Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Algebraic curves of GL2-type
Jerome William HoffmanAtkin Memorial Conference
Louisiana State University
April 29-May 1, 2011
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
1 Galois representations of GL2-type
2 Explicit moduli for genus two curves
3 Humbert surfaces
4 Examples
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
For any number field K let GK = Gal(Q/K ). Let
ρ : GQ → GL(Vℓ) ∼= GL2d (Qℓ)
be a representation. Mostly we will be concerned with d = 2.Assume that
1 There is a number field E/Q of degree d and ahomomorphism E → End(Vℓ).
2 There is a finite extension K/Q such that ρ(GK ) commuteswith the image of E .
Then we have a factorization:
ρK := ρ | GK → GLE⊗Qℓ(Vℓ) ∼= GL2(E⊗Qℓ) ⊂ GL(Vℓ) ∼= GL2d (Qℓ).
We say that ρK has an E -linear structure. or ρ has anGL2-structure over K .
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
For any number field K let GK = Gal(Q/K ). Let
ρ : GQ → GL(Vℓ) ∼= GL2d (Qℓ)
be a representation. Mostly we will be concerned with d = 2.Assume that
1 There is a number field E/Q of degree d and ahomomorphism E → End(Vℓ).
2 There is a finite extension K/Q such that ρ(GK ) commuteswith the image of E .
Then we have a factorization:
ρK := ρ | GK → GLE⊗Qℓ(Vℓ) ∼= GL2(E⊗Qℓ) ⊂ GL(Vℓ) ∼= GL2d (Qℓ).
We say that ρK has an E -linear structure. or ρ has anGL2-structure over K .
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
For any number field K let GK = Gal(Q/K ). Let
ρ : GQ → GL(Vℓ) ∼= GL2d (Qℓ)
be a representation. Mostly we will be concerned with d = 2.Assume that
1 There is a number field E/Q of degree d and ahomomorphism E → End(Vℓ).
2 There is a finite extension K/Q such that ρ(GK ) commuteswith the image of E .
Then we have a factorization:
ρK := ρ | GK → GLE⊗Qℓ(Vℓ) ∼= GL2(E⊗Qℓ) ⊂ GL(Vℓ) ∼= GL2d (Qℓ).
We say that ρK has an E -linear structure. or ρ has anGL2-structure over K .
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example: Abelian varieties of GL2-type. For an abelian variety A/klet Vℓ(A) be its Tate module: a 2 dim A-dimenional representationof Gk .Theorem. (Ribet + Serre’s conjecture)
1 Let A/Q be a Q-simple abelian variety. Suppose thatE = Q⊗ EndQ(A) is a number field of degree = dim A. Thenthe Tate module Vℓ(A) defines a representation of Gal(Q/Q)with values in GL2(E ⊗Qℓ). Moreover A is isogenous to aQ-simple factor of J1(N) for some N ≥ 1.
2 Let C/Q be a elliptic curve. Then C is a quotient of J1(N)Qfor some N ≥ 1 if and only if C is a Q-curve, i.e., C isisogenous to each of its conjugates σC , σ ∈ Gal(Q/Q).
Problem: Explicitly construct Q-curves.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example: Abelian varieties of GL2-type. For an abelian variety A/klet Vℓ(A) be its Tate module: a 2 dim A-dimenional representationof Gk .Theorem. (Ribet + Serre’s conjecture)
1 Let A/Q be a Q-simple abelian variety. Suppose thatE = Q⊗ EndQ(A) is a number field of degree = dim A. Thenthe Tate module Vℓ(A) defines a representation of Gal(Q/Q)with values in GL2(E ⊗Qℓ). Moreover A is isogenous to aQ-simple factor of J1(N) for some N ≥ 1.
2 Let C/Q be a elliptic curve. Then C is a quotient of J1(N)Qfor some N ≥ 1 if and only if C is a Q-curve, i.e., C isisogenous to each of its conjugates σC , σ ∈ Gal(Q/Q).
Problem: Explicitly construct Q-curves.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example: Abelian varieties of GL2-type. For an abelian variety A/klet Vℓ(A) be its Tate module: a 2 dim A-dimenional representationof Gk .Theorem. (Ribet + Serre’s conjecture)
1 Let A/Q be a Q-simple abelian variety. Suppose thatE = Q⊗ EndQ(A) is a number field of degree = dim A. Thenthe Tate module Vℓ(A) defines a representation of Gal(Q/Q)with values in GL2(E ⊗Qℓ). Moreover A is isogenous to aQ-simple factor of J1(N) for some N ≥ 1.
2 Let C/Q be a elliptic curve. Then C is a quotient of J1(N)Qfor some N ≥ 1 if and only if C is a Q-curve, i.e., C isisogenous to each of its conjugates σC , σ ∈ Gal(Q/Q).
