Definite Quaternion Algebras over Function Fields and Brandt Matrices
Fu-Tsun Wei, National Tsing Hua University , and Jing Yu, National Taiwan University.
March 18, 2010, Waseda University
Jing Yu Quaternion Algebras over function fields
Notations
k : rational function field Fq(t), q is power of p, p an odd prime.
A : polynomial ring Fq[t].
∞ : infinite place, corresponding to the valuation of the degree.
k∞: Fq((1t)), i.e., the completion of k at ∞.
P : monic irreducible in A, i.e. finite prime.
¯k∞: a fixed algebraic closure of k∞.
¯k : the algebraic closure of k inside k∞. Fq(t) : the algebraic closure of Fq inside ¯k.
v∞: the valuation on k∞ s.t. v∞(a) = − deg(a) for all a ∈ A.
For us : k, A, k∞ play the role of Q, Z, and R respectively.
Jing Yu Quaternion Algebras over function fields
Definite quaternion algebras
Let P0 be a fixed finite prime, D be the (“definite”) quaternion algebra over k which ramifies only at ∞ and P0.
Let R ⊂ D be a maximal order (A- rank 4).
Interested in left ideals I of R inside D.
The left ideal classes can be put into 1-1 correspondence with isomorphism classes of rank 2 supersingular Drinfeld A-modules in A-characteristic P0.
Let RI be the right order of I, and set w(I) = #(RI)×/(q − 1).
If [φ] is class of Drinfeld A-modules corresponds to I, w(φ) = w(I) counts its automorphisms, then Mass Formula (Gekeler) says
X
[φ]
1
w(φ) = qdeg P0 − 1
q2− 1 = ζA(−1)(1 − qdeg P0).
Jing Yu Quaternion Algebras over function fields
Definite quaternion algebras
Let P0 be a fixed finite prime, D be the (“definite”) quaternion algebra over k which ramifies only at ∞ and P0.
Let R ⊂ D be a maximal order (A- rank 4).
Interested in left ideals I of R inside D.
The left ideal classes can be put into 1-1 correspondence with isomorphism classes of rank 2 supersingular Drinfeld A-modules in A-characteristic P0.
Let RI be the right order of I, and set w(I) = #(RI)×/(q − 1).
If [φ] is class of Drinfeld A-modules corresponds to I, w(φ) = w(I) counts its automorphisms, then Mass Formula (Gekeler) says
X
[φ]
1
w(φ) = qdeg P0 − 1
q2− 1 = ζA(−1)(1 − qdeg P0).
Jing Yu Quaternion Algebras over function fields
Definite quaternion algebras
Let P0 be a fixed finite prime, D be the (“definite”) quaternion algebra over k which ramifies only at ∞ and P0.
Let R ⊂ D be a maximal order (A- rank 4).
Interested in left ideals I of R inside D.
The left ideal classes can be put into 1-1 correspondence with isomorphism classes of rank 2 supersingular Drinfeld A-modules in A-characteristic P0.
Let RI be the right order of I, and set w(I) = #(RI)×/(q − 1).
If [φ] is class of Drinfeld A-modules corresponds to I, w(φ) = w(I) counts its automorphisms, then Mass Formula (Gekeler) says
X
[φ]
1
w(φ) = qdeg P0 − 1
q2− 1 = ζA(−1)(1 − qdeg P0).
Jing Yu Quaternion Algebras over function fields
Drinfeld A-modules
Let (L, ι) (denoted by L simply) be an A-field, i.e. a field L together with Fq-algebra homomorphism ι : A → L.
The kernel of ι is called the A-characteristic of L. This
A-characteristic is a prime ideal (P ), here P is a prime (monic irreducible) in A or zero.
Consider the twist polynomial ring : ( τ (x) = xq) L{τ } = EndFq(Ga/L)
A rank 2 Drinfeld A-modules φ over L with A-characteristic P is an Fq-algebra homomorphism φ : A → L{τ }, which satisfies
φt= ι(t) + gτ + ∆τ2, ∆ 6= 0.
Jing Yu Quaternion Algebras over function fields
Drinfeld A-modules
Let (L, ι) (denoted by L simply) be an A-field, i.e. a field L together with Fq-algebra homomorphism ι : A → L.
The kernel of ι is called the A-characteristic of L. This
A-characteristic is a prime ideal (P ), here P is a prime (monic irreducible) in A or zero.
Consider the twist polynomial ring : ( τ (x) = xq) L{τ } = EndFq(Ga/L)
A rank 2 Drinfeld A-modules φ over L with A-characteristic P is an Fq-algebra homomorphism φ : A → L{τ }, which satisfies
φt= ι(t) + gτ + ∆τ2, ∆ 6= 0.
Jing Yu Quaternion Algebras over function fields
Supersingular Drinfeld A-modules
Let φ and φ0 be two Drinfeld modules. A morphism u : φ → φ0 over L is an element u ∈ L{τ } such that for all a ∈ A
uφa= φ0au.
We have accordingly endomorphisms, isomorphisms, and automorphisms of Drinfeld modules. A non-zero morphism is called an isogeny.
Given φ of rank 2 over L, and prime P ∈ A. The P -torsion of φ φ[P ] = {x ∈ L : φP(x) = 0},
where L is fixed algebraic closure of L, is a finite A-module isomorphic to (A/(P ))2, if P is not the A-characteristic of L.
In case the A-characteristic is (P0) 6= 0, either φ[P0] ∼= A/(P ) or φ is supersingular, i.e. φ[P0] = 0.
