FU-TSUN WEI
Abstract. We propose a concept of half integral weight in the global function field context, and construct natural families of functions with given weight. An analogue of Shimura correspondence (between weight 2 functions and weight 32 functions) via theta series from “definite” quaternion algebras over function fields is then established. From the study of Fourier coefficients of these theta series, we arrive at a Waldspurger-type formula.
This formula is then applied toL-series coming from elliptic curves over function fields.
Keywords:Function field, Half integral weight, Metaplectic form, Theta series, Special value ofL-series
MSC code:MSC 11R58, MSC 11F37, MSC 11F27, MSC 11F67
1
Introduction
Let k be the rational function field Fq(t) with odd characteristic. Fix the infinite place
∞ of k, and denote byk∞ the completion ofk at ∞. The Kubota 2-cocycle on GL2(k∞), which is defined via Hilbert n-th symbol for n|q−1, gives non-trivial central extension of GL2(k∞), calledmetaplectic group. The aim of this paper is to study families of functions on the metaplectic groupG, so-calledmetaplectic forms. We focus on the case whenn= 2and present an analogue of classical theory of half integral weight modular forms.
In the function field context, we take the Iwahori Hecke operator at ∞ to be our “non- Euclidean Laplacian.” Functions onGof weight κ2 are the eigenfunctions of this operator with eigenvalueq1−κ4. From the norm form on lattices of pure quaternions in “definite” quaternion algebra over k (i.e. ramified at ∞), we construct theta series which are functions on G of weight 32. The action of Hecke operators on these theta series can be expressed by so-called Brandt matrices. As in Eichler’s theory, these integral matrices are in fact representation of Hecke operators on the space of Drinfeld type “new” forms. This allows us to define a Shimura mapSh in §4.3 from Drinfeld type “new” forms to functions of weight 32.
From knowledge about central critical values of L-series, we arrive at the following main theorem, which gives a function field analogue of Waldspurger’s formula (cf. §4.3 for further details):
Theorem 0.1. Let N0 be a square-free ideal of A with odd number `N0 of prime factors.
Given a “normalized” Drinfeld type newform f for Γ0(N0). Suppose for each prime factor P of N0, the eigenvalue of the Hecke operator TP on f is one. Then given any irreducible polynomialD inA−k∞2 satisfying Legendre symbol DP
=−1for all primes P dividing N0, we have
L(f,0)L(f ⊗εD,0) = (q(−1)deg
D−1
4 )·(3−(−1)degD)·(f, f) 2· |D|12 ·4(`N0−1)
·m(f, D)2.
Here L(f, s)is the L-series attached to f,L(f⊗εD, s)is its twist by the quadratic character εD; (·,·) denotes Petersson inner product on the space of Drinfeld type cusp forms; and m(f, D)is the(−D)-th Fourier coefficient of the weight 32 function Sh(f).
With suitable choice ofD, the Fourier coefficientm(f, D)determines the non-vanishing of the above central critical value of L-series. This theorem can be applied toL-series coming
1This research was partially supported by National Science Council.
1
from elliptic curves over khaving split multiplicative reduction at an even number of places (including∞) and with good reduction elsewhere.
The contents of this article are as follows. We first pick the automorphy factor from the transformation law of a specific theta series in §1.1. After a brief review of the metaplectic groupG, the Iwahori Hecke operator at∞is introduced in §1.3. Similar to the classical case, integral weight functions onGare comes from functions onGL2(k∞).
In §2 we study examples of functions having integral or half integral weight. Theta series from “imaginary” (with respect to ∞) quadratic extensions over k are of weight 1. Auto- morphic forms of Drinfeld type, which can be viewed as an analogue of classical weight 2 modular forms (cf. [2] and [11]), are functions of weight 2. Furthermore, Eisenstein series give us examples of higher integral weight functions. The theta series of weight 32 from defi- nite quaternion algebra overkare introduced in §2.2. The Fourier coefficients of these series, which can be viewed as representation numbers of “three squares,” are expressed explicitly in terms of numbers of optimal embeddings of quadratic orders into the definite quaternion algebra in question.
In §3 we introduce Hecke operatorsTP2,κ2 for finite primesP on functionsf of half integral weight κ2 withκodd. The Fourier coefficients ofTP2,κ2fcan be computed as in classical theory.
The Brandt-matrix representation for the action of Hecke operators on the above theta series of weight 32 is established at the end of §3. In §4.1 we recall briefly properties of the so-called definite Shimura curves overk. The construction of the Shimura map Shis in §4.2, and the proof of our main theorem is given in §4.3. Finally we apply our theorem to families of elliptic curves overkin §4.4.
Notation We fix the following notations:
k: the rational function fieldFq(t),q=p`0 wherepis an odd prime.
A: the polynomial ringFq[t].
∞: the infinite place, which corresponds to the degree valuationv∞.
| · |: the absolute value onk∞: fora∈k∞, |a|:=q−v∞(a). π∞: t−1, a fixed uniformizer of∞.
k∞: Fq((t−1)), i.e. the completion ofk at∞.
