cm modular forms and dihedral representations - joint work...

CM modular forms and dihedral representationsjoint work with F. A. E. Nuccio

Nicolas Billerey

Laboratoire de mathématiques Blaise PascalUniversité Clermont Auvergne

BarcelonaDecember 5, 2019

Nicolas Billerey (LMBP - UCA) CM forms and dihedral representations Dec. 5, 2019 1 / 26

Table of contents

1 Introduction : The case of ∆ (mod 23)

2 Dihedral Galois representations

3 Main result and a corollary

4 Sketch of proof of the main result

5 Conclusion : The case of ∆ (mod 23)

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Ramanujan’s congruences

∆(q) = q∞∏n=1

(1− qn)24 =∑n≥1

τ(n)qn ∈ Z[[q]]

be the unique normalized cuspform of weight 12 and level 1.

It was conjectured by Ramanujan in 1916 that for every prime p 6= 23, wehave

τ(p) ≡

0 (mod 23) if p is a quadratic non residue modulo 23 ;2 (mod 23) if p = u2 + 23v2 ;−1 (mod 23) if p is a quadratic residue modulo 23,

but p 6= u2 + 23v2.

These congruences were first proved in 1930 by Wilton using themodularity of ∆.

∆(q) = q∞∏n=1

(1− qn)24 =∑n≥1

τ(n)qn ∈ Z[[q]]

τ(p) ≡

but p 6= u2 + 23v2.

∆(q) = q∞∏n=1

(1− qn)24 =∑n≥1

τ(n)qn ∈ Z[[q]]

τ(p) ≡

but p 6= u2 + 23v2.

CM forms

Recall the definition of a CM form.

Definition

Let ν be a non trivial Dirichlet character. A newform g =∑n≥1

cnqn has

complex multiplication by ν if

ν(p)cp = cp

for all primes p in a set of primes of density 1.

RemarksIf a newform g has complex multiplication (CM) by a Dirichlet character ν,then the field K fixed by the kernel of ν is imaginary quadractic. We alsosay that g has CM by K .

CM forms

Definition

cnqn has

ν(p)cp = cp

RemarksIf a newform g has complex multiplication (CM) by a Dirichlet character ν,then the field K fixed by the kernel of ν is imaginary quadractic.

We alsosay that g has CM by K .

CM forms

Definition

cnqn has

ν(p)cp = cp

RemarksIf a newform g has complex multiplication (CM) by a Dirichlet character ν,then the field K fixed by the kernel of ν is imaginary quadractic. We alsosay that g has CM by K .

A CM form modulo 23

Ramanujan’s congruences imply in particular that

τ(p) ≡ τ(p)

(−23p

)(mod 23),

for every prime p 6= 23.

Hence the (non-CM) modular form ∆ looks like a CM form modulo 23.

Question - First version

Does there exist a CM form g =∑n≥1

cnqn such that ‘g ≡ ∆ (mod 23)’, i.e.

there exists a prime ideal L dividing 23 in a sufficiently large number fieldsuch that for all but finitely primes p we have

cp ≡ τ(p) (mod L) ?

A CM form modulo 23

τ(p) ≡ τ(p)

(−23p

)(mod 23),

A CM form modulo 23

τ(p) ≡ τ(p)

(−23p

)(mod 23),

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Galois representations

Consider a prime ` and fix throughout a place of Q above ` with residuefield F`.

Given a newform g =∑n≥1

cnqn of weight k ≥ 2, level N ≥ 1 and

Nebentypus character χ : (Z/NZ)× → C×, denote by

ρg ,` : Gal(Q/Q)→ GL2(F`)

the unique semi-simple residual Galois representation associated with f byDeligne.

Recall that ρg ,` is unramified outside N` and for every prime p - N`, thecharacteristic polynomial of ρg ,`(Frobp) is

X 2 − cpX + χ(p)pk−1.

Galois representation associated with ∆ (mod 23)

In the case of g = ∆ and ` = 23, the construction of ρ∆,23 dates back toSerre and Swinnerton-Dyer.

The quadratic field K = Q(√−23) is unramified away from 23 and has

class number 3. Let H be its Hilbert class field. The extension H/Q isGalois and Gal(H/Q) ' D3.