Jing Yu Quaternion Algebras over function fields
Supersingular Drinfeld A-modules
Let φ and φ0 be two Drinfeld modules. A morphism u : φ → φ0 over L is an element u ∈ L{τ } such that for all a ∈ A
uφa= φ0au.
We have accordingly endomorphisms, isomorphisms, and automorphisms of Drinfeld modules. A non-zero morphism is called an isogeny.
Given φ of rank 2 over L, and prime P ∈ A. The P -torsion of φ φ[P ] = {x ∈ L : φP(x) = 0},
where L is fixed algebraic closure of L, is a finite A-module isomorphic to (A/(P ))2, if P is not the A-characteristic of L.
In case the A-characteristic is (P0) 6= 0, either φ[P0] ∼= A/(P ) or φ is supersingular, i.e. φ[P0] = 0.
Jing Yu Quaternion Algebras over function fields
Quaternion algebras as endomorphism algebras
Supersingular Drinfeld A-modules φ are always definable over finite A-field L, in fact, quadratic extension of FP0 := A/(P0).
If φ is of rank 2, EndL(φ) ⊗Ak = D = D(P0, ∞) is a quaternion division algebra over k. This quaternion algebra is “definite”, in the sense it splits at primes differ from the characteristic P0 and ∞.
Then EndL(φ) is a maximal order in D. Left ideal classes of EndL(φ) correspond bijectively to the isomorphism classes of rank 2 supersingular Drinfeld A-modules over L = FP0.
The group G = Gal(FP0/FP0) acts on the left ideal classes by acting on the corresponding supersingular Drinfeld A-modules, the types (i.e. conjugacy classes) of maximal orders in D correspond bijectively to the orbits of isomorphism classes of supersingular Drinfeld A-modules under the action of G.
Jing Yu Quaternion Algebras over function fields
Quaternion algebras as endomorphism algebras
Supersingular Drinfeld A-modules φ are always definable over finite A-field L, in fact, quadratic extension of FP0 := A/(P0).
If φ is of rank 2, EndL(φ) ⊗Ak = D = D(P0, ∞) is a quaternion division algebra over k. This quaternion algebra is “definite”, in the sense it splits at primes differ from the characteristic P0 and ∞.
Then EndL(φ) is a maximal order in D. Left ideal classes of EndL(φ) correspond bijectively to the isomorphism classes of rank 2 supersingular Drinfeld A-modules over L = FP0.
The group G = Gal(FP0/FP0) acts on the left ideal classes by acting on the corresponding supersingular Drinfeld A-modules, the types (i.e. conjugacy classes) of maximal orders in D correspond bijectively to the orbits of isomorphism classes of supersingular Drinfeld A-modules under the action of G.
Jing Yu Quaternion Algebras over function fields
Quaternion algebras as endomorphism algebras
Supersingular Drinfeld A-modules φ are always definable over finite A-field L, in fact, quadratic extension of FP0 := A/(P0).
If φ is of rank 2, EndL(φ) ⊗Ak = D = D(P0, ∞) is a quaternion division algebra over k. This quaternion algebra is “definite”, in the sense it splits at primes differ from the characteristic P0 and ∞.
Then EndL(φ) is a maximal order in D. Left ideal classes of EndL(φ) correspond bijectively to the isomorphism classes of rank 2 supersingular Drinfeld A-modules over L = FP0.
The group G = Gal(FP0/FP0) acts on the left ideal classes by acting on the corresponding supersingular Drinfeld A-modules, the types (i.e. conjugacy classes) of maximal orders in D correspond bijectively to the orbits of isomorphism classes of supersingular Drinfeld A-modules under the action of G.
Jing Yu Quaternion Algebras over function fields
Brandt matrices
Fix miximal order R. For left ideal I, set I−1 = {b ∈ D : IbI ⊂ I}, a right ideal for R whose left order is the right order of I.
Let {I1, ..., In} be left ideals of R representing the distinct ideal classes, with I1= R. Let Ri be the right order of Ii, and
wi = #(R×i )/(q − 1). Let Mij = Ij−1Ii, which is a left ideal of Rj with right order Ri. For any element b ∈ Mij, Nr(b) denotes its reduced norm, and define Nij = f /g where f and g are the unique monic polynomials in A s.t. the quotients Nr(b)/Nij are all in A with no common factor.
For each monic m ∈ A, let
Bij(m) = #{b ∈ Mij : (Nr(b)/Nij) = (m)}
(q − 1)wj and B(m) = (Bij(m)) ∈ Matn(Z).
Also set B(0) = (Bij(0)), with Bij(0) = (q−1)w1
j.
Jing Yu Quaternion Algebras over function fields
Supersingular Drinfeld Modules and Brandt Matrices
For each i, let φi be a supersingular Drinfeld module rank 2 corresponding to Ii. Then End(φi) ∼= Ri. Moreover, one has
Mij ∼= Hom(φi, φj), b 7→ ujbu−1i , where ui : φ1→ φi is the isogeny corresponding to Ii.
Note that given two isogenies u and u0 from φi to φj, the finite A-submodule scheme ker(u) and ker(u0) are equal if and only if u0 = αu, where α ∈ Aut(φj). Any finite A-submodule scheme C of φi is the kernel of some isogeny with height h, 0 ≤ h ≤ 2.
The Euler-Poincar´e characteristic of C is the ideal (P0hd1d2), if C(L) ∼= A/(d1) × A/(d2).