Problem: Explicitly construct Q-curves.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example: Abelian varieties of GL2-type. For an abelian variety A/klet Vℓ(A) be its Tate module: a 2 dim A-dimenional representationof Gk .Theorem. (Ribet + Serre’s conjecture)
1 Let A/Q be a Q-simple abelian variety. Suppose thatE = Q⊗ EndQ(A) is a number field of degree = dim A. Thenthe Tate module Vℓ(A) defines a representation of Gal(Q/Q)with values in GL2(E ⊗Qℓ). Moreover A is isogenous to aQ-simple factor of J1(N) for some N ≥ 1.
2 Let C/Q be a elliptic curve. Then C is a quotient of J1(N)Qfor some N ≥ 1 if and only if C is a Q-curve, i.e., C isisogenous to each of its conjugates σC , σ ∈ Gal(Q/Q).
Problem: Explicitly construct Q-curves.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Theorem. (Shimura) Let f =∑
anqn be a normalized cuspidal
Hecke eigenform of weight 2 on Γ1(N) for some N ≥ 1. Then thereis an abelian variety Af defined over Q with an action of the fieldE = Q(..., an , ...). This is a quotient of J1(N). We havedim Af = [E : Q], E = EndQ(Af )⊗Q, and thus Vℓ(Af ) is ofGL2-type over Q.Example. N = 29. There exist Hecke eigenformsf , f ∈ S2(29, (∗/29)) with field of coefficients E = Q(
√−5).
Shimura’s Af is two-dimensional, and in fact it is isogenous toC × σC for an elliptic Q-curve C/Q(
√5). Also we have an isogeny
Af∼= Jac(X ) where X is the genus 2 curve
y2 = −(x3 + 4x2 + x + 2)(8x3 − 4x2 − 9x − 14).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Theorem. (Shimura) Let f =∑
anqn be a normalized cuspidal
Hecke eigenform of weight 2 on Γ1(N) for some N ≥ 1. Then thereis an abelian variety Af defined over Q with an action of the fieldE = Q(..., an , ...). This is a quotient of J1(N). We havedim Af = [E : Q], E = EndQ(Af )⊗Q, and thus Vℓ(Af ) is ofGL2-type over Q.Example. N = 29. There exist Hecke eigenformsf , f ∈ S2(29, (∗/29)) with field of coefficients E = Q(
√−5).
Shimura’s Af is two-dimensional, and in fact it is isogenous toC × σC for an elliptic Q-curve C/Q(
√5). Also we have an isogeny
Af∼= Jac(X ) where X is the genus 2 curve
y2 = −(x3 + 4x2 + x + 2)(8x3 − 4x2 − 9x − 14).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Theorem. (Shimura) Let f =∑
anqn be a normalized cuspidal
Hecke eigenform of weight 2 on Γ1(N) for some N ≥ 1. Then thereis an abelian variety Af defined over Q with an action of the fieldE = Q(..., an , ...). This is a quotient of J1(N). We havedim Af = [E : Q], E = EndQ(Af )⊗Q, and thus Vℓ(Af ) is ofGL2-type over Q.Example. N = 29. There exist Hecke eigenformsf , f ∈ S2(29, (∗/29)) with field of coefficients E = Q(
√−5).
Shimura’s Af is two-dimensional, and in fact it is isogenous toC × σC for an elliptic Q-curve C/Q(
√5). Also we have an isogeny
Af∼= Jac(X ) where X is the genus 2 curve
y2 = −(x3 + 4x2 + x + 2)(8x3 − 4x2 − 9x − 14).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Note that for a given ρ, there can be more than one pair E ,K suchthat ρK has an E -linear structure.Examples are given by quaternion structures. Here d = 2. That is,assume:
1 there is a quaternion algebra B/Q and a homomorphismB → End(Vℓ) and
2 a finite extension K/Q such that the image of B commuteswith ρK .
Then ρK has an E -linear structure for every quadratic subfieldE ⊂ B .
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Note that for a given ρ, there can be more than one pair E ,K suchthat ρK has an E -linear structure.Examples are given by quaternion structures. Here d = 2. That is,assume:
1 there is a quaternion algebra B/Q and a homomorphismB → End(Vℓ) and
2 a finite extension K/Q such that the image of B commuteswith ρK .
Then ρK has an E -linear structure for every quadratic subfieldE ⊂ B .
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Note that for a given ρ, there can be more than one pair E ,K suchthat ρK has an E -linear structure.Examples are given by quaternion structures. Here d = 2. That is,assume:
1 there is a quaternion algebra B/Q and a homomorphismB → End(Vℓ) and
2 a finite extension K/Q such that the image of B commuteswith ρK .
Then ρK has an E -linear structure for every quadratic subfieldE ⊂ B .