Consider the group morphism

σ : Gal(Q/Q) � Gal(H/Q) ' D3 → GL2(Z) � PGL2(F23)

where the second arrow is the unique 2-dimensional irreduciblerepresentation of D3. It defines an irreducible representation, unramifiedoutside 23 and with image isomorphic to D3.

Class Field Theory and Ramanujan’s congruences then imply that

σ ' ρ∆,23.

Dihedral Galois representationsDefinitionLet ρ : Gal(Q/Q)→ GL2(F`) be an irreducible Galois representation. Wesay that ρ is a dihedral Galois representation if its projective image isisomorphic to the dihedral group Dn of order 2n with n ≥ 3 coprime to `.

On one hand, Galois representations associated with CM forms are,generally, dihedral.On the other hand, dihedral Galois representations are known to bemodular.

Hence the following natural question.

Question - Second versionLet ρ : Gal(Q/Q)→ GL2(F`) be a dihedral Galois representation. Doesthere exist a newform g with CM such that

ρ ' ρg ,` ?

On one hand, Galois representations associated with CM forms are,generally, dihedral.

On the other hand, dihedral Galois representations are known to bemodular.

ρ ' ρg ,` ?

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Serre’s parametersTo a given (continuous) irreducible Galois representation

ρ : Gal(Q/Q)→ GL2(F`)

Serre attaches a triple (N(ρ), k(ρ), ε(ρ)) consisting of

an integer N(ρ) ≥ 1 called the level such that N(ρ) is coprime to `and for every prime p 6= `

ρ is ramified at p if and only if p divides N(ρ) ;

an integer k(ρ) in the range[2, `2 − 1

]if ` > 2 and in {2, 4}

for ` = 2, called the weight and depending only on ρ locally at ` ;a character ε(ρ) : (Z/N(ρ)Z)× → F×` such that

det ρ = ε(ρ)χk(ρ)−1`

with χ` the mod ` cyclotomic character.

Serre attaches a triple (N(ρ), k(ρ), ε(ρ)) consisting ofan integer N(ρ) ≥ 1 called the level such that N(ρ) is coprime to `and for every prime p 6= `

]if ` > 2 and in {2, 4}

for ` = 2, called the weight and depending only on ρ locally at ` ;

a character ε(ρ) : (Z/N(ρ)Z)× → F×` such that

]if ` > 2 and in {2, 4}

Lack of optimality

Recall Serre’s famous modularity conjecture.

Serre’s conjectureEvery odd irreducible Galois representation ρ arises from a newform ofweight k(ρ), level N(ρ) and Nebentypus character a lift of ε(ρ).

For the Galois representation ρ∆,23 associated with ∆ and ` = 23, Serre’sparameters are (

N(ρ∆,23), k(ρ∆,23), ε(ρ∆,23))

= (1, 12, 1).

In particular, we cannot expect a given dihedral Galois representation toarise from a CM form of Serre’s ‘optimal’ type. Therefore, we have to relaxon (at least) one of these parameters.

Our choice was to seek for a CM form of ‘optimal’ weight k(ρ).

Lack of optimality

N(ρ∆,23), k(ρ∆,23), ε(ρ∆,23))

= (1, 12, 1).

Lack of optimality

N(ρ∆,23), k(ρ∆,23), ε(ρ∆,23))

= (1, 12, 1).

In particular, we cannot expect a given dihedral Galois representation toarise from a CM form of Serre’s ‘optimal’ type.

Therefore, we have to relaxon (at least) one of these parameters.

Lack of optimality

N(ρ∆,23), k(ρ∆,23), ε(ρ∆,23))

= (1, 12, 1).

Lack of optimality

N(ρ∆,23), k(ρ∆,23), ε(ρ∆,23))

= (1, 12, 1).

The setting

Let ρ : Gal(Q/Q)→ GL2(F`) be a dihedral Galois representation. Wedenote by

Pρ : Gal(Q/Q)ρ−→ GL2(F`) � PGL2(F`)

its projectivization.

By assumption Pρ(Gal(Q/Q)

)' Dn with Dn dihedral group of order 2n

with n ≥ 3 and coprime to `. Denote by Cn the unique cyclic subgroup oforder n in Dn.

Let K be the quadratic field cut out by the kernel of the character

ω : Gal(Q/Q)Pρ−→ Dn � Dn/Cn ' {±1}.

It is unramified outside N`.

The setting

Main result

Theorem (B.-Nuccio)With the previous notation, assume that

the field K is quadratic imaginary ;we have 2 ≤ k(ρ) ≤ `− 1 and ` ≥ 5.