O∞: Fq[[t−1]], i.e. the valuation ring in k∞. P : a finite prime (place) ofk.
kP : the completion ofkat the finite primeP.
AP : the closure ofA inkP. kˆ: Q0
PkP, the finite adele ring ofk.
Aˆ: Q
PAP.
ψ∞: a fixed additive character on k∞: for y = P
iaiπ∞i ∈ k∞, we define ψ∞(y) := exp
2π√
−1
p ·TrFq/Fp(−a1) .
We identify non-zero ideals of A with the monic polynomials in A by using the same notation.
1. Half integral weight In the classical case, the theta series
X
n∈Z
exp(2π√
−1n2z)
is a modular form of weight 12 for the congruence subgroup Γ0(4). Shimura ([12]) uses the automorphy factor of this theta series to develop the theory of half integral weight modular forms. In this section, starting from an explicit theta series, we propose a concept of half integral weight in the function field setting which has direct applications to arithmetic of function fields.
1.1. Theta series. For(x, y)∈k×∞×k∞, define θ(x, y) :=X
b∈A
φ∞(b2xt2)ψ∞(b2y)
Here φ∞is the characteristic function of O∞. Given(x, y)ink∞× ×k∞ andγ= a b c d
! in GL2(A)withcy+d6= 0. TheMöbius transform
γ◦(x, y) :=
(ad−bc)x (cy+d)2 ,ay+b
cy+d
.
To state the transformation law ofθ, we recall the following symbols:
1. Forβ ∈k×∞, theroot number ofβ ω(β) :=
1 ifv∞(β)is even, ε·sgn(β)q−12 ifv∞(β)is odd.
Here sgnis a sign functiononk∞× (with respect toπ∞): for y∈k×∞ y= sgn(y)·u·π∞v∞(y)
whereu∈1 +π∞O∞,sgn(y)∈F×q ;εis the sign of the followingGauss sum:
ε:=q−12 X
∈Fq
q−12 exp 2π√
−1
p TrFq/Fp()
.
We point out thatω(−β) =ω(β)−1 andω(β)2= (−1)q−12 v∞(β). 2. Forα,β∈k∞×, theHilbert quadratic symbol
(α, β)∞:=
1 ifαX2+βY2=Z2 has a non-trivial solution,
−1 otherwise.
It can be checked that forα, β∈k×∞, (α, β)∞=ω(α)ω(β)/ω(αβ).
3. Givenγ= a b c d
!
∈SL2(A), theKubota symbol ofγ
µ(γ) :=
d c
ifcd6= 0, 1 otherwise.
Here ··
is theLegendre quadratic symbol.
Basing on Poisson summation formula, the transformation law of θis worked out:
Proposition 1.1. (cf. [16] IV.2.1)Letx∈k×∞ andy∈k∞. For anyγ= a b c d
!
inSL2(A) with v∞(cx)> v∞(cy+d), one has
θ(γ◦(x, y)) =|cy+d|12(γ, x, y)θ(x, y)
with
(γ, x, y) =
µ(γ)ω(cy+d)(c, cy+d)∞, if c6= 0,
1, otherwise.
This leads to a function on themetaplectic coverGofGL2(k∞).
1.2. Extension Gof GL2(k∞) by the circle groupS1. Giveng= a b c d
!
inGL2(k∞).
Let X(g) := c if c is non-zero and d if c is zero. Kubota (cf. [8]) introduced a 2-cocycle σ: GL2(k∞)×GL2(k∞)→ {±1}which is defined by
σ(g, g0) =
X(gg0)
X(g) , X(gg0) det(g)X(g0)
∞
.
This gives us an extensionGofGL2(k∞)byS1:
1−→S1−→G−→GL2(k∞)−→1 whereS1={z∈C,|z|= 1}.
The mapµ˜: SL2(A)→Ggiven byγ7−→γ˜= (γ, µ(γ))is a group monomorphism (cf. [6]).
The groupGL2(O∞)also can be embedded intoGby the homomorphism%˜: GL2(O∞)→G mapping γ∞ toγ˜∞= (γ∞, %(γ∞))where forγ∞= a b
c d
!
∈GL2(O∞),
%(γ∞) :=
(c, d/detγ∞)∞, if0<|c|<1,
1, otherwise.
Furthermore, since σ(z1, z2) = (z1, z2)∞ forz1, z2∈k×∞, we embedk×∞intoGvia the homo- morphismω˜:k∞× →Gdefined byz7−→z˜= (z, ω(z)−1).Note thatω(k˜ ∞×)is not in the center ofG.
For any congruence subgroupΓ = Γ(1)0 (N)inSL2(A)whereN is a non-zero ideal ofAand Γ(1)0 (N) =
( a b c d
!
∈SL2(A) :c≡0 modN )
,
one hasGL2(k∞) = Γ·H∞·Γ1∞·k×∞, where Γ1∞:=
( a b c d
!
∈GL2(O∞) :c≡d−1≡0 modπ∞O∞ )
,H∞:= k×∞ k∞
0 1
! .
Let kf×∞ := ˜ω(k×∞), Γ := ˜e µ(Γ), gΓ1∞ := ˜%(Γ1∞), H∞ := {(h, ξ) : h ∈ H∞,ξ ∈ S1}. Then G=kf×∞·Γe·H∞·gΓ1∞.