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
From now on, we take d = 2Suppose ρ has an E -linear structure defined over K (so[E : Q] = 2).Let λ, λ be the places of E lying over the prime number ℓ (possiblyλ = λ ). Then we get a decomposition E ⊗Qℓ = ⊕
λ,λEλ hence
ρK ⊗ E = ρK ,λ ⊕ ρK ,λ, L(ρK , s) = L(ρK ,λ, s)L(ρK ,λ, s).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
From now on, we take d = 2Suppose ρ has an E -linear structure defined over K (so[E : Q] = 2).Let λ, λ be the places of E lying over the prime number ℓ (possiblyλ = λ ). Then we get a decomposition E ⊗Qℓ = ⊕
λ,λEλ hence
ρK ⊗ E = ρK ,λ ⊕ ρK ,λ, L(ρK , s) = L(ρK ,λ, s)L(ρK ,λ, s).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
From now on, we take d = 2Suppose ρ has an E -linear structure defined over K (so[E : Q] = 2).Let λ, λ be the places of E lying over the prime number ℓ (possiblyλ = λ ). Then we get a decomposition E ⊗Qℓ = ⊕
λ,λEλ hence
ρK ⊗ E = ρK ,λ ⊕ ρK ,λ, L(ρK , s) = L(ρK ,λ, s)L(ρK ,λ, s).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Special case: Suppose that:
1 K/Q is also a quadratic extension.
2 The generator s ∈ Gal(K/Q) induces an isomorphisms : ρK ,λ
∼= ρK ,λ.
Thenρ = Ind
QK (ρK ,λ)
thereforeL(ρ, s) = L(ρK ,λ, s).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Special case: Suppose that:
1 K/Q is also a quadratic extension.
2 The generator s ∈ Gal(K/Q) induces an isomorphisms : ρK ,λ
∼= ρK ,λ.
Thenρ = Ind
QK (ρK ,λ)
thereforeL(ρ, s) = L(ρK ,λ, s).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Let X be a projective nonsingular curve of genus 2 defined over Q.We assume an equation y2 = f (x) where f (x) ∈ Q[x] has degree 5or 6 with distinct roots.Then
Vℓ = H1(X ⊗Q,Qℓ) = H1(Jac(X )⊗Q,Qℓ)
gives a four dimensional representation of GQ. These are ofGL2-type if the Jacobian Jac(X ) has extra endomorphisms.For any abelian variety A defined over a field k we letEnd0
k(A) := Endk(A)⊗Q be the semisimple Q-algebra ofendomorphisms defined over k ; End0(A) = End0
k(A)
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Let X be a projective nonsingular curve of genus 2 defined over Q.We assume an equation y2 = f (x) where f (x) ∈ Q[x] has degree 5or 6 with distinct roots.Then
Vℓ = H1(X ⊗Q,Qℓ) = H1(Jac(X )⊗Q,Qℓ)
gives a four dimensional representation of GQ. These are ofGL2-type if the Jacobian Jac(X ) has extra endomorphisms.For any abelian variety A defined over a field k we letEnd0
k(A) := Endk(A)⊗Q be the semisimple Q-algebra ofendomorphisms defined over k ; End0(A) = End0
k(A)
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Let X be a projective nonsingular curve of genus 2 defined over Q.We assume an equation y2 = f (x) where f (x) ∈ Q[x] has degree 5or 6 with distinct roots.Then
Vℓ = H1(X ⊗Q,Qℓ) = H1(Jac(X )⊗Q,Qℓ)
gives a four dimensional representation of GQ. These are ofGL2-type if the Jacobian Jac(X ) has extra endomorphisms.For any abelian variety A defined over a field k we letEnd0
k(A) := Endk(A)⊗Q be the semisimple Q-algebra ofendomorphisms defined over k ; End0(A) = End0
k(A)
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Then if A is a two dimensional absolutely simple abelian varietydefined over a number field k then End0(A) is one of the following:
1 Q;
2 a real quadratic field E/Q;
3 an indefinite quaternion algebra B/Q;
4 a quartic CM field K/Q.
Items 2 (RM) and 3 (QM) lead to representations of GL2-type.How to write down genus two curves over Q such that Jac(X ) hasRM or QM?
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Then if A is a two dimensional absolutely simple abelian varietydefined over a number field k then End0(A) is one of the following:
1 Q;
2 a real quadratic field E/Q;
3 an indefinite quaternion algebra B/Q;
4 a quartic CM field K/Q.
Items 2 (RM) and 3 (QM) lead to representations of GL2-type.How to write down genus two curves over Q such that Jac(X ) hasRM or QM?
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Then if A is a two dimensional absolutely simple abelian varietydefined over a number field k then End0(A) is one of the following:
1 Q;
2 a real quadratic field E/Q;
3 an indefinite quaternion algebra B/Q;
4 a quartic CM field K/Q.
Items 2 (RM) and 3 (QM) lead to representations of GL2-type.How to write down genus two curves over Q such that Jac(X ) hasRM or QM?
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Then if A is a two dimensional absolutely simple abelian varietydefined over a number field k then End0(A) is one of the following:
1 Q;
2 a real quadratic field E/Q;
3 an indefinite quaternion algebra B/Q;
4 a quartic CM field K/Q.
Items 2 (RM) and 3 (QM) lead to representations of GL2-type.How to write down genus two curves over Q such that Jac(X ) hasRM or QM?
