i = i j 1 × 2 #Rr (ai ) if i = j.

Given m ∈ N and a ∈ J (R), let

2 tm(a) = {b ∈ J (R): b ⊂ a, [a : b] = m }.

Deﬁnitions Application Quaternions HMF Application II

Quaternionic modular forms

Let B be a quaternion algebra over Q ramiﬁed at ∞. Let R ⊂ B be an Eichler order of level N.

Let J (R) be the set of left R-ideals and let {[a1],..., [an]} be ideal classes representatives. Let M(R) be the C-v.s. spanned by these representatives.

Ariel Pacetti Half integral weight modular forms Given m ∈ N and a ∈ J (R), let

2 tm(a) = {b ∈ J (R): b ⊂ a, [a : b] = m }.

Deﬁnitions Application Quaternions HMF Application II

Quaternionic modular forms

Let B be a quaternion algebra over Q ramiﬁed at ∞. Let R ⊂ B be an Eichler order of level N.

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Quaternionic modular forms

Let B be a quaternion algebra over Q ramiﬁed at ∞. Let R ⊂ B be an Eichler order of level N.

Given m ∈ N and a ∈ J (R), let

2 tm(a) = {b ∈ J (R): b ⊂ a, [a : b] = m }.

Ariel Pacetti Half integral weight modular forms 1 are self adjoint (all of them). 2 commute with each other. Pn 1 Let e0 = [ai ]. It is an eigenvector for the Hecke i=1 hai ,ai i operators. Denote by S(R) its orthogonal complement.

Proposition The Hecke operators satisfy:

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Hecke operators

For m ∈ N, the Hecke operators Tm : M(R) → M(R) is

X [b] T ([a]) = . m hb, bi b∈tm(a)

Deﬁnitions Application Quaternions HMF Application II

Hecke operators

For m ∈ N, the Hecke operators Tm : M(R) → M(R) is

X [b] T ([a]) = . m hb, bi b∈tm(a)

Proposition The Hecke operators satisfy:

Ariel Pacetti Half integral weight modular forms 2 commute with each other. Pn 1 Let e0 = [ai ]. It is an eigenvector for the Hecke i=1 hai ,ai i operators. Denote by S(R) its orthogonal complement.

Deﬁnitions Application Quaternions HMF Application II

Hecke operators

For m ∈ N, the Hecke operators Tm : M(R) → M(R) is

X [b] T ([a]) = . m hb, bi b∈tm(a)

Proposition The Hecke operators satisfy: 1 are self adjoint (all of them).

Ariel Pacetti Half integral weight modular forms Pn 1 Let e0 = [ai ]. It is an eigenvector for the Hecke i=1 hai ,ai i operators. Denote by S(R) its orthogonal complement.

Deﬁnitions Application Quaternions HMF Application II

Hecke operators

For m ∈ N, the Hecke operators Tm : M(R) → M(R) is

X [b] T ([a]) = . m hb, bi b∈tm(a)

Proposition The Hecke operators satisfy: 1 are self adjoint (all of them). 2 commute with each other.

Ariel Pacetti Half integral weight modular forms It is an eigenvector for the Hecke operators. Denote by S(R) its orthogonal complement.

Deﬁnitions Application Quaternions HMF Application II

Hecke operators

For m ∈ N, the Hecke operators Tm : M(R) → M(R) is

X [b] T ([a]) = . m hb, bi b∈tm(a)

Proposition The Hecke operators satisfy: 1 are self adjoint (all of them). 2 commute with each other. Pn 1 Let e0 = [ai ]. i=1 hai ,ai i

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Hecke operators

For m ∈ N, the Hecke operators Tm : M(R) → M(R) is

X [b] T ([a]) = . m hb, bi b∈tm(a)

Proposition The Hecke operators satisfy: 1 are self adjoint (all of them). 2 commute with each other. Pn 1 Let e0 = [ai ]. It is an eigenvector for the Hecke i=1 hai ,ai i operators. Denote by S(R) its orthogonal complement.