The entry Bij(m) is exactly the number of finite A-submodule schemes C of φi whose Euler-Poincar´e characteristic is (m) and φi/C ∼= φj.
Jing Yu Quaternion Algebras over function fields
Supersingular Drinfeld Modules and Brandt Matrices
For each i, let φi be a supersingular Drinfeld module rank 2 corresponding to Ii. Then End(φi) ∼= Ri. Moreover, one has
Mij ∼= Hom(φi, φj), b 7→ ujbu−1i , where ui : φ1→ φi is the isogeny corresponding to Ii.
Note that given two isogenies u and u0 from φi to φj, the finite A-submodule scheme ker(u) and ker(u0) are equal if and only if u0 = αu, where α ∈ Aut(φj). Any finite A-submodule scheme C of φi is the kernel of some isogeny with height h, 0 ≤ h ≤ 2.
The Euler-Poincar´e characteristic of C is the ideal (P0hd1d2), if C(L) ∼= A/(d1) × A/(d2).
The entry Bij(m) is exactly the number of finite A-submodule schemes C of φi whose Euler-Poincar´e characteristic is (m) and φi/C ∼= φj.
Jing Yu Quaternion Algebras over function fields
Supersingular Drinfeld Modules and Brandt Matrices
For each i, let φi be a supersingular Drinfeld module rank 2 corresponding to Ii. Then End(φi) ∼= Ri. Moreover, one has
Mij ∼= Hom(φi, φj), b 7→ ujbu−1i , where ui : φ1→ φi is the isogeny corresponding to Ii.
Note that given two isogenies u and u0 from φi to φj, the finite A-submodule scheme ker(u) and ker(u0) are equal if and only if u0 = αu, where α ∈ Aut(φj). Any finite A-submodule scheme C of φi is the kernel of some isogeny with height h, 0 ≤ h ≤ 2.
The Euler-Poincar´e characteristic of C is the ideal (P0hd1d2), if C(L) ∼= A/(d1) × A/(d2).
The entry Bij(m) is exactly the number of finite A-submodule schemes C of φi whose Euler-Poincar´e characteristic is (m) and φi/C ∼= φj.
Jing Yu Quaternion Algebras over function fields
About Brandt matrices
(1) The row sumsP
jBij(m) are independent of i and equal to σ(m)P0 :=X
m0
qdeg(m0)
sum is over all monic polynomial m0|m which is prime to P0. (2) If (m, m0) = 1, then B(m)B(m0) = B(mm0).
(3) If B(P0) 6= 1, it is a permutation matrix of order 2 and B(P0`) = B(P0)`.
(4) If P 6= P0 is another monic prime, then for ` ≥ 2, B(P`) = B(P`−1)B(P ) − qdeg(P )B(P`−2).
(5) The B(m) generate a commutative subring B of Matn(Z).
(6) For all i, j the symmetry relation
wjBij(m) = wiBji(m).
(7) The algebra B ⊗ZQ is semisimple, and isomorphic to a product of totally real number fields.
Jing Yu Quaternion Algebras over function fields
About Brandt matrices
(1) The row sumsP
jBij(m) are independent of i and equal to σ(m)P0 :=X
m0
qdeg(m0)
sum is over all monic polynomial m0|m which is prime to P0. (2) If (m, m0) = 1, then B(m)B(m0) = B(mm0).
(3) If B(P0) 6= 1, it is a permutation matrix of order 2 and B(P0`) = B(P0)`.
(4) If P 6= P0 is another monic prime, then for ` ≥ 2, B(P`) = B(P`−1)B(P ) − qdeg(P )B(P`−2).
(5) The B(m) generate a commutative subring B of Matn(Z).
(6) For all i, j the symmetry relation
wjBij(m) = wiBji(m).
(7) The algebra B ⊗ZQ is semisimple, and isomorphic to a product of totally real number fields.
Jing Yu Quaternion Algebras over function fields
About Brandt matrices
(1) The row sumsP
jBij(m) are independent of i and equal to σ(m)P0 :=X
m0
qdeg(m0)
sum is over all monic polynomial m0|m which is prime to P0. (2) If (m, m0) = 1, then B(m)B(m0) = B(mm0).
(3) If B(P0) 6= 1, it is a permutation matrix of order 2 and B(P0`) = B(P0)`.
(4) If P 6= P0 is another monic prime, then for ` ≥ 2, B(P`) = B(P`−1)B(P ) − qdeg(P )B(P`−2).
(5) The B(m) generate a commutative subring B of Matn(Z).
(6) For all i, j the symmetry relation
wjBij(m) = wiBji(m).
(7) The algebra B ⊗ZQ is semisimple, and isomorphic to a product of totally real number fields.
Jing Yu Quaternion Algebras over function fields
About Brandt matrices
(1) The row sumsP
jBij(m) are independent of i and equal to σ(m)P0 :=X
m0
qdeg(m0)
sum is over all monic polynomial m0|m which is prime to P0. (2) If (m, m0) = 1, then B(m)B(m0) = B(mm0).
(3) If B(P0) 6= 1, it is a permutation matrix of order 2 and B(P0`) = B(P0)`.
(4) If P 6= P0 is another monic prime, then for ` ≥ 2, B(P`) = B(P`−1)B(P ) − qdeg(P )B(P`−2).
(5) The B(m) generate a commutative subring B of Matn(Z).
(6) For all i, j the symmetry relation
wjBij(m) = wiBji(m).
(7) The algebra B ⊗ZQ is semisimple, and isomorphic to a product of totally real number fields.
