On Theta Series In Characteristic 2
∗
Chih-Yun Chuang
National Tsing Hua University
Hsinchu, Taiwan, R. O. China
E-mail: [email protected]
Ting-Fang Lee
National Tsing Hua University
Hsinchu, Taiwan, R. O. China
E-mail: [email protected]
Abstract
Let k = Fq(t), with q even. We derive transformation law for
natural theta series in characteristic two. These theta series come from left ideal classes of Eichler orders in a definite quaternion algebra over k with respect to the place at infinity. We also show that these theta series give rise to automorphic forms of Drinfeld type, and are connected with the problem of representing polynomials as specific integral quadratic form of other polynomials.
∗Research supported by NSC:M99-2119-M-002-024 and National Center for Theoretical
Key words: theta series, function fields of characteristic 2, definite
quater-nion algebra with respect to ∞, Eichler orders, automorphic forms of Drinfeld type.
1
Introduction
This papers aims at explicit construction of “modular forms”in characteristic two. We start with a global function field, in particular the rational function
field k of one variable t over Fq, with q a power of 2. With the infinite place ∞ singled out, interesting arithmetic can arise by taking any Eichler order in a definite quaternion algebra D over k. Here “definite”means the algebra in question ramifies at the place ∞.
We have natural quadratic forms of four variables coming from the norm forms of ideals. If the Eichler order chosen has class number n, for 1 ≤ i, j ≤
n, we introduce theta series θij into this characteristic 2 world. These series are generating functions for the representation numbers of polynomials as a
specific integral quadratic form in other polynomials. They are written as Fourier series on a ∞-adic upper half plane (cf. Weil [8]). Our main result
is a transformation law for these theta series, with respect to action of the discrete group Γ0(N0) ⊂ SL2(Fq[t]), where N0 is the “level”of the chosen Eichler order. This transformation law allows us to view our theta series as automorphic functions on the locally compact group GL2(k∞), where k∞ is the completion of k at ∞. Moreover, they are the so-called automorphic forms of Drinfeld type (cf. [3] and [6]). These automorphic forms are very
2 modular forms. Particularly, one may apply our explicitly constructed
theta series to the study of elliptic curves over even characteristic function fields which is one of our motivation. We also believe that our approach can
lead to effective construction of all automorphic forms of Drinfeld type in characteristic two.
The odd characteristic case of theta series from definite quaternion al-gebras has been treated by Wei-Yu in [9]. We do the characteristic 2 game
here. Theta series are introduced in Section 2. The method for establishing transformation law is of course Fourier analysis, following Ruck (cf. [5]),
and parallels closely the classical lines of Hecke and Eichler. In Section 3 and Section 4 this work is carried out. In section 5, we show that our theta
series give rise to automorphic forms of Drinfeld type. In the final section, we illustrate our theta series by giving analogue of formulas for the sums of
two squares, and four squares. Here in characteristic two, the natural sum of squares have to be with extra xy terms. However, the resulting formulas
for the representation numbers are just as neat as the classical formulas of Jacobi.
k : the rational function field Fq(t) with q even.
℘ : the additive homomorphism defined by ℘(x) = x2+ x. A : the polynomial ring Fq[t].
∞ : the infinite place corresponds to the degree valuation v∞. π∞ : t−1, a fixed uniformizer at ∞
k∞ : Fq((t−1)), i.e.the completion of k at ∞. O∞ : Fq[[t−1]].
P : a finite prime(place) of k.
kP : the completion of k at the finite prime P . AP : the closure of A in kP.
ˆ
k : Q0
P kP, the finite adele ring of k. ˆ
A : Q
P AP.
We identify ideals of A with the monic polynomials in A by using the
same notation.
2
Brandt Matrices and Theta Series
Given a quaternion algebra D over k ramified at ∞ (call D definite). Let
N− be the product of the finite ramified primes of D. We first note
Lemma 2.1. Any definite quaternion algebra over k can be written as D =
k + ku + kv + kuv, where k(u)/k is inert at ∞ and v2 = b with odd degree b ∈ A.
Proof. We have D∞ is division,it contains F2q. Take ζ0 ∈ Fq− ℘(Fq). Let u be the root of x2 + x = ζ0 +
1
extension inert at ∞, and at all primes dividing N−. (cf.[4]) Hence L can be embedded in D.
For this L, we can find b ∈ A and v2 = b such that D = L + Lv. This b lies be odd degree. If deg b is even, b must in N (L∗∞) which contradicts to the fact that D∞ is division.