Ariel Pacetti Half integral weight modular forms If N has valuation 1 at p, the new subspace lies in the image. In general, considering other orders, any weight 2 form which has a non-principal series prime is in the image (J-L).

Moreover,

Deﬁnitions Application Quaternions HMF Application II

Basis problem

Theorem (Eichler)

There is a natural map of T0-modules S(R) × S(R) → S2(N).

Ariel Pacetti Half integral weight modular forms In general, considering other orders, any weight 2 form which has a non-principal series prime is in the image (J-L).

Deﬁnitions Application Quaternions HMF Application II

Basis problem

Theorem (Eichler)

There is a natural map of T0-modules S(R) × S(R) → S2(N). Moreover, If N has valuation 1 at p, the new subspace lies in the image.

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Basis problem

Theorem (Eichler)

There is a natural map of T0-modules S(R) × S(R) → S2(N). Moreover, If N has valuation 1 at p, the new subspace lies in the image. In general, considering other orders, any weight 2 form which has a non-principal series prime is in the image (J-L).

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Basis problem

Theorem (Eichler)

S(R) / S2(N) :

Shim + S3/2(4N)

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Basis problem

Theorem (Eichler)

S(R) / S2(N) :

Shim + Õ S3/2(4N)

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Basis problem

Theorem (Eichler)

S(R) / S2(N) :

Shim $+ Õ S3/2(4N)

Ariel Pacetti Half integral weight modular forms If a ∈ J (R), and d ∈ N, let

ad (a) = #{[x] ∈ Rr (a)/Z : ∆(x) = −d}.

For d ∈ N0, let ed ∈ M(R) be given by

n X ad (ai ) ed = [ai ]. hai , ai i i=1

Deﬁnitions Application Quaternions HMF Application II

Ternary forms

In B, the quadratic form ∆(x) = Tr(x)2 − 4 N(x) is a quadratic negative deﬁnite form invariant under translation, hence a form in B/Q.

Ariel Pacetti Half integral weight modular forms For d ∈ N0, let ed ∈ M(R) be given by

n X ad (ai ) ed = [ai ]. hai , ai i i=1

Deﬁnitions Application Quaternions HMF Application II

Ternary forms

In B, the quadratic form ∆(x) = Tr(x)2 − 4 N(x) is a quadratic negative deﬁnite form invariant under translation, hence a form in B/Q. If a ∈ J (R), and d ∈ N, let

ad (a) = #{[x] ∈ Rr (a)/Z : ∆(x) = −d}.

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Ternary forms

In B, the quadratic form ∆(x) = Tr(x)2 − 4 N(x) is a quadratic negative deﬁnite form invariant under translation, hence a form in B/Q. If a ∈ J (R), and d ∈ N, let

ad (a) = #{[x] ∈ Rr (a)/Z : ∆(x) = −d}.

For d ∈ N0, let ed ∈ M(R) be given by

n X ad (ai ) ed = [ai ]. hai , ai i i=1

Ariel Pacetti Half integral weight modular forms The image lies in the Kohnen space. Θ(v) is cuspidal iﬀ v is cuspidal.

S(R) / S2(N) :

Θ Shim $+ Õ S3/2(4N)

Theorem (P.,Tornar´ıa)

The map Θ is T0-linear . Furthermore,

Deﬁnitions Application Quaternions HMF Application II

Theta map

Let Θ : M(R) → M3/2(4N(R), χR ) be given by X d Θ(v)(z) = hv, ed iq . d≥0

Ariel Pacetti Half integral weight modular forms The image lies in the Kohnen space. Θ(v) is cuspidal iﬀ v is cuspidal.