Jing Yu Quaternion Algebras over function fields
About Brandt matrices
(1) The row sumsP
jBij(m) are independent of i and equal to σ(m)P0 :=X
m0
qdeg(m0)
sum is over all monic polynomial m0|m which is prime to P0. (2) If (m, m0) = 1, then B(m)B(m0) = B(mm0).
(3) If B(P0) 6= 1, it is a permutation matrix of order 2 and B(P0`) = B(P0)`.
(4) If P 6= P0 is another monic prime, then for ` ≥ 2, B(P`) = B(P`−1)B(P ) − qdeg(P )B(P`−2).
(5) The B(m) generate a commutative subring B of Matn(Z).
(6) For all i, j the symmetry relation
wjBij(m) = wiBji(m).
(7) The algebra B ⊗ZQ is semisimple, and isomorphic to a product of totally real number fields.
Jing Yu Quaternion Algebras over function fields
About Brandt matrices
(1) The row sumsP
jBij(m) are independent of i and equal to σ(m)P0 :=X
m0
qdeg(m0)
sum is over all monic polynomial m0|m which is prime to P0. (2) If (m, m0) = 1, then B(m)B(m0) = B(mm0).
(3) If B(P0) 6= 1, it is a permutation matrix of order 2 and B(P0`) = B(P0)`.
(4) If P 6= P0 is another monic prime, then for ` ≥ 2, B(P`) = B(P`−1)B(P ) − qdeg(P )B(P`−2).
(5) The B(m) generate a commutative subring B of Matn(Z).
(6) For all i, j the symmetry relation
wjBij(m) = wiBji(m).
(7) The algebra B ⊗ZQ is semisimple, and isomorphic to a product of totally real number fields.
Jing Yu Quaternion Algebras over function fields
Class numbers of imaginary quadratic fields
Let a be an element in k ⊂ k∞. If a 6= 0, then we define (a > 0 if a ∈ (k∞×)2,
a < 0 if a ∈ k×∞− (k∞×)2.
If d ∈ A with d < 0 let h(d) be class number of Od= A[√ d] and let u(d) = #(O×d/F×q) (u(d) = q + 1 or 1).
For a ∈ A with a < 0 the Hurwitz class number is given by
H(a) = X
df2=a,f monic
h(d) u(d).
HP0(a) =
0 if P0 splits in Oa,
2
q−1H(a) if P0 is inert in Oa,
1
q−1H(a) if P0 ramified but prime to conductor of Oa, HP0(a/P02) if P0 divides the conductor of Oa.
Jing Yu Quaternion Algebras over function fields
Class numbers of imaginary quadratic fields
Let a be an element in k ⊂ k∞. If a 6= 0, then we define (a > 0 if a ∈ (k∞×)2,
a < 0 if a ∈ k×∞− (k∞×)2.
If d ∈ A with d < 0 let h(d) be class number of Od= A[√ d] and let u(d) = #(O×d/F×q) (u(d) = q + 1 or 1).
For a ∈ A with a < 0 the Hurwitz class number is given by
H(a) = X
df2=a,f monic
h(d) u(d).
HP0(a) =
0 if P0 splits in Oa,
2
q−1H(a) if P0 is inert in Oa,
1
q−1H(a) if P0 ramified but prime to conductor of Oa, HP0(a/P02) if P0 divides the conductor of Oa.
Jing Yu Quaternion Algebras over function fields
Trace formula
We have analogue of Eichler’s trace formula,
tr B(m) = X
m0∈A,(m0)=(m)
X
s∈A,s2≤4m0
HP0(s2− 4m0)
,
for all monic polynomial m ∈ A.
Set also HP0(0) = (q−1)(qqd−12−1), then Mass formula amounts to tr B(0) = HP0(0).
Jing Yu Quaternion Algebras over function fields
Theta series
Fix addtive characters as σ : Fq → C×, and ψ∞: k∞→ C×, σ(ξ) = exp(2πip trFq/Fp(ξ)),
ψ∞(y) = σ(Res∞(ydt)).
Let n be the class number of the maximal order R, choose representatives Ii, i = 1, · · · , n, of the left ideal classes, and set Mij = Ij−1Ii. For x ∈ k∞×, y ∈ k∞, define Theta Series for D,
θij(x, y) = X
b∈Mij
φ∞(Nr(b)
Nij xt2) · ψ∞(Nr(b) Nij y), where φ∞ is the characteristic function of O∞, and Nij = f /g where f and g are the unique monic polynomials in A s.t. the quotients Nr(b)/Nij are all in A with no common factor.
Jing Yu Quaternion Algebras over function fields
Theta series
Fix addtive characters as σ : Fq → C×, and ψ∞: k∞→ C×, σ(ξ) = exp(2πip trFq/Fp(ξ)),
ψ∞(y) = σ(Res∞(ydt)).
Let n be the class number of the maximal order R, choose representatives Ii, i = 1, · · · , n, of the left ideal classes, and set Mij = Ij−1Ii. For x ∈ k∞×, y ∈ k∞, define Theta Series for D,
θij(x, y) = X
b∈Mij
φ∞(Nr(b)
Nij xt2) · ψ∞(Nr(b) Nij y), where φ∞ is the characteristic function of O∞, and Nij = f /g where f and g are the unique monic polynomials in A s.t. the quotients Nr(b)/Nij are all in A with no common factor.