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Then if A is a two dimensional absolutely simple abelian varietydefined over a number field k then End0(A) is one of the following:
1 Q;
2 a real quadratic field E/Q;
3 an indefinite quaternion algebra B/Q;
4 a quartic CM field K/Q.
Items 2 (RM) and 3 (QM) lead to representations of GL2-type.How to write down genus two curves over Q such that Jac(X ) hasRM or QM?
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Then if A is a two dimensional absolutely simple abelian varietydefined over a number field k then End0(A) is one of the following:
1 Q;
2 a real quadratic field E/Q;
3 an indefinite quaternion algebra B/Q;
4 a quartic CM field K/Q.
Items 2 (RM) and 3 (QM) lead to representations of GL2-type.How to write down genus two curves over Q such that Jac(X ) hasRM or QM?
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
The familiar invariants of an elliptic curve, e.g., j , g2, g3 arise as in-and covariants of the action of PGL(2) on binary quartic formsf (x , y) = a0x
4 + a1x3y + a2x
2y2 + a3xy3 + a4y
4. These wereworked out in the 19th century.Reason: every genus 1 curve can be expressed as a double cover ofP1 with four branch points. Then f (x , y) = 0 give the coordinatesof the branch points. This leads to coordinate systems on themoduli spaces of elliptic curves: these are the modular curves.Riemann already knew that the moduli space Mg of genus g ≥ 2curves had dimension 3g − 3, but explicit models of these asalgebraic varieties are not easy to find.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
The familiar invariants of an elliptic curve, e.g., j , g2, g3 arise as in-and covariants of the action of PGL(2) on binary quartic formsf (x , y) = a0x
4 + a1x3y + a2x
2y2 + a3xy3 + a4y
4. These wereworked out in the 19th century.Reason: every genus 1 curve can be expressed as a double cover ofP1 with four branch points. Then f (x , y) = 0 give the coordinatesof the branch points. This leads to coordinate systems on themoduli spaces of elliptic curves: these are the modular curves.Riemann already knew that the moduli space Mg of genus g ≥ 2curves had dimension 3g − 3, but explicit models of these asalgebraic varieties are not easy to find.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
The familiar invariants of an elliptic curve, e.g., j , g2, g3 arise as in-and covariants of the action of PGL(2) on binary quartic formsf (x , y) = a0x
4 + a1x3y + a2x
2y2 + a3xy3 + a4y
4. These wereworked out in the 19th century.Reason: every genus 1 curve can be expressed as a double cover ofP1 with four branch points. Then f (x , y) = 0 give the coordinatesof the branch points. This leads to coordinate systems on themoduli spaces of elliptic curves: these are the modular curves.Riemann already knew that the moduli space Mg of genus g ≥ 2curves had dimension 3g − 3, but explicit models of these asalgebraic varieties are not easy to find.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Let X be a projective nonsingular curve of genus 2 defined over afield k . Then X has a model in the form
y2 = f (x) deg f = 6, with distinct roots.
As for elliptic curves, the moduli for genus 2 curves then can beexpressed via the in- and covariants of the action of PGL(2) onbinary sextic forms. The expression of the moduli of genus 2 curvesvia projective invariants of binary sextics was done by Clebsch, andin more modern times by Igusa and Mestre.Note that dimM2 = 3, so there are three “J-invariants” j1, j2, j3.Recall also that the map X 7→ Jac(X ): M2 → A2 to the modulispace of principally polarized abelian varieties of dimension 2 is abirational correspondence.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Let X be a projective nonsingular curve of genus 2 defined over afield k . Then X has a model in the form
y2 = f (x) deg f = 6, with distinct roots.
As for elliptic curves, the moduli for genus 2 curves then can beexpressed via the in- and covariants of the action of PGL(2) onbinary sextic forms. The expression of the moduli of genus 2 curvesvia projective invariants of binary sextics was done by Clebsch, andin more modern times by Igusa and Mestre.Note that dimM2 = 3, so there are three “J-invariants” j1, j2, j3.Recall also that the map X 7→ Jac(X ): M2 → A2 to the modulispace of principally polarized abelian varieties of dimension 2 is abirational correspondence.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Let X be a projective nonsingular curve of genus 2 defined over afield k . Then X has a model in the form
y2 = f (x) deg f = 6, with distinct roots.
As for elliptic curves, the moduli for genus 2 curves then can beexpressed via the in- and covariants of the action of PGL(2) onbinary sextic forms. The expression of the moduli of genus 2 curvesvia projective invariants of binary sextics was done by Clebsch, andin more modern times by Igusa and Mestre.Note that dimM2 = 3, so there are three “J-invariants” j1, j2, j3.Recall also that the map X 7→ Jac(X ): M2 → A2 to the modulispace of principally polarized abelian varieties of dimension 2 is abirational correspondence.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Let X be a projective nonsingular curve of genus 2 defined over afield k . Then X has a model in the form
y2 = f (x) deg f = 6, with distinct roots.