S(R) / S2(N) :

Θ Shim $+ Õ S3/2(4N)

Furthermore,

Deﬁnitions Application Quaternions HMF Application II

Theta map

Let Θ : M(R) → M3/2(4N(R), χR ) be given by X d Θ(v)(z) = hv, ed iq . d≥0

Theorem (P.,Tornar´ıa)

The map Θ is T0-linear .

Ariel Pacetti Half integral weight modular forms The image lies in the Kohnen space. Θ(v) is cuspidal iﬀ v is cuspidal.

S(R) / S2(N) :

Θ Shim $+ Õ S3/2(4N)

Deﬁnitions Application Quaternions HMF Application II

Theta map

Let Θ : M(R) → M3/2(4N(R), χR ) be given by X d Θ(v)(z) = hv, ed iq . d≥0

Theorem (P.,Tornar´ıa)

The map Θ is T0-linear . Furthermore,

Ariel Pacetti Half integral weight modular forms Θ(v) is cuspidal iﬀ v is cuspidal.

S(R) / S2(N) :

Θ Shim $+ Õ S3/2(4N)

Deﬁnitions Application Quaternions HMF Application II

Theta map

Let Θ : M(R) → M3/2(4N(R), χR ) be given by X d Θ(v)(z) = hv, ed iq . d≥0

Theorem (P.,Tornar´ıa)

The map Θ is T0-linear . Furthermore, The image lies in the Kohnen space.

Ariel Pacetti Half integral weight modular forms S(R) / S2(N) :

Θ Shim $+ Õ S3/2(4N)

Deﬁnitions Application Quaternions HMF Application II

Theta map

Let Θ : M(R) → M3/2(4N(R), χR ) be given by X d Θ(v)(z) = hv, ed iq . d≥0

Theorem (P.,Tornar´ıa)

The map Θ is T0-linear . Furthermore, The image lies in the Kohnen space. Θ(v) is cuspidal iﬀ v is cuspidal.

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Theta map

Let Θ : M(R) → M3/2(4N(R), χR ) be given by X d Θ(v)(z) = hv, ed iq . d≥0

Theorem (P.,Tornar´ıa)

The map Θ is T0-linear . Furthermore, The image lies in the Kohnen space. Θ(v) is cuspidal iﬀ v is cuspidal.

S(R) / S2(N) :

Θ Shim $+ Õ S3/2(4N)

Ariel Pacetti Half integral weight modular forms Given F ∈ S2(N), how to chose R such that Θ(vF ) is non-zero? Do we have an explicit formula relating the coeﬃcients to central values of twisted L-series? It should be the case that for all fundamental discriminants d in some residue classes, the following formula should hold

2 < F , F > aF , O (d) L(F , 1)L(F , d, 1) = ? p . |d| hvF , vF i

Done by Gross if N = p. Done by B¨ochererand Schulze-Pillot if N is odd and square free. Done by P. and Tornar´ıaif N = p2.

Deﬁnitions Application Quaternions HMF Application II

Questions

Here are some questions:

Ariel Pacetti Half integral weight modular forms Do we have an explicit formula relating the coeﬃcients to central values of twisted L-series? It should be the case that for all fundamental discriminants d in some residue classes, the following formula should hold

2 < F , F > aF , O (d) L(F , 1)L(F , d, 1) = ? p . |d| hvF , vF i

Done by Gross if N = p. Done by B¨ochererand Schulze-Pillot if N is odd and square free. Done by P. and Tornar´ıaif N = p2.

Deﬁnitions Application Quaternions HMF Application II

Questions

Here are some questions:

Given F ∈ S2(N), how to chose R such that Θ(vF ) is non-zero?

Ariel Pacetti Half integral weight modular forms It should be the case that for all fundamental discriminants d in some residue classes, the following formula should hold

2 < F , F > aF , O (d) L(F , 1)L(F , d, 1) = ? p . |d| hvF , vF i

Done by Gross if N = p. Done by B¨ochererand Schulze-Pillot if N is odd and square free. Done by P. and Tornar´ıaif N = p2.