Then, the representation ρ is modular and arises from a newform withcomplex multiplication by K of weight k(ρ) and level

N ′ =

{N(ρ) if ` is unramified in K ;`2N(ρ) otherwise.

Moreover, the following properties hold :1 if ` is ramified in K , then ` ∈ {2k(ρ)− 1, 2k(ρ)− 3} ;2 if ε(ρ) is trivial then, ρ arises from a newform with CM by K ,

weight k(ρ), level dividing N ′ and trivial Nebentypus character.

Main result

Theorem (B.-Nuccio)With the previous notation, assume that

the field K is quadratic imaginary ;we have 2 ≤ k(ρ) ≤ `− 1 and ` ≥ 5.

Then, the representation ρ is modular and arises from a newform withcomplex multiplication by K of weight k(ρ) and level

N ′ =

{N(ρ) if ` is unramified in K ;`2N(ρ) otherwise.

Moreover, the following properties hold :1 if ` is ramified in K , then ` ∈ {2k(ρ)− 1, 2k(ρ)− 3} ;2 if ε(ρ) is trivial then, ρ arises from a newform with CM by K ,

weight k(ρ), level dividing N ′ and trivial Nebentypus character.

Application to abelian varieties of GL2-type

Let A/Q be a simple abelian variety. Denote by EndQ(A) its ring ofendomorphisms defined over Q.

Assume that A is of GL2-type in the terminology of Ribet, meaning thatE = EndQ(A)⊗Q is a number field of degree dim(A). For every primeideal λ in the integer ring of E above ` with residue field Fλ, denote by

ρA,λ : Gal(Q/Q)→ GL2(Fλ)

the corresponding 2-dimensional Galois representation.

CorollaryWith the notation above, assume ` ≥ 5, ` - NA (the conductor of A) andthe image of ρA,λ is included in the normalizer of a non-split Cartansubgroup of GL2(Fλ). Then, ρA,λ arises from a CM newform of weight 2and level N(ρA,λ) (dividing NA).

Assume that A is of GL2-type in the terminology of Ribet, meaning thatE = EndQ(A)⊗Q is a number field of degree dim(A).

For every primeideal λ in the integer ring of E above ` with residue field Fλ, denote by

Remarks

For elliptic curves the corollary was proved by Chen in 2002.

The case of weight k(ρ) = 2 in our main result was proved in 2011 byNualart.The main difference in the higher weight situation lies in the factthat K might ramify at ` ;Our result is optimal : cf. the case of ∆ (mod 23).

Remarks

For elliptic curves the corollary was proved by Chen in 2002.The case of weight k(ρ) = 2 in our main result was proved in 2011 byNualart.

The main difference in the higher weight situation lies in the factthat K might ramify at ` ;Our result is optimal : cf. the case of ∆ (mod 23).

Remarks

For elliptic curves the corollary was proved by Chen in 2002.The case of weight k(ρ) = 2 in our main result was proved in 2011 byNualart.The main difference in the higher weight situation lies in the factthat K might ramify at ` ;

Our result is optimal : cf. the case of ∆ (mod 23).

Remarks

For elliptic curves the corollary was proved by Chen in 2002.The case of weight k(ρ) = 2 in our main result was proved in 2011 byNualart.The main difference in the higher weight situation lies in the factthat K might ramify at ` ;Our result is optimal : cf. the case of ∆ (mod 23).

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Hecke-Shimura construction

Let K be an imaginary quadratic field of disciriminant −D and integerring OK . Denote by AK the ring of adèles of K . Let k ≥ 2 be an integer.

A Grössencharacter of K of weight k is a continuous homomorphism

δ : A×K → C×

which is trivial on K× and whose infinite type is δ∞(z) = z1−k .

View such a δ as homomorphism

{fractional ideals of K coprime to m} → C×

where m is the conductor of δ and define

η(m) = δ(mOK )/mk−1, for m ∈ Z.

Then η induces a Dirichlet character modulo M = NK/Q(m).

δ : A×K → C×

Theorem (Hecke-Shimura)

The q-series gδ =∑

a integral(a,m)=1

δ(a)qNK/Q(a) is a newform of weight k , level MD

and Nebentypus character η(−D·)with complex multiplication by K .

Furthermore, all CM newforms arise this way.