As for elliptic curves, the moduli for genus 2 curves then can beexpressed via the in- and covariants of the action of PGL(2) onbinary sextic forms. The expression of the moduli of genus 2 curvesvia projective invariants of binary sextics was done by Clebsch, andin more modern times by Igusa and Mestre.Note that dimM2 = 3, so there are three “J-invariants” j1, j2, j3.Recall also that the map X 7→ Jac(X ): M2 → A2 to the modulispace of principally polarized abelian varieties of dimension 2 is abirational correspondence.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Analytic moduli.Let
H2 = {τ =
(
τ1 τ2τ2 τ3
)
∈ M2(C) | Im(τ) > 0}
the Siegel space of genus 2.As an analytic space, Aan
2 = Γ\H2 where
Γ = {g =
(
A B
C D
)
∈ Sp4(Z)}
acting via τ 7→ (Aτ + B)(Cτ + D)−1.We get coverings by taking subgroups of finite index in Γ(congruence subgroups!)
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Analytic moduli.Let
H2 = {τ =
(
τ1 τ2τ2 τ3
)
∈ M2(C) | Im(τ) > 0}
the Siegel space of genus 2.As an analytic space, Aan
2 = Γ\H2 where
Γ = {g =
(
A B
C D
)
∈ Sp4(Z)}
acting via τ 7→ (Aτ + B)(Cτ + D)−1.We get coverings by taking subgroups of finite index in Γ(congruence subgroups!)
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Analytic moduli.Let
H2 = {τ =
(
τ1 τ2τ2 τ3
)
∈ M2(C) | Im(τ) > 0}
the Siegel space of genus 2.As an analytic space, Aan
2 = Γ\H2 where
Γ = {g =
(
A B
C D
)
∈ Sp4(Z)}
acting via τ 7→ (Aτ + B)(Cτ + D)−1.We get coverings by taking subgroups of finite index in Γ(congruence subgroups!)
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Algebraic moduli. A natural set of coordinates on the covering ofM2 given by level 2 structure is gotten by taking the cross ratios ofthe roots ei , i = 1, ..., 6 of
y2 = f (x) = (x − e1)...(x − e6).
level 2 structure = an ordering of the 6 roots e1, ..., e6:Sp4(Z/2) = S6.So we can describe subvarieties of M2 by equationsH(e1, ..., e6) = 0, etc. Cross-ratios in the ei can be expressed bycertain Siegel modular functions (ratios of thetanullwerte.)
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Algebraic moduli. A natural set of coordinates on the covering ofM2 given by level 2 structure is gotten by taking the cross ratios ofthe roots ei , i = 1, ..., 6 of
y2 = f (x) = (x − e1)...(x − e6).
level 2 structure = an ordering of the 6 roots e1, ..., e6:Sp4(Z/2) = S6.So we can describe subvarieties of M2 by equationsH(e1, ..., e6) = 0, etc. Cross-ratios in the ei can be expressed bycertain Siegel modular functions (ratios of thetanullwerte.)
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
We are interested in the subvariety H∆ ⊂ M2 of those genus 2curves X where Jac(X ) has endomorphisms by an order in a realquadratic field Q(
√∆). We have dim H∆ = 2 and these are called
Humbert surfaces.
1 When the integer ∆ is a square, H∆ is a product of modularcurves; Jac(X ) factors into 2 elliptic curves for X ∈ H∆.
2 When ∆ is a not a square, H∆ is a Hilbert modular surface forthe field Q(
√∆).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
We are interested in the subvariety H∆ ⊂ M2 of those genus 2curves X where Jac(X ) has endomorphisms by an order in a realquadratic field Q(
√∆). We have dim H∆ = 2 and these are called
Humbert surfaces.
1 When the integer ∆ is a square, H∆ is a product of modularcurves; Jac(X ) factors into 2 elliptic curves for X ∈ H∆.
2 When ∆ is a not a square, H∆ is a Hilbert modular surface forthe field Q(
√∆).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
We are interested in the subvariety H∆ ⊂ M2 of those genus 2curves X where Jac(X ) has endomorphisms by an order in a realquadratic field Q(
√∆). We have dim H∆ = 2 and these are called
Humbert surfaces.
1 When the integer ∆ is a square, H∆ is a product of modularcurves; Jac(X ) factors into 2 elliptic curves for X ∈ H∆.
2 When ∆ is a not a square, H∆ is a Hilbert modular surface forthe field Q(
√∆).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Explicit equations for Humbert surfaces were written down for∆ = 5, 8 by G. Humbert. Modern treatment given by P. Bending,Hashimoto, Hirzebruch, Murabayashii, R. M. Wilson, Sakai,Shephard-Barron, R. Taylor, van der Geer.Algorithms to compute equations of general Humbert surfaces havebeen developed by B. Runge, D, Gruenewald.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Explicit equations for Humbert surfaces were written down for∆ = 5, 8 by G. Humbert. Modern treatment given by P. Bending,Hashimoto, Hirzebruch, Murabayashii, R. M. Wilson, Sakai,Shephard-Barron, R. Taylor, van der Geer.Algorithms to compute equations of general Humbert surfaces havebeen developed by B. Runge, D, Gruenewald.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
The analytic equations of Humbert surfaces are quite simple: Eachpoint of H∆ is Sp4(Z)-equivalent to a point τ ∈ H2 which satisfies
aτ1 + bτ2 + τ3 = 0, a, b,∈ Z, b2 − 4a = ∆, b = 0 or 1.