Deﬁnitions Application Quaternions HMF Application II

Questions

Here are some questions:

Given F ∈ S2(N), how to chose R such that Θ(vF ) is non-zero? Do we have an explicit formula relating the coeﬃcients to central values of twisted L-series?

Ariel Pacetti Half integral weight modular forms Done by Gross if N = p. Done by B¨ochererand Schulze-Pillot if N is odd and square free. Done by P. and Tornar´ıaif N = p2.

Deﬁnitions Application Quaternions HMF Application II

Questions

Here are some questions:

Given F ∈ S2(N), how to chose R such that Θ(vF ) is non-zero? Do we have an explicit formula relating the coeﬃcients to central values of twisted L-series? It should be the case that for all fundamental discriminants d in some residue classes, the following formula should hold

2 < F , F > aF , O (d) L(F , 1)L(F , d, 1) = ? p . |d| hvF , vF i

Ariel Pacetti Half integral weight modular forms Done by B¨ochererand Schulze-Pillot if N is odd and square free. Done by P. and Tornar´ıaif N = p2.

Deﬁnitions Application Quaternions HMF Application II

Questions

Here are some questions:

2 < F , F > aF , O (d) L(F , 1)L(F , d, 1) = ? p . |d| hvF , vF i

Done by Gross if N = p.

Ariel Pacetti Half integral weight modular forms Done by P. and Tornar´ıaif N = p2.

Deﬁnitions Application Quaternions HMF Application II

Questions

Here are some questions:

2 < F , F > aF , O (d) L(F , 1)L(F , d, 1) = ? p . |d| hvF , vF i

Done by Gross if N = p. Done by B¨ochererand Schulze-Pillot if N is odd and square free.

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Questions

Here are some questions:

2 < F , F > aF , O (d) L(F , 1)L(F , d, 1) = ? p . |d| hvF , vF i

Done by Gross if N = p. Done by B¨ochererand Schulze-Pillot if N is odd and square free. Done by P. and Tornar´ıaif N = p2.

Ariel Pacetti Half integral weight modular forms + If α ∈ GL2 (F ), deﬁne the factor of automorphy Y j(α, z) = j(τ(α), zτ ). τ∈a

+ a GL2 (F ) acts on H component-wise.

Let OF denotes the ring of integers of F . If r, n are ideals, let

a b + × Γ(r, n) = α = c d ∈ GL2 (F ) : det(α) ∈ OF and −1 a, d ∈ OF , b ∈ r , c ∈ rn .

Let {b1,..., br } be representatives for the narrow class group.Deﬁne r r M M M2(n) = M2(Γ(bi , n)) S2(n) = S2(Γ(bi , n)) i=1 i=1

Deﬁnitions Application Quaternions HMF Application II

Hilbert modular forms

Let F be a totally real number ﬁeld, and a = {τ : F ,→ R}.

Ariel Pacetti Half integral weight modular forms + If α ∈ GL2 (F ), deﬁne the factor of automorphy Y j(α, z) = j(τ(α), zτ ). τ∈a

Let OF denotes the ring of integers of F . If r, n are ideals, let

a b + × Γ(r, n) = α = c d ∈ GL2 (F ) : det(α) ∈ OF and −1 a, d ∈ OF , b ∈ r , c ∈ rn .

Let {b1,..., br } be representatives for the narrow class group.Deﬁne r r M M M2(n) = M2(Γ(bi , n)) S2(n) = S2(Γ(bi , n)) i=1 i=1

Deﬁnitions Application Quaternions HMF Application II

Hilbert modular forms

Let F be a totally real number ﬁeld, and a = {τ : F ,→ R}. + a GL2 (F ) acts on H component-wise.

Ariel Pacetti Half integral weight modular forms Let OF denotes the ring of integers of F . If r, n are ideals, let

a b + × Γ(r, n) = α = c d ∈ GL2 (F ) : det(α) ∈ OF and −1 a, d ∈ OF , b ∈ r , c ∈ rn .