Back to the theta series θ in §1.1. Given ˜g ∈ G, write ˜g as z˜˜γ(h, ξ)˜γ∞, where z ∈k∞×, γ∈SL2(A),γ∞∈Γ1∞,h= x y
0 1
!
, andξ∈S1. We extendθ to a functionΘonGby Θ(˜g) :=|x|14θ(x, y)·ξ.
The transformation law ofθimplies thatΘis a well-defined function onG, and forz∈k∞×, γ∈SL2(A),γ∞∈Γ1∞,ξ∈S1
Θ(˜z˜γ(1, ξ)˜g˜γ∞) =ξΘ(˜g).
This leads to a concept of half integral weight in the function field setting.
1.3. Half integral weight and weight operators. Here we are interested in functionsf on the coset spaceG/gΓ1∞. Forκ∈N, define
Te∞,κ
2f(˜g) :=qκ4−1· X
v∈Fq
f ˜g π∞ v
0 1
! ,1
! .
Since
gΓ1∞
π∞ 0
0 1
! ,1
!
gΓ1∞= a
v∈Fq
π∞ v
0 1
! ,1
! gΓ1∞,
Te∞,κ
2f is also a function onG/gΓ1∞. The metaplectic formΘis of weight 12 under the following definition:
Definition 1.2. f is of weight κ2 if for allξ∈S1 andg˜∈G, f((1, ξ)˜g) =ξκf(˜g)andTe∞,κ
2f =f.
1.3.1. Fourier expansion. Supposef is a function onG/gΓ1∞ satisfying that forz∈k×∞ f(˜z˜g) =χ∞(z)f(˜g)
where χ∞ is a characterχ∞ : k∞× →C× trivial on 1 +π∞O∞. We call χ∞ the “central ” character off.
Given a weight κ2 functionf onG/gΓ1∞ with “central” characterχ∞. Suppose there exists a Dirichlet character χN : (A/N A)× → C× such that for all γ = a b
c d
!
∈ Γ(1)0 (N),
f(˜γ˜g) =χN(d)f(˜g). Then forr∈Zandu∈k∞, theFourier expansion off is:
f
π∞r u
0 1
! ,1
!
= X
degλ+2≤r
f∗(r, λ)ψ∞(λu)
where
f∗(r, λ) = Z
A\k∞
f
πr∞ u
0 1
! ,1
!
ψ∞(−λu)du.
The Haar measure taken here is normalized so that R
A\k∞du = 1. Since f is of weight κ2, one hasf∗(r+ 1, λ) =q−κ4f∗(r, λ)for allλ∈Awithdegλ+ 2≤r.
Define functionϕf onk×∞×k∞by ϕf(x, y) :=|x|−κ4f
x y 0 1
! ,1
!
= X
λ∈A,degλ+2≤v∞(x)
ϕ∗f(λ)ψ∞(λy)
whereϕ∗f(λ) :=qκr4 f∗(r, λ)for anyr≥degλ+ 2. Then we have the following transformation law forϕf: given(x, y)∈k∞× ×k∞and γ= a b
c d
!
in Γ(1)0 (N)withv∞(cx)> v∞(cy+d), ϕf γ◦(x, y)
=χN(d)χ∞(cy+d)−1 |cy+d|κ2(γ, x, y)κ
·ϕf(x, y).
1.3.2. Integral weight. Whenκ is even andh is a weight κ2 (=:ν) function on G/gΓ1∞ with
“central” characterχ∞, let
h0(g) :=h(g,1)forg∈GL2(k∞).
Thenh0 is a function onGL2(k∞)/Γ1∞ such that forz∈k×∞andg∈GL2(k∞) h0(zg) = (−1)(q−1)ν2 v∞(z)·χ∞(z)·h0(g),
and
T∞,νh0(g) :=qν2−1 X
v∈Fq
h0 g π∞ v
0 1
!!
=h0(g).
Conversely, given a functionf onGL2(k∞)/Γ1∞such thatf(zg) =χ∞(z)f(g)forz∈k∞×, g∈GL2(k∞)andT∞,νf =f. Let
f0(g, ξ) =ξνf(g)for all(g, ξ)∈G.
Thenf0 is a weightν function with “central” characterχ∞·(−1)(q−1)ν2 v∞(·). This tells us that functions of weightν onG/eΓ1∞ are in fact induced from functions onGL2(k∞)/Γ1∞fixed by the “integral weight” operatorT∞,ν.
2. Natural family of functions having weight
In this section we give examples of functions having weight. A functionfof integral weight ν can be viewed as a function onGL2(k∞)/Γ1∞ which is fixed by the operatorT∞,ν in §1.3.2.
We start with integral weight examples.
2.1. Functions of integral weight. 1. Theta series of imaginary quadratic function fields.
Given a square-free polynomialD in Asuch that K=k(√
D)is “imaginary” quadratic field (i.e. ∞ does not split in K). Let a0 be a fractional ideal in K such that the ideal norm NK/k(a0) = (λ0)forλ0∈k×. The theta seriesθa0,λ0 onk∞××k∞introduced by Rück [10] is:
θa0,λ0(x, y) := X
µ∈a0
φ∞(NK/k(µ)
λ0 xt2)ψ∞(NK/k(µ) λ0 y).