Choose a maximal order R of D. For any finite prime P , let RP = R ⊗AAP, DP = D ⊗kkP, and ˆR = R ⊗AA, ˆˆ D = D ⊗kˆk.
For an ideal N+ in A coprime to N−, we consider an Eichler order (cf. [1] and [7]) RN+,N− of level N+, which is the subring
{x ∈ R|ϕP(x) ≡ ∗ ∗ 0 ∗ mod N+AP, ∀P |N+},
where for each P |N+, ϕ
P : DP → Mat2(kP) is an isomorphism such that ϕ−1P (Mat2(AP)) = RP.
The localizations of Eichler order (RN+,N−)P = RN+,N−⊗AAP, are given
by (RN+,N−)P = RP for P - N+, x ∈ RP : ϕP(x) ≡ ∗ ∗ 0 ∗ mod N +A P for P |N+.
Given a left ideal I of RN+,N−. Let RI be the right order of I in D, and
set I−1 = {b ∈ D|IbI ⊂ I} which is a right ideal for RN+,N− whose left order
is the right order of I. The reduced norm of I is denoted by N(I), which is
Suppose the class number of RN+,N− is n. Let {I1, ..., In} be a set of left
ideals of RN+,N− representing the distinct ideal classes, with I1 = RN+,N−.
Let wi := #(R×i )/(q − 1) where Ri is the right order of Ii. Note all these orders Ri are also Eichler order of level N+. Consider Mij = Ij−1Ii, which is a left ideal of Rj with right order Ri. Choose a generator Nij = f /g of N(Mij) where f and g are relatively prime monic elements in A. Define for each monic polynomial m in A,
BN+,N−
ij (m) :=
#{b ∈ Mij|(N(b)/Nij) = (m)} (q − 1)wj
,
and the m-th Brandt matrix
BN+,N−(m) := (BN+,N−
ij (m))1≤i,j≤n ∈ Matn(Z). We use the notations Bij(m) and B(m) instead of BN
+,N −
ij (m) and BN
+,N −(m)
when N+ and N− are fixed. We also write Bij(m) = 1 (q − 1)wj X (λ)=(m) Bij0 (λ) with Bij0 = #{b ∈ Mij|N(b)/Nij = λ}.
In the special case when N− = P0 and N+= 1. We recall
Proposition 2.2. For each monic m ∈ A, the row sums P
jBij(m) are independent of i and equals to
σ(m)P0 := X m0|m monic P0-m qdeg(m0) = X m0|m monic P0-m |m0|. Proof. cf. [2]
Define additive character σ : Fq → C× by σ() = exp(πiTFq/F2()) and
ψ∞: k∞ → C× by ψ∞(y) = σ(Res∞(ydt)). We introduce (cf. [5] and [9])
Definition 2.3. For each (i, j) 1 ≤ i, j ≤ n, theta series θij is the function on (x, y) ∈ k∞× × k∞ θij(x, y) := X b∈Mij φ∞( N(b) Nij xt2)ψ∞( N(b) Nij y),
where φ∞ is the characteristic function of O∞.
Remark. It is easy to check the following properties: (1) θij(x, y) = X λ∈A deg(λ)≤v∞(x)−2 B0ij(λ)ψ∞(λy). (2) θij(x, y + h) = θij(x, y) for h ∈ A. (3) θij(α, βx + y) = θij(x, y) for α ∈ O×∞, β ∈ O∞.
3
Fourier Analysis
Let D∞ = D ⊗k k∞, which is a division algebra. For α ∈ D∞, ¯α denotes its conjugate. Let V∞ by the valuation on D∞ such that V∞ = 2v∞ on k∞. Fix α, β ∈ k∞× with v∞(α) > v∞(β) − 2. Let Φα,β : D∞ → C defined by Φα,β(w) = φ∞(N(w)α)ψ∞(N(w)β).
Define [·, ·] : D∞× D∞ → C× by [w, w∗] = ψ∞(T(ww∗)). Here T is the reduced trace on D∞. Let dw be a Haar measure on D∞. Define the Fourier transform of Φα,β by Φ∗α,β(w∗) = Z D∞ Φα,β(w)[w, w∗]dw. Lemma 3.1. If v∞(N(w∗) α β2 < 0), then Φ ∗ α,β(w ∗) = 0.