Let {b1,..., br } be representatives for the narrow class group.Deﬁne r r M M M2(n) = M2(Γ(bi , n)) S2(n) = S2(Γ(bi , n)) i=1 i=1

Deﬁnitions Application Quaternions HMF Application II

Hilbert modular forms

Let F be a totally real number ﬁeld, and a = {τ : F ,→ R}. + a + GL2 (F ) acts on H component-wise. If α ∈ GL2 (F ), deﬁne the factor of automorphy Y j(α, z) = j(τ(α), zτ ). τ∈a

Ariel Pacetti Half integral weight modular forms Let {b1,..., br } be representatives for the narrow class group.Deﬁne r r M M M2(n) = M2(Γ(bi , n)) S2(n) = S2(Γ(bi , n)) i=1 i=1

Deﬁnitions Application Quaternions HMF Application II

Hilbert modular forms

Let F be a totally real number ﬁeld, and a = {τ : F ,→ R}. + a + GL2 (F ) acts on H component-wise. If α ∈ GL2 (F ), deﬁne the factor of automorphy Y j(α, z) = j(τ(α), zτ ). τ∈a

Let OF denotes the ring of integers of F . If r, n are ideals, let

a b + × Γ(r, n) = α = c d ∈ GL2 (F ) : det(α) ∈ OF and −1 a, d ∈ OF , b ∈ r , c ∈ rn .

Ariel Pacetti Half integral weight modular forms Deﬁne r r M M M2(n) = M2(Γ(bi , n)) S2(n) = S2(Γ(bi , n)) i=1 i=1

Deﬁnitions Application Quaternions HMF Application II

Hilbert modular forms

Let F be a totally real number ﬁeld, and a = {τ : F ,→ R}. + a + GL2 (F ) acts on H component-wise. If α ∈ GL2 (F ), deﬁne the factor of automorphy Y j(α, z) = j(τ(α), zτ ). τ∈a

Let OF denotes the ring of integers of F . If r, n are ideals, let

a b + × Γ(r, n) = α = c d ∈ GL2 (F ) : det(α) ∈ OF and −1 a, d ∈ OF , b ∈ r , c ∈ rn .

Let {b1,..., br } be representatives for the narrow class group.

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Hilbert modular forms

Let F be a totally real number ﬁeld, and a = {τ : F ,→ R}. + a + GL2 (F ) acts on H component-wise. If α ∈ GL2 (F ), deﬁne the factor of automorphy Y j(α, z) = j(τ(α), zτ ). τ∈a

Let OF denotes the ring of integers of F . If r, n are ideals, let

a b + × Γ(r, n) = α = c d ∈ GL2 (F ) : det(α) ∈ OF and −1 a, d ∈ OF , b ∈ r , c ∈ rn .

Let {b1,..., br } be representatives for the narrow class group.Deﬁne r r M M M2(n) = M2(Γ(bi , n)) S2(n) = S2(Γ(bi , n)) i=1 i=1

Ariel Pacetti Half integral weight modular forms There are Hecke operators indexed by integral ideals (satisfying the same properties). The action can be given in terms of q-expansion. There is a theory of new subspaces.

Deﬁnitions Application Quaternions HMF Application II

Main properties

The forms have q-expansions indexed by integral ideals.

Ariel Pacetti Half integral weight modular forms The action can be given in terms of q-expansion. There is a theory of new subspaces.

Deﬁnitions Application Quaternions HMF Application II

Main properties

The forms have q-expansions indexed by integral ideals. There are Hecke operators indexed by integral ideals (satisfying the same properties).

Ariel Pacetti Half integral weight modular forms There is a theory of new subspaces.