For any g∈GL2(k∞), writeg as γ x y 0 1
!
γ∞z whereγ= a b c d
!
inΓ(1)0 (D), (x, y)in k∞× ×k∞, γ∞ inΓ1∞, and zin k∞×. Let
Θa0,λ0(g) :=|x|12 · d
D
θa0,λ0(x, y)·δz.
Here δ : k∞× → {±1} is the local norm symbol at ∞, i.e. δz = 1 if z ∈ k∞× is a norm of an element in K∞ = k∞(√
D) and −1 otherwise. Then from the transformation law of θa0,λ0 (cf. [10] Proposition 5.1),Θa0,λ0 is a weight 1 function onGL2(k∞)satisfying that for γ= a b
c d
!
∈Γ(1)0 (D),z∈k×∞,g∈GL2(k∞), γ∞∈Γ1∞,
Θa0,λ0(γgγ∞z) = d
D
Θa0,λ0(g)δz.
2. Automorphic forms of Drinfeld type. Given a non-zero idealN ofA. Let Γ0(N) :=
( a b c d
!
∈GL2(A) :c≡0 modN )
.
An automorphic formf of Drinfeld type forΓ0(N)is aC-valued function on the double coset spaceΓ0(N)\GL2(k∞)/Γ∞k∞× satisfying the followingharmonic property: for any elementg in GL2(k∞),
f g 0 1
π∞ 0
!!
=−f(g)and X
µ∈GL2(O∞)/Γ∞
f(gµ) = 0.
Here Γ∞ is the Iwahori subgroup (
a b c d
!
∈GL2(O∞) :c≡0 modπ∞O∞ )
.
The harmonicity implies thatf is of weight2.
Remark. The space of weight 2 functions onΓ0(N)\GL2(k∞)/Γ∞k∞× is generated by auto- morphic forms of Drinfeld type forΓ0(N)together with a “non-harmonic” functionE on the double coset spaceGL2(A)\GL2(k∞)/Γ∞k∞× satisfying that forg∈GL2(k∞)
E(g) +E(g 0 1 π∞ 0
! ) = q
1−q,
and the Fourier expansion ofE is:
E π∞r u
0 1
!
=q−r+2
1
1−q2 + X
06=λ∈A degλ≤r+2
σ(λ)ψ∞(λu)
where r∈Zandu∈k∞. Hereσis the divisor functionσ(λ) := P
monicm|λ
|m|.RestrictingE to the subgroupH∞, it agrees with theimproper Eisenstein series of Gekeler (cf. [1] 5.5).
3. Principal series and Eisenstein series. Given an integer ν. Consider theprincipal series π(| · |ν−12 ,| · |1−ν2 )(cf. [7] Chap. I §3), which is the space of smooth functionsf onGL2(k∞) satisfying that for any elementg∞∈GL2(k∞), aandd∈k∞×,b∈k∞,
f
a b 0 d
! g∞
!
=|a|ν2 · |d|−ν2 f(g∞).
The subspace of functionsf inπ(| · |ν−12 ,| · |1−ν2 )such that
f(g∞γ∞) =f(g∞)for allγ∞∈GL2(O∞)
is spanned by the functionϕν where forxandwin k×∞,y in k∞,γ∞ inGL2(O∞), ϕν
x y 0 w
! γ∞
!
:=|x|ν2|w|−ν2.
Define
Φν(g∞) :=ϕν(g∞)−qν2−1ϕν(g∞ 1 0 0 π∞
! )
for any g∞ ∈GL2(k∞). ThenΦν is a function of weightν (with trivial central character) with
Φν
x y 0 1
!
=|x|ν2(1−qν−1)andΦν
x y 0 1
!
0 1
π∞ 0
!!
=|x|ν2(qν2 −qν2−1).
Fix an integerν >2. Let B(A) :=
( a b c d
!
∈GL2(A) :c= 0 )
.
Then for anyγ∈B(A)andg∞∈GL2(k∞), one has Φν(γg∞) = Φν(g∞).
Consider the Eisenstein series
Eν(g∞) := X
γ∈B(A)\GL2(A)
Φν(γg∞), forg∞∈GL2(k∞).
ThenEν is a well-defined weightν function onGL2(A)\GL2(k∞)/Γ∞k∞×.
2.2. Functions of half integral weight. In §1.3 we note thatΘis a function of weight 12. Here we give weight 32 functions via theta series from reduced norm form on lattices of pure quaternions inside quaternion algebras.
LetD=DN− be a “definite” quaternion algebra overk(i.e.Dis ramified at∞) whereN− is the product of finite ramified primes ofD. Choose an idealN+ofAwhich is prime toN−. AnEichler order RN+,N− of type (N+, N−)is an (A-)order inD such that the (AP-)order (RN+,N−)P :=RN+,N−⊗AAP is maximal inDP :=D ⊗kkP for allP -N+, and forP |N+ there exist an isomorphismϕP :DP ∼= Mat2(kP)such that
ϕP (RN+,N−)P
= (
a b c d
!