Proof. Since TF
q2/Fq(Fq2) = Fq, we can choose ∈ Fq2 with σ(TFq2/Fq()) 6= 1
and let w0 = tw∗. Then Φ∞(T(w0w∗)) 6= 1, and
V∞(w0) = v∞(N(w0)) ≥ 3 + v∞(α) − 2v∞(β) = 3 + 1
2V∞(α) − V∞(β). Since v∞(α) > v∞(β) − 2, one gets
V∞(w0) + 1
2V∞(α) ≥ 0, V∞(w0) + 1
2V∞(β) ≥ 2. Now check the following equality directly,
Φ∗α,β(w∗) = Z D∞ φ∞(N(w + w0)α)ψ∞(N(w + w0)β + T((w + w0)w∗)dw =φ∞(T(w0w∗)) Z D∞ [φ∞(N(w + w0)α) · ψ∞(N(w)β + N(w0)β + T(w ¯w0)β + T(ww∗))]dw If φ∞(N(w)α) = 1, then V∞(w) ≥ −12V∞(α) and so φ∞(N(w +w0)α) = 1. Furthermore, v∞(T(w ¯w0)β) ≥ 1 2V∞(ww0β) ≥ 3 2. Thus v∞(T(w ¯w0)β) ≥ 2 and ψ∞(N(w + w0)β) = ψ∞(N(w)β). If φ∞(N(w)α) is 0, then V∞(w) < − 1 2V∞(α) and therefore φ∞(N(w + w0)α) = 0 From above equalities one gets Φ∗α,β(w∗) = ψ∞(T(w0w∗))Φ∗α,β(w
∗) and so Φ∗α,β(w∗) = 0. Lemma 3.2. If v∞(N(w∗) α β2 ≥ 0), then Φ∗α,β(w∗) = ψ∞(N(w∗) 1 β) Z D∞ φ∞(N(w)α)ψ∞(N(w)β)dw.
Proof. Since v∞(N(w∗) α β2) ≥ 0, one gets V∞( ¯ w∗ β ) ≥ − 1 2V∞(α). Thus Φ∗α,β(w∗) = Z D∞ φ∞(N(w + ¯ w∗ β )α)ψ∞(N(w + ¯ w∗ β )β + T((w + ¯ w∗ β )w ∗ )dw =φ∞(N(w∗) 1 β) Z D∞ φ∞(N(w + ¯ w∗ β )α)ψ∞(N(w)β)dw =φ∞(N(w∗) 1 β) Z D∞ φ∞(N(w)α)ψ∞(N(w)β)dw Let S(α, β, dw) := Z D∞ φ∞(N(w)α)ψ∞(N(w)β)dw.
Lemma 3.3. For v∞(α) > v∞(β) − 2, we have
S(α, β, dw) = −q2v∞(β)−3dw(O
D∞) where OD∞ is the maximal order of D∞.
Proof. Let β = N(u) for some u ∈ D∞×. Then d(w/u) = q2V∞(u)d(w) =
q2v∞(β)dw. Thus S(α, β, dw) = q2v∞(β) Z D∞ φ∞(N(w) α β)ψ∞(N(w))dw.
Let n0 := v∞(β) − v∞(α). Write D∞ = k∞+ k∞a + k∞b + k∞ab, where a ∈ Fq∗− ℘(F∗q), b2 = t, ab = b¯a. If φ∞(N(w) α β) = 1, w can be written as w = ∞ X i=n0
ib−i where i ∈ Fq2 = Fq(a) and ψ∞(N(w)) = σ(
X i+j=2 i¯j). Φ∗α,β(w∗) =q2v∞(β)( Z V∞(w)≥3−n0 dw) X 1∈Fq2 σ(N(1)) = X (n0,...0) X (2,...2−n0) σ(TF q2/Fq( 0 X i=n0 i¯2−i)).
Note that X (2,...2−n0) σ(TF q2/Fq( 0 X i=n0 i¯2−i)) = 0 ,if (n0, ..., 0) 6= (0, ..., 0), q2(1−n0) ,if ( n0, ..., 0) = (0, ..., 0). So S(α, β, dw) =q2v∞(β)q2(1−n0)q2(n0−3)dw(O D∞) X 1∈F2q σ(N(1)) =q2(v∞(β))−2 X 1∈F2q σ(N(1))dw(OD∞). Since X 1∈F2q
σ(N(1)) = −q, the result holds.
Combine the above lemmas, we obtain the following propositions.
Proposition 3.4. Let α, β ∈ k∞∗ with v∞(α) > v∞(β) − 2. The Fourier transform of the function Φα,β(w) = φ∞(N(w)α)ψ∞(N(w)β) is
S(α, β, dw)φ∞(N(w∗) α β2)ψ∞(N(w ∗ )1 β).