We want equations in the algebraic moduli. Humbert’s constructionis based on Poncelet’s theorem.Given two projective plane conics C and D, if an n-gon can beinscribed in C in such a way that each edge of the polygon istangent to D (i.e., the n-gon is circumscribed about D) then, givenany point P ∈ C , there is an n-gon inscribed in C andcircumscribed about D which passes through P .
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
The analytic equations of Humbert surfaces are quite simple: Eachpoint of H∆ is Sp4(Z)-equivalent to a point τ ∈ H2 which satisfies
aτ1 + bτ2 + τ3 = 0, a, b,∈ Z, b2 − 4a = ∆, b = 0 or 1.
We want equations in the algebraic moduli. Humbert’s constructionis based on Poncelet’s theorem.Given two projective plane conics C and D, if an n-gon can beinscribed in C in such a way that each edge of the polygon istangent to D (i.e., the n-gon is circumscribed about D) then, givenany point P ∈ C , there is an n-gon inscribed in C andcircumscribed about D which passes through P .
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
The analytic equations of Humbert surfaces are quite simple: Eachpoint of H∆ is Sp4(Z)-equivalent to a point τ ∈ H2 which satisfies
aτ1 + bτ2 + τ3 = 0, a, b,∈ Z, b2 − 4a = ∆, b = 0 or 1.
We want equations in the algebraic moduli. Humbert’s constructionis based on Poncelet’s theorem.Given two projective plane conics C and D, if an n-gon can beinscribed in C in such a way that each edge of the polygon istangent to D (i.e., the n-gon is circumscribed about D) then, givenany point P ∈ C , there is an n-gon inscribed in C andcircumscribed about D which passes through P .
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
The analytic equations of Humbert surfaces are quite simple: Eachpoint of H∆ is Sp4(Z)-equivalent to a point τ ∈ H2 which satisfies
aτ1 + bτ2 + τ3 = 0, a, b,∈ Z, b2 − 4a = ∆, b = 0 or 1.
We want equations in the algebraic moduli. Humbert’s constructionis based on Poncelet’s theorem.Given two projective plane conics C and D, if an n-gon can beinscribed in C in such a way that each edge of the polygon istangent to D (i.e., the n-gon is circumscribed about D) then, givenany point P ∈ C , there is an n-gon inscribed in C andcircumscribed about D which passes through P .
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
The analytic equations of Humbert surfaces are quite simple: Eachpoint of H∆ is Sp4(Z)-equivalent to a point τ ∈ H2 which satisfies
aτ1 + bτ2 + τ3 = 0, a, b,∈ Z, b2 − 4a = ∆, b = 0 or 1.
We want equations in the algebraic moduli. Humbert’s constructionis based on Poncelet’s theorem.Given two projective plane conics C and D, if an n-gon can beinscribed in C in such a way that each edge of the polygon istangent to D (i.e., the n-gon is circumscribed about D) then, givenany point P ∈ C , there is an n-gon inscribed in C andcircumscribed about D which passes through P .
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Proof (Cayley): Consider the dual projective plane:(P2)∗ = the variety of lines in P2.Let D∗ ⊂ (P2)∗ be the variety of lines tangent to D. This is also aconic.Define
E (C ,D) := {(P , ℓ) ∈ C × D∗ | P ∈ ℓ},the incidence correspondence.Then E (C ,D) is an elliptic curve, and a Poncelet n-goncorresponds to a point of order n on E (C ,D).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Proof (Cayley): Consider the dual projective plane:(P2)∗ = the variety of lines in P2.Let D∗ ⊂ (P2)∗ be the variety of lines tangent to D. This is also aconic.Define
E (C ,D) := {(P , ℓ) ∈ C × D∗ | P ∈ ℓ},the incidence correspondence.Then E (C ,D) is an elliptic curve, and a Poncelet n-goncorresponds to a point of order n on E (C ,D).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Proof (Cayley): Consider the dual projective plane:(P2)∗ = the variety of lines in P2.Let D∗ ⊂ (P2)∗ be the variety of lines tangent to D. This is also aconic.Define
E (C ,D) := {(P , ℓ) ∈ C × D∗ | P ∈ ℓ},the incidence correspondence.Then E (C ,D) is an elliptic curve, and a Poncelet n-goncorresponds to a point of order n on E (C ,D).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Proof (Cayley): Consider the dual projective plane:(P2)∗ = the variety of lines in P2.Let D∗ ⊂ (P2)∗ be the variety of lines tangent to D. This is also aconic.Define
E (C ,D) := {(P , ℓ) ∈ C × D∗ | P ∈ ℓ},the incidence correspondence.Then E (C ,D) is an elliptic curve, and a Poncelet n-goncorresponds to a point of order n on E (C ,D).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Humbert, Mestre: Given a Poncelet n-gon, one constructs ahyperelliptic curve X together with an action of Q(ζm + ζ−1
m ) onJac(X ). X is a double cover of C branched over the vertices of then-gon.When n = 5, 8 this curve X has genus 2, with endomorphisms byQ(
√5) and Q(
√2) respectively.