∈Mat2(AP) :c≡0 modN+AP
) .
A left ideal I of RN+,N− is a rank 4 A-lattice in D such that for all finite primes P, IP :=I⊗AAP =RN+,N−·gP for somegP ∈ D×P.
Let I1, ..., In be representatives of left ideal classes of R. For eachi, let Ri be the right order ofIi. Consider theA-latticeSiof pure quaternions inRi, i.e.Si:={b∈Ri: Tr(b) = 0}.
For(x, y)∈k×∞×k∞, define the following theta series forSi: ϑi(x, y) := 1
2 X
b∈Si
φ∞(Nr(b)xt2)ψ∞(Nr(b)y).
Proposition 2.1. (cf. [17] Proposition 3.1)LetN0=N+·N−. Forγ= a b c d
!
∈Γ(1)0 (N0)
with v∞(cx)> v∞(cy+d), we have
ϑi(γ◦(x, y)) =|cy+d|32(γ, x, y)3ϑi(x, y).
Giveng˜∈G. Write˜gasz˜˜γ(h, ξ)˜γ∞, wherezink×∞,γinΓ(1)0 (N0),γ∞inΓ1∞,h= x y 0 1
!
in H∞, andξin S1. For eachi, we extendϑi to a function Θi onG: Θi(˜g) :=|x|34 ·ϑi(x, y)·ξ3.
ThenΘi is of weight 32 and forz∈k∞×,γ∈Γ(1)0 (N0),γ∞∈Γ1∞, ξ∈S1 Θi(˜zγ(1, ξ)˜˜ g˜γ∞) =ξ3Θi(˜g).
Givenλ∈A withλ6= 0, the Fourier coefficient Θ∗i(r, λ) =
0 ifdegλ+ 2> ror−λ∈(k∞×)2,
1
2·q−34r·#{b∈Si: Nr(b) =λ} ifdegλ+ 2≤rand−λ /∈k∞2 .
Whendegλ+2≤rand−λ /∈k∞2 ,Θ∗i(r, λ)can be expressed by number ofoptimal embeddings intoRi. Letd∈A−k2∞. An optimal embedding of the quadratic orderOd:=A[√
d]intoRi
is an embedding ιofk(√
d)intoDso that ι k(√
d)
∩Ri=ι(Od).
For any α∈R×i , α−1ια is also an optimal embedding ofOd into Ri. Let wi:= #(R×i /F×q), u(d) := #(Od×/F×q), and hi(d) denote the number of optimal embeddings of Od into Ri
modulo conjugation by elements inR×i . We have Proposition 2.2. For06=λ∈Awith degλ+ 2≤r,
Θ∗i(r, λ) =q−34r· wi
2
X
−λ=df2,f monic
hi(d) u(d).
3. Hecke operators
Here we assume κ = 2ν + 1 where ν ∈ Z≥0 and study Hecke operators on weight κ2 functions.
For06=m∈A, consider the double cosetΓ^1(N)
1 0
0 m
! ,1
!
Γ^1(N)inGwhere
Γ1(N) = (
γ∈SL2(A) :γ≡ 1 ∗ 0 1
!
modN )
.
One has
Γ^1(N)
1 0
0 m
! ,1
!
Γ^1(N) =a
i
Γ^1(N)
1 0 0 m
! ,1
!
·˜γi
whereγ˜i are right coset representatives of the following subgroup inΓ^1(N):
1 0
0 m
! ,1
!−1
Γ^1(N)
1 0 0 m
! ,1
!
\Γ^1(N).
Let f be a weight κ2 function f on G/gΓ1∞ with “central” character χ∞ onk∞×. Suppose f(˜γ˜g) =f(˜g)for allγ∈Γ1(N). For each monic polynomialmin A, define
f Γ^1(N)
1 0
0 m
! ,1
!
Γ^1(N)˜g
! :=X
i
f
1 0 0 m
! ,1
!
˜ γi˜g
! .
Lemma 3.1. Let m be a monic polynomial in A. Suppose the prime-to-N part of the ideal (m)is not a square ideal of A, then for allg˜∈G
f Γ^1(N)
1 0 0 m
! ,1
!
Γ^1(N)˜g
!
= 0.
Definition 3.2. Letf be a weight κ2 functionf onG/gΓ1∞ with “central” character χ∞ on k∞×. Supposef(˜γ˜g) =f(˜g)for allγ∈Γ1(N). Given any primeP ofA. We define for˜g∈G
TP2,κ2f(˜g) :=|P|(κ2−2)·f Γ^1(N)
1 0 0 P2
! ,1
!
Γ^1(N)˜g
! .
As in the classical case, we have
Proposition 3.3. LetχN : (A/N A)× be a Dirichlet character. Given a weight κ2 functionf
on Gwith “central” character χ∞ such thatf(˜γ˜g) =χN(d)f(˜g)for γ= a b c d
!
∈Γ(1)0 (N)
andg˜∈G. Then for each prime P,
TP2,κ2f
πr∞ u
0 1
! ,1
!