Deﬁnitions Application Quaternions HMF Application II

Main properties

The forms have q-expansions indexed by integral ideals. There are Hecke operators indexed by integral ideals (satisfying the same properties). The action can be given in terms of q-expansion.

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Main properties

The forms have q-expansions indexed by integral ideals. There are Hecke operators indexed by integral ideals (satisfying the same properties). The action can be given in terms of q-expansion. There is a theory of new subspaces.

Ariel Pacetti Half integral weight modular forms + and deﬁne the factor of automorphy for γ ∈ GL2 (F ), θ(γz) J(γ, z) = j(γ, z). θ(z)

For n an integral ideal in OF , let

−1 −1 Γ[2˜ δ, n] = Γ[2 δ, n] ∩ SL2(F ).

Deﬁnitions Application Quaternions HMF Application II

Half integral weight HMF

Let ! X Y 2 θ(z) = eπiτ(ξ) zτ ,

ξ∈OF τ∈a

Ariel Pacetti Half integral weight modular forms For n an integral ideal in OF , let

−1 −1 Γ[2˜ δ, n] = Γ[2 δ, n] ∩ SL2(F ).

Deﬁnitions Application Quaternions HMF Application II

Half integral weight HMF

Let ! X Y 2 θ(z) = eπiτ(ξ) zτ ,

ξ∈OF τ∈a + and deﬁne the factor of automorphy for γ ∈ GL2 (F ), θ(γz) J(γ, z) = j(γ, z). θ(z)

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Half integral weight HMF

Let ! X Y 2 θ(z) = eπiτ(ξ) zτ ,

ξ∈OF τ∈a + and deﬁne the factor of automorphy for γ ∈ GL2 (F ), θ(γz) J(γ, z) = j(γ, z). θ(z)

For n an integral ideal in OF , let

−1 −1 Γ[2˜ δ, n] = Γ[2 δ, n] ∩ SL2(F ).

Ariel Pacetti Half integral weight modular forms The theory is more involved, and there is a Fourier expansion attached to each ideal in the narrow class group. There is a theory of Hecke operators as in the classical case. There is a formula relating the Hecke operators with the Fourier expansion at diﬀerent ideals.

We denote by M3/2(4n, ψ) the v.s. of such forms, and by S3/2(4n, ψ) the cuspidal ones.

Deﬁnitions Application Quaternions HMF Application II

Half integral weight HMF

Deﬁnition If ψ is a Hecke character of conductor n, a Hilbert modular form of parallel weight 3/2, level 4n and character ψ, is a holomorphic function f on Ha satisfying:

f (γz) = ψ(d)J(γ, z)f (z) ∀ γ ∈ Γ[2˜ −1δ, 4n].

Deﬁnitions Application Quaternions HMF Application II

Half integral weight HMF

Deﬁnition If ψ is a Hecke character of conductor n, a Hilbert modular form of parallel weight 3/2, level 4n and character ψ, is a holomorphic function f on Ha satisfying:

f (γz) = ψ(d)J(γ, z)f (z) ∀ γ ∈ Γ[2˜ −1δ, 4n].

We denote by M3/2(4n, ψ) the v.s. of such forms, and by S3/2(4n, ψ) the cuspidal ones.

Ariel Pacetti Half integral weight modular forms There is a theory of Hecke operators as in the classical case. There is a formula relating the Hecke operators with the Fourier expansion at diﬀerent ideals.

Deﬁnitions Application Quaternions HMF Application II

Half integral weight HMF

Deﬁnition If ψ is a Hecke character of conductor n, a Hilbert modular form of parallel weight 3/2, level 4n and character ψ, is a holomorphic function f on Ha satisfying:

f (γz) = ψ(d)J(γ, z)f (z) ∀ γ ∈ Γ[2˜ −1δ, 4n].

Ariel Pacetti Half integral weight modular forms There is a formula relating the Hecke operators with the Fourier expansion at diﬀerent ideals.