Proposition 3.5. Let h ∈ k∗∞, ρ ∈ D∞. For α, β ∈ k∞∗ with v∞(α) > v∞(β) − 2. Define Ψα,β := Φα,β(ρ + hw). Then Ψ∗α,β(w) = q4v∞(h)S(α, β, dw)φ ∞(N( w∗ h ) α β2)ψ∞(N( w∗ h ) 1 β)ψ∞(T( ρw∗ h )).
Given 1 ≤ i, j ≤ n, we choose Haar measure dw with dw(D∞/Mij) = 1.
Lemma 3.6. S(α, β, dw) = −q2v∞(β)−deg(N0)+2v∞(Nij)=: S(α, β, M
ij). Proof. By Lemma 2.1, we may assume D = k + ku + kv + kuv where v2 = b for some b ∈ A with odd degree and there exist G, D1, Q ∈ A such that
u2 + Gu = D
1Q and degD1=2degG−degQ. Let R = A + Au + Av + Auv. We can find a suitable a ∈ A such that RN+,N− ⊃ aR and RN+,N− ⊃ aMij.
By comparing the discriminants we can get the formula
dw(D∞/Mij) = dw(D∞/R)q−2v∞(Nij)−v∞(
N0 G2b).
Note that D∞/R = {x1+ x2u + x3v + x4uv|xi ∈ k∞, v∞(xi) > 0}. Let M = {x1+ x2u + x3v + x4uv|xi ∈ k∞, v∞(xi) ≥ 0}. Then dw(D∞/R) = q−4dw(M ). Since OD∞ consists of x1+ x2u + x3v + x4uv with the conditions v∞(x1) ≥ 0,
v∞(x2) ≥ −v∞(G), v∞(x3) ≥ −12v∞(b), v∞(x4) ≥ −v∞(G) −12v∞(b), we have
dw(OD∞) = q
−2degG−degb+3dw(D ∞/R). Therefore, S(α, β, Mij) = −q2v∞(β)−deg(N0)+2v∞(Nij).
Let ˜Mij be the dual lattice of Mij, i.e. ˜
Mij = {w ∈ D∞ : T(wµ) ∈ A for all µ ∈ ˜Mij}.
We can apply the Poisson summation formula
X µ∈Mij Ψα,β(µ) = X µ∗∈ ˜M ij Ψ∗α,β(µ∗) and get
Theorem 3.7. Let h ∈ k∗∞, ρ ∈ D∞. For α, β ∈ k∞∗ with v∞(α) > v∞(β)−2. Then X µ∈Mij φ∞(N(ρ + hµ)α)ψ∞(N(ρ + hµ)β) =q4(v∞(h))S(α, β, M ij) X µ∗∈ ˜M ij φ∞(N( µ∗ h ) α β2)ψ∞(N( µ∗ h ) 1 β)ψ∞(T( ρµ∗ h )).
Let (x, y) ∈ K∞∗ × K∞, and M ⊂ D∞ be a discrete A-lattice, NM ∈ k such that NM · A is the fractional ideal of A generated by N(µ) for µ ∈ M . For h ∈ A with h 6= 0, ρ ∈ M , define ”partial theta” series :
θ(x, y, M, NM, h, ρ) := X µ∈M µ≡ρmod hM φ∞( Nxt2 NMh )ψ∞( N(µ)y NMh ).
Note that θij(x, y) = θ(x, y, Mij, Nij, 1, 0), and
θ(x, y, M, NM, h, ρ) := X µ∈M φ∞(N(ρ + hµ)α)ψ∞(N(ρ + hµ)β), with α = xt 2 NMh , β = y NMh .
Proposition 3.8. Let x, y ∈ K∞∗ with v∞(x) > v∞(y), 0 6= h ∈ Fq[t] and κ ∈ ˜Mij. Then θ(x y2, 1 y, ˜Mij, N −1 0 , h, κ) =S( xt 2 NijN0h , y NijN0h , Mij)−1 X µ∈Mij/hMij φ∞(T( µ∗ h ) α β2)ψ∞(N( µ∗ h) 1 β)ψ∞(T( ρµ∗ h )).
Proof. By Proposition 3.7, We have
θ(x, y, Mij, Nij, h, ρ) =q4(v∞(h))S(α, β, M ij) X µ∗∈ ˜M ij φ∞(N( µ∗ h ) α β2)ψ∞(N( µ∗ h ) 1 β)ψ∞(T( ρµ∗ h )). Multiply this by ψ∞(T( ρκ
h )) for κ ∈ ˜Mij and sum over ρ ∈ Mij/hMij, we obtain X ρ∈Mij/hMij ψ∞(T( ρκ h ))θ(x, y, Mij, Nij, h, ρ) =q4(v∞(h))S(α, β, M ij) X µ∗∈ ˜M ij φ∞(N( µ∗ h) α β2)ψ∞(N( µ∗ h ) 1 β) X ρ∈Mij/hMij ψ∞(T( ρ(κ + µ∗) h )).