= X
degλ+2≤r
(TP2,κ2f)∗(r, λ)ψ∞(λu)
where
(TP2,κ2f)∗(r, λ) = χ∞(P2)·
|P|κ2f∗(r+ 2 degP, P2λ)
+χN(P)χ∞(P)
(−1)νλ P
|P|(ν−1)·
f∗(r, λ)
+χN(P2)|P|κ−2·
|P|−κ2f∗(r−2 degP, λ P2)
.
Here we setf∗(r−2 degP,Pλ2) = 0if P2-λandχN(P) = 0 ifP|N.
Proof. For sake of completeness we sketch the argument.
Consider the double cosetΓ1(N) 1 0 0 P2
!
Γ1(N), which is equal to
S
degv<2 degP
Γ1(N)av ifP|N,
S
degv<2 degP
Γ1(N)av
!
∪ S
h6=0,degh<degP
Γ1(N)bh
!
∪Γ1(N)τ ifP -N.
Here av = 1 v 0 P2
!
,bh=ςP P h 0 P
!
,τ=ςP2
P2 0 0 1
!
,ςmis an element inSL2(A)such
that ςm≡ m−1 0
0 m
!
modN. The contribution ofav is:
|P|(κ2−2)· X
degv<2 degP
f
1 v 0 P2
! ,1
·
πr∞ u
0 1
! ,1
!
= X
degλ+2≤r
χ∞(P2)·
|P|κ2f∗(r+ 2 degP, P2λ)
ψ∞(λu).
WhenP divides N, χN(P) =χN(P2) = 0 and so the formula holds.
Assume P is prime toN. The contribution ofτ is:
|P|(κ2−2)f ˜ςP2
πr−2 deg∞ P P2u
0 1
! ,1
!
= X
degλ+2≤r
χN(P2)|P|(κ−2)
|P|−κ2f∗(r−2 degP, λ P2)
·ψ∞(λu).
For the contribution of bh with h 6= 0 and degh < degP, let γh, γh0 ∈ Γ1(N) such that γh0 1 0
0 P2
!
γh=bh.
Thenγ˜h0
1 0 0 P2
! ,1
˜ γh= ˜ςP
P h 0 P
! ,
h P
and
|P|(κ2−2)f
1 0 0 P2
! ,1
˜ γh
π∞r u
0 1
! ,1
!
=
ω(P)κh P
|P|−12ψ∞(λh P )
· X
degλ+2≤r
χN(P)χ∞(P)|P|(ν−1)f∗(r, λ)ψ∞(λu)
Sinceω(P)2ν =(−1)ν
P
and|P|−12 P
h6=0,degh<degP
h P
ψ∞(λhP) =
λ P
ω(P)−1,
|P|(κ2−2) X
h6=0,degh<degP
f
1 0 0 P2
! ,1
˜ γh
π∞r u
0 1
! ,1
!
= X
degλ+2≤r
χN(P)χ∞(P)|P|(ν−1)(−1)νλ P
f∗(r, λ)
ψ∞(λu).
Combining these we get the formula for the Fourier coefficients ofTP2,κ2f.
From lattices of pure quaternions in the right orders of left idealsI1, . . . , Inof Eichler order RN+,N− inD(N−)we have introduced weight 32theta seriesΘifor1≤i≤nin §2.2. Suppose N+= 1, i.e.R=RN+,N− is a maximal order (andN0=N−). The action of Hecke operators onΘi can be expressed by so-calledBrandt matrices.
Given a left idealIofRN+,N−, the setI−1={b∈ D:IbI⊂I}is a right ideal forRN+,N−
whose left order is the right order of I. The reduced ideal norm of I is denoted byNr(I), which is the fractional ideal of Agenerated by the reduced norm of elements inI. For each monic polynomial m in A, the m-th Brandt matrix B(m) :=
Bij(m)
1≤i,j≤n in Matn(Z) where
Bij(m) := #{b∈Ij−1Ii: (Nr(b)/Nij) = (m)}
(q−1)wj .
Here Nij is the monic generator of the reduced ideal norm Nr(Ij)−1Nr(Ii) of Ij−1Ii, and wj= #(R×j/F×q)where Rj is the right order ofIj.
Proposition 3.4. For every prime P ofA,
TP2,32Θi=X
j
Bij(P)Θj=wi
X
j
Bji(P)( 1 wjΘj).
Proof. For each monic polynomialminA,wjBij(m) =wiBji(m)and so the second equality is clear. We only need to show the first equality.
When P | N0, TP2,32Θ∗i(r, λ) = |P|32Θ∗i(r+ 2 degP, P2λ) for λ∈ A with degλ+ 2≤ r.
By Proposition 2.2 and the fact that hi(P2d) = 0 for any d and prime P | N0, we get
|P|32Θ∗i(r+ 2 degP, P2λ) = Θ∗i(r, λ)and soTP2,32Θi = Θi.
On the other hand, B(P)is a permutation matrix of order 2, and the entry Bij(P) = 1 impliesRi∼=Rj. Therefore the proposition holds forP|N0.