Jing Yu Quaternion Algebras over function fields
Automorphy of theta series
For each a ∈ A, let Bij0 (a) = #{b ∈ Mij : Nr(b)/Nij = a}. Then (q − 1)wj· Bij(m) = X
(a)=(m)
B0ij(a).
We may rewrite the theta series as θij(x, y) = X
a∈A,deg(a)≤v∞(x)−2
Bij0 (a)ψ∞(ay).
One has θij(x, y + a) = θij(x, y) for a ∈ A.
Also θij(αx, βx + y) = θij(x, y) for α ∈ O×∞, β ∈ O∞. For g = a b
c d
!
∈ SL2(A). Assume v∞(x) > v∞(y), v∞(cx) > v∞(cy + d), and c ≡ 0 (mod P0). Then
θij(g ◦ (x, y)) = q−2v∞(cy+d)· θij(x, y).
Jing Yu Quaternion Algebras over function fields
Automorphy of theta series
For each a ∈ A, let Bij0 (a) = #{b ∈ Mij : Nr(b)/Nij = a}. Then (q − 1)wj· Bij(m) = X
(a)=(m)
B0ij(a).
We may rewrite the theta series as θij(x, y) = X
a∈A,deg(a)≤v∞(x)−2
Bij0 (a)ψ∞(ay).
One has θij(x, y + a) = θij(x, y) for a ∈ A.
Also θij(αx, βx + y) = θij(x, y) for α ∈ O×∞, β ∈ O∞. For g = a b
c d
!
∈ SL2(A). Assume v∞(x) > v∞(y), v∞(cx) > v∞(cy + d), and c ≡ 0 (mod P0). Then
θij(g ◦ (x, y)) = q−2v∞(cy+d)· θij(x, y).
Jing Yu Quaternion Algebras over function fields
Functions on ∞-adic space
Introducing complex-valued functions on GL2(k∞) : θij0 (g) = q−v∞(x)θij(x, y)
where g = γ x y 0 1
!
γ∞α for some γ ∈ Γ0(P0) ∩ SL2(A), γ∞∈ Γ∞, α ∈ k∞×. Moreover, let
Θij(g) = X
∈F×q
θ0ij
1
! g
! .
Then Θij are complex-valued functions on the double coset space Γ0(P0)\ GL2(k∞)/Γ∞k∞×.
Jing Yu Quaternion Algebras over function fields
Definite Shimura curves
Let Y be the genus 0 curve over k associated with the quaternion algebra D, which is defined by:
Y (M ) = {x ∈ D ⊗kM : tr(x) = Nr(x) = 0}/M×. Here M is any k-algebra. The group D× acts on Y by conjugation. If K is a quadratic extension of k, then one can identify Y (K) = Hom(K, D).
To each embedding f : K → D we let y = yf be the image of the unique K-line on the quadric {x ∈ D ⊗kK : tr(x) = N (x) = 0} on which conjugation by f (K×) acts by the character a 7→ a/σ(a), σ is the non-trivial automorphism of K/k. Note that yf is one of the 2 fixed points of f (K×) acting on Y (K); the other is the image of the line where conjugation acts by the character a 7→ σ(a)/a.
Jing Yu Quaternion Algebras over function fields
Definite Shimura curves
Let Y be the genus 0 curve over k associated with the quaternion algebra D, which is defined by:
Y (M ) = {x ∈ D ⊗kM : tr(x) = Nr(x) = 0}/M×. Here M is any k-algebra. The group D× acts on Y by conjugation. If K is a quadratic extension of k, then one can identify Y (K) = Hom(K, D).
To each embedding f : K → D we let y = yf be the image of the unique K-line on the quadric {x ∈ D ⊗kK : tr(x) = N (x) = 0} on which conjugation by f (K×) acts by the character a 7→ a/σ(a), σ is the non-trivial automorphism of K/k. Note that yf is one of the 2 fixed points of f (K×) acting on Y (K); the other is the image of the line where conjugation acts by the character a 7→ σ(a)/a.
Jing Yu Quaternion Algebras over function fields
Notations continued
kP : completion of k at a finite prime P . AP : closure of A in kP.
RP : = R ⊗AAP, KP := K ⊗kkP, and DP := D ⊗kkP. k :ˆ Q0
PkP, the finite adele ring of k.
R :ˆ =Q
P RP, K =ˆ Q0
P KP, and ˆD =Q0 PDP. For quadratic order Od⊂ K one has
Oˆ×d\ ˆK×/K×∼= Pic Od.
For left ideal classes of the maximal order R, one has bijection with double cosets in
Rˆ×\ ˆD×/D×.
Jing Yu Quaternion Algebras over function fields
Notations continued
kP : completion of k at a finite prime P . AP : closure of A in kP.
RP : = R ⊗AAP, KP := K ⊗kkP, and DP := D ⊗kkP. k :ˆ Q0
PkP, the finite adele ring of k.
R :ˆ =Q
P RP, K =ˆ Q0
P KP, and ˆD =Q0 PDP. For quadratic order Od⊂ K one has
Oˆ×d\ ˆK×/K×∼= Pic Od.
For left ideal classes of the maximal order R, one has bijection with double cosets in
Rˆ×\ ˆD×/D×.
Jing Yu Quaternion Algebras over function fields
Special points
Our definite Shimura curve XP0 is defined as
ˆR×\ ˆD×× Y /D×.
This is union of curves of genus 0, with components in bijection with the left ideal classes of R. Thus if there are n left ideal classes, Pic(XP0) ∼= Zn, generated by ei, i = 1, . . . , n, which are classes of degree 1 on each component of XP0.