Since X ρ∈Mij/hMij ψ∞(T( ρ(κ + µ∗) h )) = q4v∞(h) if µ∗+ κ ∈ h ˜M ij, 0 , otherwise.
The proposition follows by replacing x with x N0
, and y with y N0
.
We also need the following transformations which can be very easily de-duced. Lemma 3.9. Let x ∈ K∞∗ , y ∈ K∞ (1) For b ∈ A we get θ(x, y + b, M, NM, h, ρ) = ψ∞( N(ρ)b NMh )θ(x, y, M, NM, h, ρ). (2) For h0 ∈ A, h0 6= 0 θ(x, y, M, NM, h, ρ) = X ρ0∈M/h0hM ρ0≡ρ mod hM θ(h0x, h0y, M, NM, h0h, ρ0).
4
Transformation of the Theta Series
Let (x, y) ∈ k∞∗ × k∞, γ = a b c d
∈ GL2(A). Suppose cy + d 6= 0. Define
γ ◦ (x, y) = (x(ad + bc) (cy + d)2 , ay + d cy + d). Lemma 4.1. Let γ = a b c d
v∞(x) > v∞(y), and v∞(cx) > v∞(cy + d). Let 1 ≤ i, j ≤ n. Then θij(γ ◦ (x, y)) =S( Nijxt2 y2 , Nij(cy + d) dy , ˜Mij) −1S(xt2 Nij , y Nij , Mij)−1 ( X κ∈Mij/dMij ψ∞( N(κ)b Nijd ))θij(x, y). Proof. Put u = x y2, v = 1 y. Then θij(γ ◦ (x, y)) = θ( u (c + dv)2, b d + 1 d(c + dv), Mij, Nij, 1, 0) = X κ∈Mij/dMij θ( du (c + dv)2, b + 1 c + dv, Mij, Nij, d, κ) = X κ∈Mij/dMij ψ∞( N(κ)b Nijd )θ( du (c + dv)2, b + 1 c + dv, Mij, Nij, d, κ)
Since v∞(cx) > v∞(cy + d), we have v∞(du) > v∞(dv + c) and
θij(γ ◦ (x, y)) =S(Nijut2, Nij(v + c d, ˜Mij) −1 ( X κ∈Mij/dMij ψ∞( N(κ)b Nijd ) · ( X ρ∈ ˜Mij/d ˜Mij ψ∞( T(ρκ) d ))θ( du N0 ,dv + c N0 , ˜Mij, Nij−1N −1 0 , d, ρ). Since c N0 ∈ F
q[t], we see that θij(γ ◦ (x, y)) is given by
S(Nijut2, Nij(v + c d, ˜Mij) −1 X ρ∈ ˜Mij/d ˜Mij [θ(du N0 , dv N0 , ˜Mij, Nij−1N −1 0 , d, ρ) · X κ∈Mij/dMij ψ∞( N(κ)b Nijd + T(ρκ) d + N(ρ)cNij d )]
Note that cNijρ ∈ M¯ ij. Replacing κ by κ + cNijρ the last summand¯ equals to
Nb Nijd
Since aT(ρκ) + NijacN(ρ) ∈ A, we have θij(γ ◦ (x, y)) =S(Nijut2, Nij(v + c d, ˜Mij)) −1· X ρ∈ ˜Mij/d ˜Mij ψ∞( N(κ)b Nijd ) · θ(du N0 , dv N0 , ˜Mij, Nij−1N −1 0 , d, ρ). Now u = x y2, v = 1 y, by Proposition 3.8 we obtain θij(γ ◦ (x, y)) =S( Nijxt2 y2 , Nij(cy + d) dy , ˜Mij) −1S(xt2 Nij , y Nij , Mij)−1 · ( X κ∈Mij/dMij ψ∞( N(κ)b Nijd ))θij(x, y).
Lemma 4.2. Let 1 ≤ i, j ≤ n, b, d ∈ A are relatively prime with (d, N0) = 1. Then X κ∈Mij/dMij ψ∞( N(κ)b Nijd ) = q2deg(d).
Proof. This is proved by four steps:
(1) X κ∈Mij/dMij ψ∞( N(κ)b Nijd ) = X κ∈Rj/dRj ψ∞( N(κ) d ) =: S(d). (2)S(dd0) = S(d)S(d0) when d and d0 are relatively prime. (3)S(Pm) = S(pm−2) · q4deg(P ) for any prime P - N
0, m ≥ 2. (4)S(1) = 1 and S(P ) = q2deg(P ) for P - N
0.