In order to prove the main result we have to pin down a Grössencharacterwith

good reduction properties ;weight k(ρ) ;controlled ramification.

a integral(a,m)=1

good reduction properties ;

weight k(ρ) ;controlled ramification.

a integral(a,m)=1

good reduction properties ;weight k(ρ) ;

controlled ramification.

a integral(a,m)=1

good reduction properties ;weight k(ρ) ;controlled ramification.

Construction of the appropriate Grössencharacter

Let ρ : Gal(Q/Q)→ GL2(F`) be a dihedral Galois representation. With thenotation of the theorem, we have

ρ ' IndGal(Q/K)

Gal(Q/Q)(ϕ), for some character ϕ : Gal(Q/K )→ F×` .

The composition map

α : A×KrecK−−→ Gal(Q/K )ab ϕ−→ F×`

lift−→ C×

defines a Grössencharacter but with trivial type at infinity.

ρ ' IndGal(Q/K)

The composition map

lift−→ C×

ρ ' IndGal(Q/K)

The composition map

lift−→ C×

Put k = k(ρ) and let λ be the prime ideal dividing ` in K induced by ourchoice of a place of Q above `.

Starting from α and using Class Field Theory, we construct aGrössencharacter δ of K such that

has weight k ;reduces to ϕ ;has the same conductor as α away from λ

has controlled ramification at λ with respect to that of ϕ and K .

has weight k ;

reduces to ϕ ;has the same conductor as α away from λ

has weight k ;reduces to ϕ ;

has the same conductor as α away from λ

Study of the ramification of ϕ and K at `

Here we denote by I` a inertia group at ` in Gal(Q/Q) and by Iλ its inertiasubgroup at λ in Gal(Q/K ).

Proposition1 Assume that ρ|I` is reducible. Then,

I either `OK = λ · λ is split in K , and in this case ϕ is unramified at λand tamely ramified at λ with ϕ|Iλ = χk−1

` ;I or ` is ramified in K and in this case, ` = 2k − 1 and ϕ is unramified

at λ.2 Assume that ρ|I` is irreducible. Then,

I either ` is inert in K ;I or ` is ramified in K and ` = 2k − 3.

In both cases, ϕ is ramified at λ and ϕ|Iλ = ψk−12 with ψ2 a

fundamental character of level 2.

I or ` is ramified in K and in this case, ` = 2k − 1 and ϕ is unramifiedat λ.

2 Assume that ρ|I` is irreducible. Then,

at λ.

2 Assume that ρ|I` is irreducible. Then,

I either ` is inert in K ;

I or ` is ramified in K and ` = 2k − 3.

End of the proof

Take δ to be the suitable modification of

lift−→ C×

and attach to it the CM newform gδ.

Conclude that gδ has the required properties using the formula for theconductor of an induced representation and the previous proposition.

End of the proof

Take δ to be the suitable modification of

lift−→ C×

and attach to it the CM newform gδ.

Conclude that gδ has the required properties using the formula for theconductor of an induced representation and the previous proposition.

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Back to the example of ∆ (mod 23)

According to the theorem, there exists a newform g of weight 12,level 529 = 232 and trivial Nebentypus with CM by K = Q(

√−23) such

thatρ∆,23 ' ρg ,23.

Using the construction of Hecke-Shimura, we can directly compute the firstcoefficients of g . The class group of K has order 3 and is generated by anideal above 2.

We define δ to be a Grössencharacter of K with weight 12 andconductor

√−23OK such that, in the previous notation, η =

(−23·).

We find that

gδ = q + (−21b2 − 4b + 84)q2 + (53b2 + 251b − 212)q3 + · · ·

where b a root of X 3 − 6X − 3.

√−23) such

thatρ∆,23 ' ρg ,23.

(−23·).

We find that

gδ = q + (−21b2 − 4b + 84)q2 + (53b2 + 251b − 212)q3 + · · ·

√−23) such

thatρ∆,23 ' ρg ,23.

(−23·).

We find that

gδ = q + (−21b2 − 4b + 84)q2 + (53b2 + 251b − 212)q3 + · · ·

√−23) such

thatρ∆,23 ' ρg ,23.

(−23·).

We find that

gδ = q + (−21b2 − 4b + 84)q2 + (53b2 + 251b − 212)q3 + · · ·

cm modular forms and dihedral representations - joint work...

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