The special points (Gross points) on XP0 over K are points in the image of ˆR×\ ˆD×× Y (K) in XP0(K). We say the point x = (g, y) has discriminant d if f (K) ∩ g−1Rg = f (Oˆ d), where f : K → D is the embedding corresponding to y. Note that here the discriminant of a special point is well defined up to
multiplication by elements in (F×q)2.
Jing Yu Quaternion Algebras over function fields
Special points
Our definite Shimura curve XP0 is defined as
ˆR×\ ˆD×× Y /D×.
This is union of curves of genus 0, with components in bijection with the left ideal classes of R. Thus if there are n left ideal classes, Pic(XP0) ∼= Zn, generated by ei, i = 1, . . . , n, which are classes of degree 1 on each component of XP0.
The special points (Gross points) on XP0 over K are points in the image of ˆR×\ ˆD×× Y (K) in XP0(K). We say the point x = (g, y) has discriminant d if f (K) ∩ g−1Rg = f (Oˆ d), where f : K → D is the embedding corresponding to y. Note that here the discriminant of a special point is well defined up to
multiplication by elements in (F×q)2.
Jing Yu Quaternion Algebras over function fields
Special points
Our definite Shimura curve XP0 is defined as
ˆR×\ ˆD×× Y /D×.
This is union of curves of genus 0, with components in bijection with the left ideal classes of R. Thus if there are n left ideal classes, Pic(XP0) ∼= Zn, generated by ei, i = 1, . . . , n, which are classes of degree 1 on each component of XP0.
The special points (Gross points) on XP0 over K are points in the image of ˆR×\ ˆD×× Y (K) in XP0(K). We say the point x = (g, y) has discriminant d if f (K) ∩ g−1Rg = f (Oˆ d), where f : K → D is the embedding corresponding to y. Note that here the discriminant of a special point is well defined up to
multiplication by elements in (F×q)2.
Jing Yu Quaternion Algebras over function fields
Hecke correspondences
Given P . Let T be the Bruhat-Tits tree of PGL2(kP). The vertices are the classes of AP-lattices in kP2, and two such vertices are adjacent if the “distance”between the lattice classes is 1.
The Hecke correspondence tP sends vertex v to the formal sum of its qdeg(P )+ 1 neighbors on the tree.
Identifying PGL2(AP)\ PGL2(kP) with vertices of the Bruhat-Tits tree, for gP ∈ PGL2(AP)\ PGL2(kP) one has:
tP(gP) = X
deg(u)≤deg(P )
1 u 0 P
!
gP + P 0 0 1
! gP.
When P 6= P0, one has R×P\DP×/kP×∼= PGL2(AP)\ PGL2(kP).
On the other hand R×P
0\DP×
0/kP×
0 has two elements, just let tP0
sends one element to the other.
Jing Yu Quaternion Algebras over function fields
Hecke correspondences
Given P . Let T be the Bruhat-Tits tree of PGL2(kP). The vertices are the classes of AP-lattices in kP2, and two such vertices are adjacent if the “distance”between the lattice classes is 1.
The Hecke correspondence tP sends vertex v to the formal sum of its qdeg(P )+ 1 neighbors on the tree.
Identifying PGL2(AP)\ PGL2(kP) with vertices of the Bruhat-Tits tree, for gP ∈ PGL2(AP)\ PGL2(kP) one has:
tP(gP) = X
deg(u)≤deg(P )
1 u 0 P
!
gP + P 0 0 1
! gP.
When P 6= P0, one has R×P\DP×/kP×∼= PGL2(AP)\ PGL2(kP).
On the other hand R×P
0\DP×
0/kP×
0 has two elements, just let tP0
sends one element to the other.
Jing Yu Quaternion Algebras over function fields
Hecke correspondences
Given P . Let T be the Bruhat-Tits tree of PGL2(kP). The vertices are the classes of AP-lattices in kP2, and two such vertices are adjacent if the “distance”between the lattice classes is 1.
The Hecke correspondence tP sends vertex v to the formal sum of its qdeg(P )+ 1 neighbors on the tree.
Identifying PGL2(AP)\ PGL2(kP) with vertices of the Bruhat-Tits tree, for gP ∈ PGL2(AP)\ PGL2(kP) one has:
tP(gP) = X
deg(u)≤deg(P )
1 u 0 P
!
gP + P 0 0 1
! gP.
When P 6= P0, one has R×P\DP×/kP×∼= PGL2(AP)\ PGL2(kP).
On the other hand R×P
0\DP×
0/kP×
0 has two elements, just let tP0
sends one element to the other.
Jing Yu Quaternion Algebras over function fields
Correspondence on Shimura curve
View XP0 as ( ˆR×\ ˆD×/ˆk×) × Y /(D×/k×). This leads to global Hecke correspondence tP on XP0 for all P .
As tP and tP0 are commute for any prime P and P0, one defines tm for every ideal (m) of A:
tmm0 = tmtm0, if m and m0 are relatively prime, tP` = tP`−1tP − qdeg PtP`−2, for P 6= P0,
t`P0 = t`P0.
Let T be the Z algebra generated by all tm, m ∈ A monic. Then T∼= B as Z-algebras. Passing to Pic(XP0), one shows that, for the basis ei, 1 ≤ i ≤ n :
tmei=
n
X
j=1
Bij(m)ej.
Jing Yu Quaternion Algebras over function fields
Correspondence on Shimura curve
View XP0 as ( ˆR×\ ˆD×/ˆk×) × Y /(D×/k×). This leads to global Hecke correspondence tP on XP0 for all P .