(1) After multiplying an element in D∗ we can assume Mij is a proper left ideal of Rj and (Nij, d) = 1. Hence Rj/dRj ∼= Mij/dMij. For d1, d2 relatively prime, we can choose a, b in A such that ad1 + ad2 = 1. Then (x, y) 7→ xbd2 + yad1 gives a well-defined map
Note that we have (d, N0) = 1 and let d = l Y i=1 Pei i , then Rj/dRj ∼= l Y i=1 Rj/PieiRj ∼= l Y i=1 Mat2(APi)/P ei i Mat2(APi).
Since the determinant map from Mat2(APi)/P
ei
i Mat2(APi) to APi/P
ei
i APi is
surjective, the map κ 7→ N(κ) mod d from Rj to A/dA is surjective. Hence there exists s ∈ Rj such that N(s) ≡ bNij mod d. Therefore
X κ∈Mij/dMij ψ∞( N(κ)b Nijd ) = X κ∈Rj/dRj ψ∞( N(κ)bNij d ) = X κ∈Rj/dRj ψ∞( N(κ) d ).
(2) Since d and d0 are relatively prime,
S(dd0) = X x∈Rj/dRj,y∈Rj/d0Rj ψ∞( N(xbd0+ yad) dd0 ) = X x∈Rj/dRj ψ∞( b2d0N(x) d ) X y∈Rj/d0Rj ψ∞( a2dN(y) d0 ) = S(d)S(d0)
(3) From the definition we have
S(Pm) = X α modPm−1 ( X w modP ψ∞( N(α + wPm−1) Pm )) = X α modPm−1 ψ∞( N(α) Pm ) X w modP ψ∞( T( ¯wα) P ) = X α modPm−2 ψ∞( N(α) Pm−2) · q 4deg(P )= S(pm−2) · q4deg(P )
(4) Now S(1) = 1. Note that Rj/P Rj ∼= Mat2(A/P A). Therefore
S(P ) = X κ∈Rj /P Rj N(κ)=0 ψ∞( N(κ) P ) + X κ∈Rj /P Rj N(κ)6=0 ψ∞( N(κ) P )
=(qdegP − 1)q2degP + (qdegP − 1)qdegP + q2degP + #(SL
2(A/P A) · X α∈(A/P A)∗ ψ∞( α P)
Since X α∈(Fq[t]/P Fq[t])∗ ψ∞( α P) = −1 we get S(P ) = q 2deg(P ).
Replace y by y + h for suitable h ∈ A, we can drop the assumption v∞(x) > v∞(y) and obtain the following result.
Theorem 4.3. For 1 ≤ i, j ≤ n. Let x ∈ K∞∗ , y ∈ K∞, γ = a b c d ∈ SL2(A). v∞(cx) > v∞(cy + d), and c ≡ 0 mod N0. Then
θ(γ ◦ (x, y)) = q−2v∞(cy+d)· θ
ij(x, y).
5
Functions on GL
2(k
∞) and Harmonicity
Set Γ0(N0) := a b c d ∈ GL2(A) : c ≡ 0 mod N0 and Γ∞ := a b c d ∈ GL2(O∞) : c ≡ 0 mod π∞O∞ .
From our theta series of section 2 we now construct functions on the double
coset space Γ0(N0)\GL2(k∞)/Γ∞k∞×.
Let 1 ≤ i, j ≤ n. For g ∈ GL2(k∞), write g as γ · x y 0 1 · γ∞· z where γ ∈ Γ0(N0), (x, y) ∈ k×∞× k∞, γ∞ ∈ Γ∞, z ∈ k∞×. Put Θij(g) := q−v∞(x)θij0 (x, y) = q −v∞(x) 1 (q − 1)wj X ∈F×q θij(x, y).
These can be checked to be well-defined functions on the above double coset
for g ∈ GL2(k∞) ˜ f (g) := f g 0 1 π∞ 0 = −f (g) and X κ∈GL2(O∞)/Γ∞ f (gκ) = 0.
A C-valued function on Γ0(N0)\GL2(k∞)/Γ∞k∞× satisfying the harmonic properties is known as automorphic form of Drinfeld type. (cf. [3] and [6])
Theorem 5.1. Let 1 ≤ i, j ≤ n, ˜Θij = −Θij and
X κ∈GL2(O∞)/Γ∞
Θ(gκ) = 0.