Suppose P - N0. For λ ∈ A and r ≥ degλ+ 2, let ci(λ) =:q34rΘi(r, λ). To prove this proposition, we need to show
ci(P2λ) +−λ P
ci(λ) +|P|ci( λ
P2) =X
j
Bij(P)cj(λ).
For each j, consider the map
{α∈Ij−1Ii: (Nr(α)/Nij) = (P)} × {a:a∈Sj,Nr(a) =λ, ∈F×q} −→Si
which is defined by
(α, a)7−→b=α∗·a·α.
Here α∗:= ¯αNij. ThenNr(b) =λP202for some 0∈F×q. Note that
#{α∈Ij−1Ii: (Nr(α)/Nij) = (P)}= (q−1)wjBij(P), and
#{a:a∈Sj,Nr(a) =λ, ∈F×q}= (q−1)cj(λ).
Given b ∈ Si with Nr(b) = λP22 for ∈ F×q . First, we consider the case when P - b, i.e. b 6= P a for any a ∈ Si. Then there exists a unique α ∈ Ij−1Ii for some j, up to R×j, with (Nr(α)/Nij) = (P) such that b = βα where β ∈ Ii−1Ij. Since Tr(b) = b+ ¯b = 0, b =−¯αβ¯=−α∗β∗ and so β∗ =aαwith a∈Sj, Nr(a) =λ and∈F×q. Therefore in this case,b=α∗(−a)αfor a uniqueα∈Ij−1Ii up toR×j.
Now, we suppose P|band consider the following two cases:
(1) Assume P | λ. Writeb as P·a for some a∈Si with Nr(a) =λ. If P - a, there exist
an element α∈Ij−1Ii for somej, up toR×j, such that (Nr(α)/Nij) = (P)and a=βαwith β ∈Ii−1Ij. Therefore
b=P·a=α∗·(αβ)·α.
IfP |a, thena=a0P for somea0∈Si withNr(a0) =Pλ2 and so b=P a0P =α∗·(α0a0α∗)·α
for anyj and any α∈Ij−1Ii with(Nr(α)/Nij) = (P). Hence we obtain X
j
Bij(P)cj(λ) = ci(λP2)−ci(λ)
+ ci(λ)−ci( λ P2)
+ (|P|+ 1)ci( λ P2)
=ci(λP2) +|P|ci( λ P2).
(2) Assume P - λ. Write b as aP where a ∈ Si with Nr(a) = λ. If b = α∗0a0α where α∈Ij−1IiwithNr(a) =λand0∈F×q, then
α∗0a0α=P a=α∗(αa).
LetI =Ii−1Ijα. Then I⊂Ri withNr(I) = (P) andI·a⊂I. We deduce that there exist a unique prime ideal p of the quadratic order A[a] so that I = Rip. Since there are only 1 + −λP
ideals ofA[a]whose ideal norm isP,b can be written asα∗0aα withNr(a) =λin 1 + −λP
ways, up toR×j ifα∈Ij−1Ii. Combining these we have that whenP -λ X
j
Bij(P)cj(λ) = ci(λP2)−ci(λ)
+ (1 +−λ P
)ci(λ) =ci(λP2) +−λ P
ci(λ).
By (1) and (2) the proposition holds.
4. Special values and an analogue of Waldspurger’s formula In this section we present a function field analogue of Waldspurger’s formula.
4.1. Definite Shimura curves and automorphic forms. Let D = DN− be a definite quaternion algebra overkand letN− be the product of finite ramified primes ofD. Choose an idealN+ ofAprime toN−. Thedefinite Shimura curveX =XN+,N− of type (N+, N−) is
RˆN×+,N−\( ˆD××Y)/D×.
Here Dˆ := D ⊗k ˆk; RˆN+,N− := RN+,N− ⊗AAˆ where RN+,N− is an Eichler order of type (N+, N−);Y is the curve of genus zero such that for eachk-algebraM,
Y(M) :={x∈ D ⊗kM :x6= 0,Tr(x) = Nr(x) = 0}/M×.
The (right) action ofD×onY is by conjugation. It is known thatX is a disjoint finite union of genus zero curves, and the components correspond canonically to left ideal classes ofRN+,N−. From now on we assume N+ = 1, i.e. RN+,N− = R is a maximal order and N0 =N−. LetI1, . . . , In be representatives of left ideal classes ofR, and letXi be the component ofX corresponding toIi. Denoteei to be the divisor class inPic(X)corresponding toXi. Then Pic(X) =⊕Zei.TheGross height pairingonPic(X)is defined by
< ei, ej>=
0 ifi6=j,
wi= #(R×i /F×q) ifi=j, and extending bi-additively.
As in the case of definite Shimura curves overQ (cf. [4]), we haveHecke correspondence tmonX for each monic polynomialminA which satisfies
Proposition 4.1. Fori= 1,2, ..., n,
tmei=
n
X
j=1
Bij(m)ej.
Here B(m) = Bij(m)
1≤i,j≤n is them-th Brandt matrix.
We point out that the Hecke correspondences are self-adjoint with respect to the Gross height pairing.