∆ = 8: −2τ1 + 2τ3 = 0. This corresponds to a Poncelet 4-gon.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Humbert, Mestre: Given a Poncelet n-gon, one constructs ahyperelliptic curve X together with an action of Q(ζm + ζ−1
m ) onJac(X ). X is a double cover of C branched over the vertices of then-gon.When n = 5, 8 this curve X has genus 2, with endomorphisms byQ(
√5) and Q(
√2) respectively.
∆ = 8: −2τ1 + 2τ3 = 0. This corresponds to a Poncelet 4-gon.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Humbert, Mestre: Given a Poncelet n-gon, one constructs ahyperelliptic curve X together with an action of Q(ζm + ζ−1
m ) onJac(X ). X is a double cover of C branched over the vertices of then-gon.When n = 5, 8 this curve X has genus 2, with endomorphisms byQ(
√5) and Q(
√2) respectively.
∆ = 8: −2τ1 + 2τ3 = 0. This corresponds to a Poncelet 4-gon.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
α δ
p
q
D
C
βγ
Quadrilateral αβγδ is
Genus 2 curve X is double cover
of C branched over p, q, α, β, γ, δ.
Humbert 8 = Poncelet 4
inscribed on conic C, tangent to conic D
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
This configuration leads to the explicit equation for H8 = 0 interms of the roots ei of the sextic f (x) where the genus 2 curve isrepresented by y2 = f (x).
H8(e1, ..., e8) =
(e3 − e1) (e3 − e2) (e3 − e4) (e4 − e2)2(e3 − e5) (e6 − e1) (e6 − e2) (e6 − e4) (e6 − e5) (e1 − e5)
2
+ (e1 − e2) (e1 − e3) (e1 − e4) (e2 − e4)2(e3 − e5)
2(e6 − e2) (e6 − e3) (e6 − e4) (e6 − e5) (e1 − e5)
+ (e3 − e1)2(e4 − e1) (e4 − e2) (e4 − e3) (e2 − e5)
2(e4 − e5) (e6 − e1) (e6 − e2) (e6 − e3) (e6 − e5)
+ (e2 − e1) (e1 − e3)2(e2 − e3) (e2 − e4) (e2 − e5) (e4 − e5)
2(e6 − e1) (e6 − e3) (e6 − e4) (e6 − e5)
+ 16 (e1 − e2) (e2 − e3) (e1 − e4) (e3 − e4) (e2 − e5) (e3 − e5)
(e4 − e5) (e1 − e6) (e2 − e6) (e3 − e6) (e4 − e6) (e1 − e5)
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
∆ = 5: −τ1 + τ2 + τ3 = 0. This corresponds to a Poncelet 5-gon.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
C
D
α
β
γ
δ
ε
Humbert 5 = Poncelet 5
q
P
P’
P" Pentagon αβγδεinscribes conic C
circumscribes conic D
Genus 2 curve X is the
double cover of C branched
above α, β, γ, δ, ε and
a point q in C intersect D.
The correspomdence
lifts to a correspondence
P −> P’+P"
2φ + φ −1=0of X with φ
in Jac(X).
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
This configuration leads to the explicit equation for H5 = 0 interms of the roots ei of the sextic f (x) where the genus 2 curve isrepresented by y2 = f (x).