Recall the Fourier expansion of functions f on Γ0(N0)\GL2(k∞)/Γ∞k×∞ (cf. [8]): for r ∈ Z and y ∈ k∞, f πr ∞ y 0 1 = X λ∈A, deg λ≤r−2 f∗(r, λ)ψ∞(λy), where f∗(r, λ) = Z A\k∞ f πr ∞ u 0 1 ψ∞(λu)du.
Here du is a Haar measure with Z
A\k∞
du = 1.
Lemma 5.2. Let 1 ≤ i, j ≤ n. Given 0 6= λ ∈ A and r ≥ deg λ + 2. Then the Fourier coefficients of Θij are given by
i. Θ∗ij(r, λ) = q−rBij(m) if λ = m, for m monic and ∈ F×q.
ii. Θ∗ij(r, 0) = qw−r
j for all r ∈ Z.
Proof. This follows directly from the remark after Definition 2.3.
Let πr
∞∈ k∞× and u ∈ k∞. Choose c, d ∈ A with c ≡ 0 mod N0, (c, d) = 1, v∞(u + dc) ≥ r + 1, and find a, b ∈ A with ad + bc = 1. Then for ` ∈ Z with ` ≤ r + 1 the following two matrices:
π` ∞ u 0 1 0 1 π∞ 0 and d b c a π∞1−` c2 a c 0 1
represent the same coset in GL2(k∞)/Γ∞k∞×. Using this fact for ` = r and ` = r + 1 one obtains ˜ Θij πr ∞ u 0 1 −q−1Θ˜ij πr+1 ∞ u 0 1 = X deg µ+2=1−r+2 deg c Θ∗ij(1−r+2 deg c, µ)ψ∞(µ a c). Set u:= d c + π r
∞ for ∈ F×q. and summing over all we get:
(q − 1) ˜Θij πr ∞ u 0 1 − X ∈F×q Θij π∞1−r c2 au+b cu+d 0 1 = q X deg µ+2=1−r+2 deg c Θ∗ij(1 − r + 2 deg c, µ)ψ∞(µ a c). Note that π1−r∞ c2 au+b cu+d 0 1 and a b c d π∞r+1 u 0 1
represent the same coset in GL2(k∞)/Γ∞k∞×. Thus one has
˜ Θij π∞r+1 u 0 1 − ˜Θij πr∞ u 0 1 = X ∈F×q Θij π∞r+1 u + π∞r 0 1 .
From the Fourier expansion of ˜Θij and Θij we have that for λ ∈ A with deg λ + 2 ≤ r,
˜
Θ∗ij(r + 1, λ) − ˜Θ∗ij(r, λ) = (q − 1)Θ∗ij(r + 1, λ),
and for deg λ + 2 = r + 1,
˜
Θ∗ij(deg λ + 2, λ) = −Θ∗ij(r + 1, λ).
Therefore ˜Θ∗ij(r, λ) = −Θ∗ij(r, λ) for λ ∈ A with λ 6= 0 and r ≥ deg λ + 2.
To compute ˜Θ∗ij(r, 0), note that
˜ Θij πr ∞ 0 0 1 = X deg λ≤r−2 ˜ Θ∗ij(r, λ) = Θ˜∗ij(r, 0) + X λ6=0,deg λ≤r−2 −Θ∗ ij(r, λ).
On the other hand, for any ∈ F×q and ` ≥ 0 the following matrices πdeg N0+` ∞ 0 0 1 · 0 1 π∞ 0 and −1 1 t`N 0 (t`N0+ 1) · π1−deg N0−`∞ (t`N 0)2 (t`N 0+1) t`N 0 0 1
represent the same coset in GL2(k∞)/Γ∞k∞×. Therefore
˜ Θij πdeg N0+` ∞ 0 0 1 = X deg λ≤deg N0+`−1 Θ∗ij(deg N0+ ` + 1, λ)ψ∞(λ t`N 0 ) = X deg λ≤deg N0+`−2 Θ∗ij(deg N0+`+1, λ)− 1 q − 1 X deg λ=deg N0+`−1 Θ∗ij(deg N0+`+1, λ).
This gives ˜ Θ∗ij(deg N0+ `, 0) = Θ∗ij(deg N0+ ` + 1, 0) + (1 + q) X λ6=0,deg λ≤deg N0+`−2 Θ∗ij(deg N0+ ` + 1, λ) − 1 q − 1 X deg λ=deg N0+`−1 Θ∗ij(deg N0+ ` + 1, λ) ! = −Θ∗ij(deg N0+ `, 0) + 1 q − 1 · qΘij πdeg N0+` ∞ 0 0 1 − Θij πdeg N0+`+1 ∞ 0 0 1 .