Deﬁnitions Application Quaternions HMF Application II

Half integral weight HMF

Deﬁnition If ψ is a Hecke character of conductor n, a Hilbert modular form of parallel weight 3/2, level 4n and character ψ, is a holomorphic function f on Ha satisfying:

f (γz) = ψ(d)J(γ, z)f (z) ∀ γ ∈ Γ[2˜ −1δ, 4n].

We denote by M3/2(4n, ψ) the v.s. of such forms, and by S3/2(4n, ψ) the cuspidal ones. The theory is more involved, and there is a Fourier expansion attached to each ideal in the narrow class group. There is a theory of Hecke operators as in the classical case.

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Half integral weight HMF

Deﬁnition If ψ is a Hecke character of conductor n, a Hilbert modular form of parallel weight 3/2, level 4n and character ψ, is a holomorphic function f on Ha satisfying:

f (γz) = ψ(d)J(γ, z)f (z) ∀ γ ∈ Γ[2˜ −1δ, 4n].

Ariel Pacetti Half integral weight modular forms As before, the image can be given in terms of eigenvalues. How do we compute preimages? ; use quaternionic forms.

Deﬁnitions Application Quaternions HMF Application II

Shimura map for HMF

Theorem (Shimura) + For each ξ ∈ F , there exists a T0 linear map

2 Shimξ : M3/2(4n, ψ) → M2(2n, ψ ).

Ariel Pacetti Half integral weight modular forms ; use quaternionic forms.

Deﬁnitions Application Quaternions HMF Application II

Shimura map for HMF

Theorem (Shimura) + For each ξ ∈ F , there exists a T0 linear map

2 Shimξ : M3/2(4n, ψ) → M2(2n, ψ ).

As before, the image can be given in terms of eigenvalues. How do we compute preimages?

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Shimura map for HMF

Theorem (Shimura) + For each ξ ∈ F , there exists a T0 linear map

2 Shimξ : M3/2(4n, ψ) → M2(2n, ψ ).

As before, the image can be given in terms of eigenvalues. How do we compute preimages? ; use quaternionic forms.

Ariel Pacetti Half integral weight modular forms Take B/F a quaternion algebra ramified at least all the infinite places, and R an Eichler order in it. Let M(R) be the C-v.s. spanned by class ideal representatives with the inner product ( 0 if i 6= j, h[a ], [a ]i = i j × × [Rr (ai ) : OF ] if i = j. Define the Hecke operators in the same way as before. −1 They commute, and the adjoint of Tp is p Tp.

Deﬁnitions Application Quaternions HMF Application II

Quaternionic HMF

We have to make some small adjustments to the classical picture.

Ariel Pacetti Half integral weight modular forms Let M(R) be the C-v.s. spanned by class ideal representatives with the inner product ( 0 if i 6= j, h[a ], [a ]i = i j × × [Rr (ai ) : OF ] if i = j. Deﬁne the Hecke operators in the same way as before. −1 They commute, and the adjoint of Tp is p Tp.

Deﬁnitions Application Quaternions HMF Application II

Quaternionic HMF

We have to make some small adjustments to the classical picture. Take B/F a quaternion algebra ramiﬁed at least all the inﬁnite places, and R an Eichler order in it.

Ariel Pacetti Half integral weight modular forms Deﬁne the Hecke operators in the same way as before. −1 They commute, and the adjoint of Tp is p Tp.

Deﬁnitions Application Quaternions HMF Application II

Quaternionic HMF

We have to make some small adjustments to the classical picture. Take B/F a quaternion algebra ramiﬁed at least all the inﬁnite places, and R an Eichler order in it. Let M(R) be the C-v.s. spanned by class ideal representatives with the inner product ( 0 if i 6= j, h[a ], [a ]i = i j × × [Rr (ai ) : OF ] if i = j.

Ariel Pacetti Half integral weight modular forms −1 They commute, and the adjoint of Tp is p Tp.