Function field analogue of Eichler’s theory. Recall that an automorphic form f of Drinfeld type for Γ0(N) is a C-valued function on Γ0(N)\GL2(k∞)/Γ∞k∞× satisfying the harmonic property in §2.1. We sayf is acusp formif for all g∞∈GL2(k∞)and γ∈GL2(A),
Z
A\k∞
f γ 1 hγu
0 1
! g∞
! du= 0.
Here the Haar measureduis normalized so thatR
A\k∞1du= 1, andhγ is a generator of the ideal ofAwhich is maximal for the property that
γ 1 hγA
0 1
!
γ−1⊂Γ0(N).
For each non-zero ideal N of A, the Petersson inner product on the C-vector space S(Γ0(N))of Drinfeld type cusp forms forΓ0(N)is a non-degenerate pairing
(f, g) :=
Z
G0(N)
f·g.
Here G0(N) = Γ0(N)\GL2(k∞)/Γ∞k∞×. The measure of each double coset [e] in G0(N)is normalized to be
d([e]) := q−1
2 · 1
#(StabΓ0(N)(e)).
Definition 4.2. A Drinfeld type cusp form f for Γ0(N)is a new form if, with respect to Petersson inner product,f is orthogonal to functions
g
d 0 0 1
! g∞
!
for all Drinfeld type cusp form g for Γ0(M) where M | N and d | (N/M). We call f a newform iff is also a Hecke eigenform.
Let Mnew(Γ0(N0)) := Snew(Γ0(N0)⊕CEN0, where Snew(Γ0(N0) is the space of Drinfeld type new forms and EN0 is an analogue of Eisenstein series with Fourier expansion given by:
forr∈Zand u∈k∞ EN0
πr∞ u
0 1
!
=q−r+2
1 q2−1
Y
P|N0
(|P| −1) + X
m∈A, monic degm+2≤r
σN0(m)X
∈F×q
ψ∞(mu)
.
Here σN0(m)is the divisor functionσN0(m) := P
m0 |m, monic (N0,m0)=1
|m0|.
As in classical Eichler’s theory, one can establish an isomorphism (cf. [18] Theorem 2.5) Mnew(Γ0(N0)) −→ Pic(X)⊗ZC
f 7−→ ef
such that Tmf 7→ tmef for all monic polynomials m in A, and < ef, ef >= 1 for each newform f normalized so that the “first” Fourier coefficientf∗(2,1)is one. Here the Tmare Hecke operators on automorphic forms for Γ0(N0).
4.2. The Shimura map Sh. Consider the definite Shimura curveX =X1,N− (i.e.N+= 1 andN0=N−). For any elemente=P
iaiei ∈Pic(X)⊗ZC, define Θe:=X
i
aiΘi
whereΘi are weight 32 function introduced in §2.2. For eacha∈A−k2∞, let
ea:=
n
X
i=1
X
a=df2,f monic
hi(d) 2u(d)
ei∈Pic(X)⊗ZQ.
From Proposition 2.2, the Fourier coefficients ofΘeare Θ∗e(r, λ) =q−34r·< e, e−λ>
for−λ6= 0∈A−k∞2 withdegλ+ 2≤r, andΘ∗e(r,0) =q−34r·dege/2.
Given any primeP ofA, Proposition 3.4 and Proposition 4.1 tells us that TP2,32Θe= Θ(tPe).
LetM3
2(Γ(1)0 (N0))be the space of weight 32 functions on the double coset space kf∞×Γ(1)0^(N0)/
G .
gΓ1∞.
SincePic(X)⊗ZCand the spaceMnew(Γ0(N0))of automorphic “new” forms of Drinfeld type forΓ0(N0)are isomorphic Hecke modules, we have the following map
Sh: Mnew(Γ0(N0)) ∼= Pic(X)⊗ZC −→ M3
2(Γ(1)0 (N0)) f 7→ ef 7−→ Θef =:Sh(f)
such that for all primeP ofA
Sh(TPf) =TP2,32 Sh(f).
This mapShcan be viewed as an analogue of Shimura correspondence. LetMnew(Γ0(N0))+ be the eigenspace ofTP with eigenvalue1forP |N0, and letMnew(Γ0(N0))−⊂Snew(Γ0(N0)) be the orthogonal component (with respect to Petersson inner product). Then the kernel of Shcontains the subspaceMnew(Γ0(N0))−.
For eachf ∈Mnew(Γ0(N0)), letef ∈Pic(X)⊗ZCbe the corresponding divisor from the Hecke module isomorphism as in §4.1. Then
q34r·Sh(f)∗(r, λ) =< ef, e−λ>
is independent of r, for−λ∈A−k2∞withdegλ+ 2≤r. This value is denoted bym(f,−λ) (i.e. theλ-th Fourier coefficient of the weight 32 functionSh(f))).
Suppose f is a normalized Drinfeld type newform for Γ0(N0). Let D be an irreducible polynomial inA−k2∞satisfying DP
=−1for all prime factorsP ofN0. Set the divisoref,D
to be< ef, eD>·ef. Summarizing, we have
Theorem 4.3. The Gross height < ef,D, ef,D>of the divisor ef,D is exactlym(f, D)2.
