On genuine characters of the metaplectic group of SL2

On genuine characters of the metaplectic group of SL

2

(o) and theta functions

Hiroshi Noguchi

Graduate school of science, Kyoto university

January 25, 2021

the Dedekind eta function η(z )

e(z) =e^2πiz for z ∈C h⊂C: the upper half plane

η(z) =e(z/24) ∏

m≥1

(1−e(mz)), z ∈h

The modular form of SL₂(Z) of weight 1/2 isη(z).

The modular form of SL2(Z) of weight 3/2 isη³(z).

χ₁₂: the primitive character mod 12 χ4: the primitive character mod 4

η(z) = 1 2

∑

m∈Z

χ12(m)e(mz/24), η³(z) = 1 2

∑

m∈Z

mχ4(m)e(mz/8)

Suppose that F =Ror F is a finite extension ofQp. x(g) =

{c c ̸= 0

d c = 0 g = (a b

c d )

∈SL2(F)

⟨·,·⟩F: the quadratic Hilbert symbol for F

c(g,h) =⟨x(g)x(gh),x(h)x(gh)⟩F: the Kubota 2-cocycle on SL₂(F) the metaplectic group of SL₂(F):

SL^2(F) ={[g, τ]|g ∈SL2(F), τ ∈ {±1}}

its multiplication law:

[g, τ][h, σ] = [gh, τ σc(g,h)], [g, τ],[h, σ]∈SL^2(F) This is a nontrivial double covering group ofSL2(F).

Put [g] = [g,1].

j˜:SL^₂(R)×h→C: an automorphy factor given by

˜j(γ,z) =





 τ√

d ifc = 0,d >0,

−τ√

d ifc = 0,d <0, τ(cz +d)^1/2 ifc ̸= 0,

γ = [g, τ]∈SL^2(R)

where−π <arg(cz+d)≤π.

(·

· )

: the Jacobi symbol c ∈Z\{0}

d ∈2Z+ 1 such that (c,d) = 1 (c

d )_∗

= ( c

|d| )

, (c

d )

∗ =t(c,d) (c

d )_∗

, t(c,d) =

{−1 c,d <0 1 otherwise.

We understand ( 0

±1

)_∗

=(₀

1

)

∗= 1, ( 0

−1

)

∗=−1.

the transformation formula of η(z) (Rademacher [5]) η(g(z)) =v_η(g)˜j([g],z)η(z), g(z) = az+b

cz+d ∈h v_η(g): the multiplier system of η(z)

=









 (d

c )_∗

e

((a+d)c −bd(c²−1)−3c 24

)

c: odd (c

d )

∗e

((a+d)c−bd(c²−1) + 3d −3−3cd 24

)

c: even.

η(z) andη³(z) are theta functions defined by a sum on Z.

Problem

F: a totally real number field o: the ring of integers ofF

When does a Hilbert modular theta series of weight 1/2 with respect to SL2(o) exist?

Feng [1] (1983) Naganuma [4] (1984)

We solve the problem above completely.

Multiplier systems for SL

₂

(o)

F: a totally real number field

Fv: the completion of F at a place v A: the adele ring ofF

SL^₂(A): the adelic metaplectic group, a double covering ofSL₂(A) There exists a canonical embeddingSL^2(Fv)→SL^2(A) for each v.

ιv: the entrywise embedding of SL2(F) into SL2(Fv) {∞1,· · · ,∞n}: the set of infinite places of F

Putιi =ι_∞i.

Assume thatv <∞.

ov: the ring of integers ofFv

p_v: the maximal ideal of o_v

ι: the embedding of SL2(F) into SL2(A)

˜

ι:SL₂(F)→SL^₂(A): a canonical embedding,g 7→([ι_v(g)])_v SL₂(Af): the finite part ofSL₂(A)

˜

ι_f :SL₂(F)→SL^₂(Af), ˜ι_f(g) = ([ι_v(g)])_v<∞×([1₂])_v|∞

j˜:SL^2(R)×h→C: an automorphy factor given by

j˜(γ,z) =





 τ√

d if c = 0,d >0,

−τ√

d if c = 0,d <0, τ(cz+d)^1/2 if c ̸= 0.