As tP and tP0 are commute for any prime P and P0, one defines tm for every ideal (m) of A:
tmm0 = tmtm0, if m and m0 are relatively prime, tP` = tP`−1tP − qdeg PtP`−2, for P 6= P0,
t`P0 = t`P0.
Let T be the Z algebra generated by all tm, m ∈ A monic. Then T∼= B as Z-algebras. Passing to Pic(XP0), one shows that, for the basis ei, 1 ≤ i ≤ n :
tmei=
n
X
j=1
Bij(m)ej.
Jing Yu Quaternion Algebras over function fields
Correspondence on Shimura curve
View XP0 as ( ˆR×\ ˆD×/ˆk×) × Y /(D×/k×). This leads to global Hecke correspondence tP on XP0 for all P .
As tP and tP0 are commute for any prime P and P0, one defines tm for every ideal (m) of A:
tmm0 = tmtm0, if m and m0 are relatively prime, tP` = tP`−1tP − qdeg PtP`−2, for P 6= P0,
t`P0 = t`P0.
Let T be the Z algebra generated by all tm, m ∈ A monic. Then T∼= B as Z-algebras. Passing to Pic(XP0), one shows that, for the basis ei, 1 ≤ i ≤ n :
tmei=
n
X
j=1
Bij(m)ej.
Jing Yu Quaternion Algebras over function fields
Gross pairing
Following B. Gross, we define a height pairing <, > on Pic(XP0) with values in Z by setting
< ei, ej >= 0, if i 6= j;
< ei, ei >= wi. This pairing gives an isomorphism of
Pic(XP0))∨ = Hom(Pic(XP0), Z) with the subgroup of Pic(XP0) ⊗ZQ with basis { ˇei= ei/wi : i = 1, ..., n}. Since wjBij(m) = wiBji(m) always hold, one has the following identity, for all classes e and e0 in Pic(XP0),
< tme, e0>=< e, tme0> .
Jing Yu Quaternion Algebras over function fields
Automorphic forms
Let O∞ be the valuation ring of k∞, with uniformizer π∞. We are interested in automorphic forms of level P0∞, i.e.
complex-valued functions on the double coset space Γ0(P0)\ GL2(k∞)/Γ∞k∞×,
where Γ0(P0) =
( a b c d
!
∈ GL2(A) : c ≡ 0 mod P0 )
,
and Γ∞= (
a b c d
!
∈ GL2(O∞) : c ∈ π∞O∞ )
.
From Brandt matrices we have constructed theta series θij. These theta series then give rise automorphic forms of Drinfeld type.
Jing Yu Quaternion Algebras over function fields
Automorphic forms
Let O∞ be the valuation ring of k∞, with uniformizer π∞. We are interested in automorphic forms of level P0∞, i.e.
complex-valued functions on the double coset space Γ0(P0)\ GL2(k∞)/Γ∞k∞×,
where Γ0(P0) =
( a b c d
!
∈ GL2(A) : c ≡ 0 mod P0 )
,
and Γ∞= (
a b c d
!
∈ GL2(O∞) : c ∈ π∞O∞ )
.
From Brandt matrices we have constructed theta series θij. These theta series then give rise automorphic forms of Drinfeld type.
Jing Yu Quaternion Algebras over function fields
Automorphic forms of Drinfeld type
An automorphic form f is of Drinfeld type if it satisfies the following harmonic properties: for any g ∈ GL2(k∞)
(1) f (g 0 1 π∞ 0
!
) = −f (g),
(2) X
κ∈GL2(O∞)/Γ∞
f (gκ) = 0.
All the functions Θij constructed from the quaternion algebra D are of Drinfeld type.
Let M (Γ(P0)) be the space of all automorphic forms of Drinfeld type of level P0∞. For each monic m ∈ A one also has Hecke operators Tm on the space M (Γ(P0)). This gives a commutative algebra of Hecke operators on automorphic forms of Drinfeld type.
This algebra is again isomorphic to the algebra of B.
Jing Yu Quaternion Algebras over function fields
Automorphic forms of Drinfeld type
An automorphic form f is of Drinfeld type if it satisfies the following harmonic properties: for any g ∈ GL2(k∞)
(1) f (g 0 1 π∞ 0
!
) = −f (g),
(2) X
κ∈GL2(O∞)/Γ∞
f (gκ) = 0.
All the functions Θij constructed from the quaternion algebra D are of Drinfeld type.
Let M (Γ(P0)) be the space of all automorphic forms of Drinfeld type of level P0∞. For each monic m ∈ A one also has Hecke operators Tm on the space M (Γ(P0)). This gives a commutative algebra of Hecke operators on automorphic forms of Drinfeld type.
This algebra is again isomorphic to the algebra of B.
Jing Yu Quaternion Algebras over function fields
Automorphic forms of Drinfeld type
An automorphic form f is of Drinfeld type if it satisfies the following harmonic properties: for any g ∈ GL2(k∞)
(1) f (g 0 1 π∞ 0
!
) = −f (g),
(2) X
κ∈GL2(O∞)/Γ∞
f (gκ) = 0.
All the functions Θij constructed from the quaternion algebra D are of Drinfeld type.
Let M (Γ(P0)) be the space of all automorphic forms of Drinfeld type of level P0∞. For each monic m ∈ A one also has Hecke operators Tm on the space M (Γ(P0)). This gives a commutative algebra of Hecke operators on automorphic forms of Drinfeld type.