To justify the claim above, note that
θij πs ∞ 0 0 1 = X deg λ≤s−2 Bij0 (λ) = #{µ ∈ Mij : V∞(µ) ≥ 2 − s + v∞(Nij)}.
Write D∞as k∞+k∞a+k∞b+k∞ab, where a2 ∈ F×q −℘(F×q), b2 = t, ab = b¯a. We have for sufficiently large s0
D∞ = Mij + bs0OD∞. Then for s ≥ s0+ 2 + v∞(Nij), θij πs+1 ∞ 0 0 1 = #{µ ∈ Mij : V∞(µ) ≥ 2 − (s + 1) + v∞(Nij)} = q2#{µ ∈ Mij : V∞(µ) ≥ 2 − s + v∞(Nij)} = q2θij πs ∞ 0 0 1 .
Therefore for sufficiently large s one has
qΘij π∞s 0 0 1 = Θij π∞s+1 0 0 1 .
Thus from the equality ˜Θ∗ij(r + 1, 0) − ˜Θ∗ij(r, 0) = (q − 1)Θ∗ij(r + 1, 0) for all r ∈ Z one gets
˜
Θ∗ij(r, 0) = −Θ∗ij(r, 0).
Comparing the Fourier coefficients we obtain ˜Θij = −Θij.
Note that the coset representations of GL2(O∞)/Γ∞ are
{ a 0 0 1 , b 0 0 1 0 π∞−1 1 0 |a, b ∈ F × q}.
Hence for all g ∈ GL2(k∞)
X κ∈GL2(O∞)/Γ∞ Θ(gκ) = X a∈F×q Θ(g a 0 0 1 )+X a∈F×q Θ(g a 0 0 1 0 π∞−1 1 0 ) = 0.
6
Analogue for Sum of Squares
Two squares. Let 0 ∈ Fq− ℘(Fq) and η2 + η = 0, then k(η) = Fq2(t).
The norm form on Fq2[t] is the quadratic form a21+ a1a2+ a220. Note that
prime P of A splits in Fq2[t] if and only if degP is even. For each λ ∈ A,
the representation number #{(a1, a2)|a21+ a1a2+ a220 = λ} is related to the number of divisors. More precisely, when λ 6= 0, write the ideal (λ) as
Y
degP odd
PrP Y degQ even
QrQ.
Then #{(a1, a2)|a21+ a1a2+ a220 = λ} is equal to
(q + 1) Y
degQ even
(rQ+ 1) = (q + 1)
X
d monic ,d|λ P -d for degP odd
when rP are even for all prime P with odd degree, and 0 otherwise. These representation numbers give the following theta series: for (x, y) ∈ k×∞× k∞
ϑ(x, y) := X
deg λ≤v∞(x)−2
#{(a1, a2) ∈ A2 : a21+ a1a2+ a220 = λ}ψ∞(λy).
Using the same methods as in section 3 and 4, we can show that this theta
function satisfies the following transformation law under γ = a b c d ∈ SL2(A):
ϑ(γ ◦ (x, y)) = (−q)−v∞(cy+d)ϑ(x, y).
We omit the details here.
This theta function can also be extended to a function on the double coset
space SL2(A)\GL2(k∞)/Γ∞k∞: for g ∈ GL2(k∞), write g = γ x y 0 1 γ∞z where γ ∈ SL2(A), γ∞ ∈ Γ∞, and z ∈ k∞, define
Θ(g) = (−q)−v∞(x)2 ϑ(x, y).
Four squares. Consider the quaternion algebra D which is of the form
k + ku + kv + kuv where u2+ u = 0 in Fq− ℘(Fq), v2 = t, and uv = v(u + 1). The ramified primes of D are t and ∞, and there is only one maximal order
R (up to conjugation)
A + Au + Av + Auv.
Note that the number of left ideal classes of R is one, and #(R×/F×q) = q + 1. We are interested in the cardinality of
for λ ∈ A. The above quadratic form is the norm form on R, and the
representation numbers are the Fourier coefficients of the theta series θ11 introduced in §2.
From Proposition 2.2, we obtain that for (x, y) ∈ k∞× × k∞,
θ11(x, y) = X λ∈A, degλ≤v∞(x)−2 σt(λ)ψ∞(λy) where σt(λ) = (q + 1) X d∈A monic, d|λ d6≡0 mod t |d|.
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Fun-damental Research, Bombay 1957.
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[5] H.-G. R¨uck, Theta Series of Imaginary Quadratic Function Fields,
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