γ = [g, τ]∈SL^₂(R)

Definition

Let Γ⊂SL2(o) be a congruence subgroup. The function v: Γ→C^× is a multiplier system of half-integral weight

⇐⇒def v(γ)∏_n

i=1j˜([ιi(γ)],zi) is an automorphy factor for Γ×hⁿ.

Lemma

A function v: Γ→C^× is a multiplier system of half-integral weight

⇐⇒v(γ1)v(γ2) =

∏n i=1

c_R(ιi(γ1), ιi(γ2))v(γ1γ2) γ1, γ2∈Γ,

where c_R(·,·) is the Kubota 2-cocycle at infinite places.

K_Γ: the closure of Γ inSL₂(Af)

ι_f :SL2(F)→ SL2(Af): the projection ofιto the finite part KΓ is a compact open subgroup and we haveι⁻_f ¹(KΓ) = Γ.

Lemma

Let λ: ˜K_Γ →C^× be a genuine character. Put v_λ(γ) =λ(˜ι_f(γ)) forγ ∈Γ.

Then v_λ is a multiplier system of half-integral weight for Γ.

Now suppose that v <∞. K₁(4)_v ={γ =

(a b c d

)

∈SL₂(o_v)|c ≡0,d ≡1(mod 4)}. sv(g) = [g,sv(g)]: the splitting onK1(4)v

s_v(g) =







1 c ∈o^×_v

⟨c,d⟩Fv c ∈p_v\{0}

⟨−1,d⟩Fv c = 0

forg = (a b

c d )

∈SL₂(o_v).

IfK_Γ⊂K₁(4)_f =∏

v<∞K₁(4)_v, a splittings:K_Γ→SL^₂(A) given by s(γ) = (sv(γ))v<∞×([12])_v|∞.

K˜Γ =s(KΓ)· {[12,±1]}: the inverse image ofKΓ in SL^2(Af) s(K_Γ)⊂SL^₂(Af) is a compact open subgroup.

Γ: a congruence subgroup v0 : Γ→C^×,v0(γ) =∏

v<∞sv(γ) Proposition

If Γ⊂Γ1(4) =

{(a b c d

)

∈SL2(o)|c ≡0,d ≡1(mod 4) }

, then v0 is a multiplier system of half-integral weight of Γ.

v: Γ→C^×: a multiplier system of half-integral weight Lemma

There exists a genuine character λ: ˜KΓ→C^× such thatvλ =v if and only if there exists a congruence subgroup Γ^′ ⊂Γ∩Γ₁(4) such that v(γ) =v₀(γ) for any γ ∈Γ^′.

Proposition

IfF ̸=Q, then any multiplier system v of half-integral weight of any congruence subgroup Γ is obtained from a genuine character of ˜K_Γ.

Kf =∏

v<∞SL2(ov): a compact open subgroup ofSL2(Af).

K˜f the inverse image ofKf in SL^2(Af) Proposition

Let v be a multiplier system of half-integral weight of SL₂(o).

There exists a genuine character λ: ˜K_f →C^× such that v=v_λ. Proposition

There exists a multiplier system of half-integral weight ofSL₂(o) if and only if 2 splits completely in F/Q.

In this case, there exists a genuine character of SL^2(ov) for any v <∞.

PutS₂ ={v<∞ |F =Q2}and T₃={v <∞ |q_v = 3}. Proposition

Let v be a multiplier system of half-integral weight of SL₂(o). Then there exist continuous functions κ_v(g) on SL₂(o_v) for each v ∈S₂∪T₃ such that

v(g) =v₀(g) ∏

v∈S2∪T3

κ_v(g).

For example, for F =Q, we havev_η(g) =v₀(g)κ₂(g)κ₃(g), where

v₀(g) =







 (d

c )_∗

c : odd (c

d )

∗ c : even, κ2(g) =

{e(₃

8[(a+d)c−3c])

c : odd e(₃

8[(b−c)d+ 3(d −1)])

c : even, κ3(g) =e(₋₁

3

[(a+d)c−bd(c²−1)]) .

The condition of the existence of a theta function

Suppose that 2 splits completely in F/Q.

