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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

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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

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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

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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

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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

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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

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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

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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

a= 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

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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

a= 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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

(21)

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

(22)

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

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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

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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

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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

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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

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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

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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

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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).

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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

!

γα 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)/Γk×.

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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

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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.

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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

×d\ ˆK×/K×∼= Pic Od.

For left ideal classes of the maximal order R, one has bijection with double cosets in

×\ ˆD×/D×.

Jing Yu Quaternion Algebras over function fields

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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

×d\ ˆK×/K×∼= Pic Od.

For left ideal classes of the maximal order R, one has bijection with double cosets in

×\ ˆD×/D×.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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)/Γk×,

where Γ0(P0) =

( a b c d

!

∈ GL2(A) : c ≡ 0 mod P0 )

,

and Γ= (

a b c d

!

∈ GL2(O) : c ∈ πO )

.

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

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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)/Γk×,

where Γ0(P0) =

( a b c d

!

∈ GL2(A) : c ≡ 0 mod P0 )

,

and Γ= (

a b c d

!

∈ GL2(O) : c ∈ πO )

.

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

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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

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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

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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

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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)

!

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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

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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

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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

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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

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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

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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

參考文獻

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