This algebra is again isomorphic to the algebra of B.
Jing Yu Quaternion Algebras over function fields
A canonical pairing
Moreover we have for all 1 ≤ i, j ≤ n and any monic m the identity,
TmΘij =X
`
Bi`(m)Θ`j.
The multiplicity one theorem for automorphic forms then implies that the theta series Θ`j generate a subspace inside M (Γ(P0)) which is a free B ⊗ C-module of rank one.
We have a pairing :
φ : Pic(XP0) × Pic(XP0) −→ M (Γ(P0)),
φ(e, e0) πr∞ u
0 1
!
= q−r+2 deg e · deg e0+
X
m monic,deg m≤r−2
< e, tme0> X
(λ)=(m)
ψ∞(λu)
!
Jing Yu Quaternion Algebras over function fields
A canonical pairing
Moreover we have for all 1 ≤ i, j ≤ n and any monic m the identity,
TmΘij =X
`
Bi`(m)Θ`j.
The multiplicity one theorem for automorphic forms then implies that the theta series Θ`j generate a subspace inside M (Γ(P0)) which is a free B ⊗ C-module of rank one.
We have a pairing :
φ : Pic(XP0) × Pic(XP0) −→ M (Γ(P0)),
φ(e, e0) πr∞ u
0 1
!
= q−r+2 deg e · deg e0+
X
m monic,deg m≤r−2
< e, tme0> X
(λ)=(m)
ψ∞(λu)
!
Jing Yu Quaternion Algebras over function fields
A canonical pairing
Moreover we have for all 1 ≤ i, j ≤ n and any monic m the identity,
TmΘij =X
`
Bi`(m)Θ`j.
The multiplicity one theorem for automorphic forms then implies that the theta series Θ`j generate a subspace inside M (Γ(P0)) which is a free B ⊗ C-module of rank one.
We have a pairing :
φ : Pic(XP0) × Pic(XP0) −→ M (Γ(P0)),
φ(e, e0) πr∞ u
0 1
!
= q−r+2 deg e · deg e0+
X
m monic,deg m≤r−2
< e, tme0> X
(λ)=(m)
ψ∞(λu)
!
Jing Yu Quaternion Algebras over function fields
Isomorphism of Hecke modules
This pairing is equivariant w.r.t. the Hecke action: for all m ∈ A.
Tmφ(e, e0) = φ(tme, e0) = φ(e, tme0).
We claim that the theta series Θ`j actually generate M (Γ(P0)). It follows that our pairing induces an isomorphism of Hecke modules:
(Pic(XP0) ⊗ZC) ⊗TC (Pic(XP0) ⊗ZC)∼= M (Γ0(P0)).
The dimension of M (Γ(P0)) therefore equals to the number of left ideal classes of R. It also equals to g(Γ(P0)) + 1 (Gekeler), where g(Γ(P0)) is the genus of the Drinfeld modular curve X0(P0).
The claim is essentially Jacquet-Langlands correspondence over the function field k.
Jing Yu Quaternion Algebras over function fields
Isomorphism of Hecke modules
This pairing is equivariant w.r.t. the Hecke action: for all m ∈ A.
Tmφ(e, e0) = φ(tme, e0) = φ(e, tme0).
We claim that the theta series Θ`j actually generate M (Γ(P0)). It follows that our pairing induces an isomorphism of Hecke modules:
(Pic(XP0) ⊗ZC) ⊗TC (Pic(XP0) ⊗ZC)∼= M (Γ0(P0)).
The dimension of M (Γ(P0)) therefore equals to the number of left ideal classes of R. It also equals to g(Γ(P0)) + 1 (Gekeler), where g(Γ(P0)) is the genus of the Drinfeld modular curve X0(P0).
The claim is essentially Jacquet-Langlands correspondence over the function field k.
Jing Yu Quaternion Algebras over function fields
Isomorphism of Hecke modules
This pairing is equivariant w.r.t. the Hecke action: for all m ∈ A.
Tmφ(e, e0) = φ(tme, e0) = φ(e, tme0).
We claim that the theta series Θ`j actually generate M (Γ(P0)). It follows that our pairing induces an isomorphism of Hecke modules:
(Pic(XP0) ⊗ZC) ⊗TC (Pic(XP0) ⊗ZC)∼= M (Γ0(P0)).
The dimension of M (Γ(P0)) therefore equals to the number of left ideal classes of R. It also equals to g(Γ(P0)) + 1 (Gekeler), where g(Γ(P0)) is the genus of the Drinfeld modular curve X0(P0).
The claim is essentially Jacquet-Langlands correspondence over the function field k.
Jing Yu Quaternion Algebras over function fields
Isomorphism of Hecke modules
This pairing is equivariant w.r.t. the Hecke action: for all m ∈ A.
Tmφ(e, e0) = φ(tme, e0) = φ(e, tme0).
We claim that the theta series Θ`j actually generate M (Γ(P0)). It follows that our pairing induces an isomorphism of Hecke modules:
(Pic(XP0) ⊗ZC) ⊗TC (Pic(XP0) ⊗ZC)∼= M (Γ0(P0)).
The dimension of M (Γ(P0)) therefore equals to the number of left ideal classes of R. It also equals to g(Γ(P0)) + 1 (Gekeler), where g(Γ(P0)) is the genus of the Drinfeld modular curve X0(P0).
The claim is essentially Jacquet-Langlands correspondence over the function field k.
Jing Yu Quaternion Algebras over function fields