There exists a genuine character λ_v :SL^₂(o_v)→C^× for anyv <∞. ψ:A/F →C^×: an additive character such that its v-component ψv(x) equals e(x) for any v | ∞

ψ_β(x) =ψ(βx) forβ ∈F^× K_f =∏

v<∞SL₂(o_v)

λ_f: a genuine character,λ_f(g) =∏

v<∞λv(gv) for g = (gv)v ∈K˜_f w = (w1,· · ·,wn)∈ {1/2,3/2}ⁿ

j^λf,w(γ,z): an automorphy factor ofSL2(o)×hⁿ

⇐⇒def j^λf,w(γ,z) = ∏

v<∞

λ_v([γ])

∏n i=1

˜j([ι_i(γ)],z_i)^wi, z = (z₁,· · · ,z_n).

M_w(SL₂(o), λ_f): the space of Hilbert modular forms onhⁿ with respect to j^λf,w(γ,z)

h(z)∈Mw(SL2(o), λf)⇐⇒^def h(γ(z)) =j^λf,w(γ,z)h(z) Here, γ(z) = (ι₁(γ)(z₁),· · ·, ι_n(γ)(z_n)).

j^λf,w(−1₂,z)̸= 1 =⇒M_w(SL₂(o), λ_f) ={0} S(Fv): the Schwartz space of Fv

ω_ψβ,v: the Weil representation of SL^₂(F_v) (ωψ_β,v,S(Fv))^λv

={f ∈S(F_v)|ω_ψβ(γ)f =λ_v(γ)f for anyγ ∈SL₂(o_v)} λ_∞,wi: a genuine character of lowest weight wi

(ω_ψβ,∞i,S(R))^λ∞,wi =Cx^wi−(1/2)e(iι_i(β)x²): a space of lowest weight vectors

K =K_f ×∏

v|∞SO(2)

w = (w₁,· · ·,w_n)∈ {1/2,3/2}ⁿ {∞1,· · · ,∞n}: the set ofv | ∞

λ: ˜K →C^×: a genuine character such that its v-component equals λv, whereλ_∞i =λ_∞,w_i for anyi,λv =µ_β orϵv if v <∞.

S2 ={v |Fv =Q2} S_∞={∞i |wi = 3/2}

Now suppose that v <∞

qv: the order of the residue field ov/pv

S₃ ={v |q_v = 3, λ_v ̸=ϵ_v}

(ω_ψβ,v,S(F_v))^λv ̸={0} ⇐⇒ord_vψ_β ≡ {

0 (mod 2) ifλv =ϵv

1 (mod 2) otherwise.

β ∈F₊^×: totally positive a: a fractional ideal ofF S(A): the Schwartz space ofA (ω_ψβ,S(A))^λ ={ϕ=∏

vϕv ∈S(A)|ϕv ∈(ω_ψβ,v,S(Fv))^λv for anyv}. Whenv ∈S2∪S3, we put

f_v(x) =







1 if x∈1 + 2pv

−1 if x∈ −1 + 2p_v 0 otherwise.

chA: the characteristic function of a setA f = ∏

v∈S2∪S3

f_v× ∏

v<∞,v∈/S2∪S3

cha⁻_v¹.

f_∞,i(x) =x^wi−(1/2)e(iιi(β)x²) for x∈R ϕ=f ×∏_n

i=1f_∞,i ∈(ω_ψβ,S(A))^λ

Now suppose that S₃ ⊂T₃ ={v <∞ |q_v = 3}. G: the set of triplets (β,S3,a) satisfying (A) and (B):

(A) |S2|+|S3|+|S_∞| ∈2Z, (B) (8β)d ∏

v∈S3

pv =a²

The equivalence relation∼onG is defined by

(β,S3,a)∼(β^′,S₃^′,a^′)⇐⇒S3 =S₃^′, β^′ =γ²β, a^′ =γa for some γ ∈F^×.

Suppose that (β,S3,a)∈Gsuch that (a,6) = 1.

We define a theta function θ_ϕ:hⁿ→Cby θ_ϕ(z) = ∑

ξ∈a⁻¹

f(ι_f(ξ)) ∏

∞i∈S_∞

ι_i(ξ)

∏n i=1

e(ι_i(βξ²)z_i), z = (z₁,· · · ,z_n).

The equivalent triplets ofG define the same theta function.

Theorem

Suppose that 2 splits completely in F/Q. Let β∈F₊^×,S₃ and w1, . . . ,wn∈ {1/2,3/2} be as above.

There exists ϕ∈(ω_ψβ,S(A))^λ such that θ_ϕ̸= 0 if and only if there exists a fractional ideal a ofF such that (β,S₃,a)∈G.