Deﬁnitions Application Quaternions HMF Application II

Quaternionic HMF

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Quaternionic HMF

Ariel Pacetti Half integral weight modular forms The image lies in the Kohnen space. Θ(v) is cuspidal iﬀ v is cuspidal.

Let Θ : M(R) → M3/2(4N(R), χR ) be given by

X ξ Θ(v)(z) = hv, eξiq . + ξ∈OF

Theorem (Sirolli)

The map Θ is T0-invariant. Furthermore,

Deﬁnitions Application Quaternions HMF Application II

Preimages

Theorem (J-L,Hida)

There is a natural map of T0-modules S(R) × S(R) → S2(N). The same remarks as in the classical case apply.

Ariel Pacetti Half integral weight modular forms The image lies in the Kohnen space. Θ(v) is cuspidal iﬀ v is cuspidal.

Theorem (Sirolli)

The map Θ is T0-invariant. Furthermore,

Deﬁnitions Application Quaternions HMF Application II

Preimages

Theorem (J-L,Hida)

There is a natural map of T0-modules S(R) × S(R) → S2(N). The same remarks as in the classical case apply.

Let Θ : M(R) → M3/2(4N(R), χR ) be given by

X ξ Θ(v)(z) = hv, eξiq . + ξ∈OF

Ariel Pacetti Half integral weight modular forms The image lies in the Kohnen space. Θ(v) is cuspidal iﬀ v is cuspidal.

Furthermore,

Deﬁnitions Application Quaternions HMF Application II

Preimages

Theorem (J-L,Hida)

There is a natural map of T0-modules S(R) × S(R) → S2(N). The same remarks as in the classical case apply.

Let Θ : M(R) → M3/2(4N(R), χR ) be given by

X ξ Θ(v)(z) = hv, eξiq . + ξ∈OF

Theorem (Sirolli)

The map Θ is T0-invariant.

Ariel Pacetti Half integral weight modular forms Θ(v) is cuspidal iﬀ v is cuspidal.

Deﬁnitions Application Quaternions HMF Application II

Preimages

Theorem (J-L,Hida)

There is a natural map of T0-modules S(R) × S(R) → S2(N). The same remarks as in the classical case apply.

Let Θ : M(R) → M3/2(4N(R), χR ) be given by

X ξ Θ(v)(z) = hv, eξiq . + ξ∈OF

Theorem (Sirolli)

The map Θ is T0-invariant. Furthermore, The image lies in the Kohnen space.

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Preimages

Theorem (J-L,Hida)

There is a natural map of T0-modules S(R) × S(R) → S2(N). The same remarks as in the classical case apply.

Let Θ : M(R) → M3/2(4N(R), χR ) be given by

X ξ Θ(v)(z) = hv, eξiq . + ξ∈OF

Theorem (Sirolli)

The map Θ is T0-invariant. Furthermore, The image lies in the Kohnen space. Θ(v) is cuspidal iﬀ v is cuspidal.

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

General picture

S(R) / S2(n) :

Θ Shimξ $ + Ö S3/2(4n)

Ariel Pacetti Half integral weight modular forms Deﬁnitions Application Quaternions HMF Application II

Example √ √ 1+ 5 Let F = Q( 5), ω = 2 , and consider the elliptic curve E : y 2 + xy + ωy = x3 − (1 + ω)x2.

This curve has conductor n = (5 + 2ω) (an ideal of norm 31). Let B/F be the quaternion algebra ramiﬁed at the two inﬁnite primes, and R an Eichler order of level n.

The space M2(R) has dimension 2 (done by Lassina). The element v = [R] − [a] is a Hecke eigenvector. If we compute θ(v), we get a form whose q-expansion is “similar” to Tunnell result. There are 5 non-trivial zero coeﬃcients with trace up to 100, and the twists of the original curve by this discriminants all have rank 2.

Ariel Pacetti Half integral weight modular forms