H={ ∏

v∈T3

p^e_v^v | ∑

v∈T3

e_v ∈2Z}

Cl⁺: the narrow ideal class group ofF Cl⁺² ={c²|c∈Cl⁺}

H: the image of¯ H inCl⁺/Cl⁺² [b]: the image of b∈Cl⁺ in Cl⁺/Cl⁺²

Theorem

Let w₁, . . . ,w_n∈ {1/2,3/2}be as above.

(1) Suppose that |S₂|+|S_∞|is even.

Then there exists (β,S3,a)∈G if and only if [d]∈H.¯ (2) Suppose that |S₂|+|S_∞|is odd.

Then there exists (β,S3,a)∈G if and only ifT3 ̸=∅ and [dpv0]∈H¯ for any fixed v₀ ∈T₃.

Theorem

The theta function θϕ is a nonzero Hilbert modular form of weight w for SL2(o) with respect to a multiplier system.

Every theta function of weight w for SL₂(o) with a multiplier system is obtained in this way.

Proposition

Let Sq:Cl→Cl⁺ be the homomorphism given by [a]7→[a²]. The number of equivalence classes of Gis equal to

[E⁺ :E²] ∑

S3⊂T3

(A)

|Sq⁻¹(d ∏

v∈S3

p_v)|,

where S₃ ranges over all subset ofT₃ satisfying |S₂|+|S₃|+|S_∞| ∈2Z. Here, E⁺ is the group of totally positive units ofF andE² is the subgroup of squares of units.

The case F is a real quadratic field

D >1: a square-free integer such thatD ≡1 mod 8 Suppose that F =Q(√

D), then 2 splits completely in F/Q. If there exists (β,S3,a)∈G, one of the followings holds.

(C1) (8β)d=a²

(C2) (8β)dp=a² such that N_F/Q(p) = 3

(C3) (8β)dp¯p=a² such that N_F/Q(p) =N_F/Q(¯p) = 3.

If|S_∞|is even, then (C1) or (C3) holds.

If|S_∞|is odd, then (C2) holds.

Proposition

Suppose that F =Q(√

D), where D>1 is a square-free integer such that D ≡1 mod 8.

There exist β ∈F₊^× and a fractional ideala satisfying (C1) ⇐⇒p ≡1 mod 4 for any prime p|D.

(C2) ⇐⇒p ≡0 or 1 mod 3 for any prime p|D.

(C3) ⇐⇒D ≡1 mod 24 and p≡1 mod 4 for any prime p|D.

Example, D = 793

PutD = 793 = 13·61.

Then Cl⁺ has order 8 and the fundamental unit ofF has norm 1.

There existβ ∈F₊^×and a fractional ideal asatisfying any condition of (C1), (C2) or (C3).

ρ= (5 +√ D)/2.

q3 = (3,1−√

D): a prime ideal which divides (3) q₂ = (2,(1 +√

D)/2): a prime ideal which divides (2) Since N_F/Q(ρ) =−3·8², we have (ρ) =q⁶₂q₃.

β =ρ/8√

D ∈F₊^× a=q³₂q3

Then we have

(8β)dq3=a².

We assume that S_∞={∞1} and thatι1 =id.

Then we have

θ_ϕ(z) = ∑

ξ∈a⁻¹

f(ι_f(ξ))ξe(βξ²z₁)e( ¯βξ¯²z₂),

where ¯ξ =a−b√

D if ξ=a+b√ D.

S₂ ={v₂,v¯₂|p_v2=q₂,p_v¯2 = (2)/q₂} S3 ={v3|pv3=q3}

f(ι_f(ξ)) = ∏

v∈S2∪S3

f_v(ι_v(ξ)) for ξ ∈a⁻¹

reference I

X. Feng, An Analog of η(z) in the Hilbert Modular Case. Journal of number theory. (1983).

W. F. Hammond, The modular groups of Hilbert and Siegel. American Journal of Mathematics. (1966).

G. Lion and M. Vergne, The Weil representation, Maslow index and Theta series. Springer Science and Business Media New York. (1980) H. Naganuma, Remarks on the modular imbedding of Hammond.

Japan. J. Math. 379 - 387 (1984)

H. Rademacher, Topics in Analytic Number Theory. Springer Berlin Heidelberg. (1973).