H5(e1, ..., e6) =
(e2 − e3)2(e5 − e1) (e5 − e2) (e5 − e3) (e5 − e4) (e1 − e6) (e2 − e6) (e3 − e6) (e4 − e6) (e1 − e4)
2
+ (e1 − e2) (e1 − e3) (e3 − e4)2(e1 − e5) (e2 − e5)
2(e2 − e6) (e3 − e6) (e4 − e6) (e5 − e6) (e1 − e4)
+ (e1 − e2)2(e4 − e1) (e4 − e2) (e4 − e3) (e4 − e5) (e5 − e3)
2(e1 − e6) (e2 − e6) (e3 − e6) (e5 − e6)
+ (e3 − e1) (e3 − e2) (e3 − e4) (e4 − e2)2(e3 − e5) (e5 − e1)
2(e1 − e6) (e2 − e6) (e4 − e6) (e5 − e6)
+ (e2 − e1) (e2 − e3) (e3 − e1)2(e2 − e4) (e2 − e5) (e4 − e5)
2(e1 − e6) (e3 − e6) (e4 − e6) (e5 − e6)
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Let B/Q be an indefinite quaternion algebra. Two-dimensionalprincipally polarized abelian varieties A with B ⊂ End(A) areparametrized by the points of the Shimura curve SB associated toB . There is a universal family X → SB of genus two curves whichhave Jacobians that have QM by B .Problem. Find explicit equations for these universal families, forsmall values of the discriminant of B .So far the only known examples are for discriminants 6, 10.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Let B/Q be an indefinite quaternion algebra. Two-dimensionalprincipally polarized abelian varieties A with B ⊂ End(A) areparametrized by the points of the Shimura curve SB associated toB . There is a universal family X → SB of genus two curves whichhave Jacobians that have QM by B .Problem. Find explicit equations for these universal families, forsmall values of the discriminant of B .So far the only known examples are for discriminants 6, 10.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Let B/Q be an indefinite quaternion algebra. Two-dimensionalprincipally polarized abelian varieties A with B ⊂ End(A) areparametrized by the points of the Shimura curve SB associated toB . There is a universal family X → SB of genus two curves whichhave Jacobians that have QM by B .Problem. Find explicit equations for these universal families, forsmall values of the discriminant of B .So far the only known examples are for discriminants 6, 10.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
The idea of Hashimoto and Murabayashii is to consider theintersection of Humbert surfaces.If a simple abelian surface A has End0(A) ⊃ Q(∆1),Q(∆2) for twodifferent real quadratic fields, then End0(A) is a quaternionalgebra. Thus
H∆1∩ H∆2
= union of Shimura curves.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
The idea of Hashimoto and Murabayashii is to consider theintersection of Humbert surfaces.If a simple abelian surface A has End0(A) ⊃ Q(∆1),Q(∆2) for twodifferent real quadratic fields, then End0(A) is a quaternionalgebra. Thus
H∆1∩ H∆2
= union of Shimura curves.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example 1 (Brumer/Hashimoto) Let
f (X ; a, b, c) = X 6 − (4 + 2b + 3c)X 5 + (2 + 2b + b2 − ac)X 4
− (6 + 4a + 6b − 2b2 + 5c + 2ac)X 3
+ (1 + b2 − ac)X 2 + (2 − 2b)X + (c + 1)
Let C (a, b, c) : Y 2 = f (X ; a, b, c), assume f (X , a, b, c) has 6distict roots. Then C (a, b, c) is a genus two curve with RM byQ(
√5). These endomorphisms are defined over Q(a, b, c). Hence if
a, b, c ,∈ Q, the curve is modular.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example 1 (Brumer/Hashimoto) Let
f (X ; a, b, c) = X 6 − (4 + 2b + 3c)X 5 + (2 + 2b + b2 − ac)X 4
− (6 + 4a + 6b − 2b2 + 5c + 2ac)X 3
+ (1 + b2 − ac)X 2 + (2 − 2b)X + (c + 1)
Let C (a, b, c) : Y 2 = f (X ; a, b, c), assume f (X , a, b, c) has 6distict roots. Then C (a, b, c) is a genus two curve with RM byQ(
√5). These endomorphisms are defined over Q(a, b, c). Hence if
a, b, c ,∈ Q, the curve is modular.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example 1 (Brumer/Hashimoto) Let
f (X ; a, b, c) = X 6 − (4 + 2b + 3c)X 5 + (2 + 2b + b2 − ac)X 4
− (6 + 4a + 6b − 2b2 + 5c + 2ac)X 3
+ (1 + b2 − ac)X 2 + (2 − 2b)X + (c + 1)
Let C (a, b, c) : Y 2 = f (X ; a, b, c), assume f (X , a, b, c) has 6distict roots. Then C (a, b, c) is a genus two curve with RM byQ(
√5). These endomorphisms are defined over Q(a, b, c). Hence if
a, b, c ,∈ Q, the curve is modular.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example 2 (P. Bending) Let K ⊂ C be a field. LetA,P 6= 0,Q,D 6= 0 in K . Define
B =Q(PA − Q) + 4P2 + 1
P2, C =
4(PA − Q)
P.
Define αi by∏2
i=0(X − αi ) = X 3 + AX 2 + BX + C . Then thegenus 2 curve
Y 2 = D
2∏
i=0
(X 2 − αiX + Pα2i + Qαi + 4P)
is defined over K and has RM by Q(√
2), also defined over K .Hence if K = Q, these curves are modular.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example 2 (P. Bending) Let K ⊂ C be a field. LetA,P 6= 0,Q,D 6= 0 in K . Define
B =Q(PA − Q) + 4P2 + 1
P2, C =
4(PA − Q)
P.
Define αi by∏2
i=0(X − αi ) = X 3 + AX 2 + BX + C . Then thegenus 2 curve
Y 2 = D
2∏
i=0
(X 2 − αiX + Pα2i + Qαi + 4P)
is defined over K and has RM by Q(√
2), also defined over K .Hence if K = Q, these curves are modular.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example 2 (P. Bending) Let K ⊂ C be a field. LetA,P 6= 0,Q,D 6= 0 in K . Define
B =Q(PA − Q) + 4P2 + 1
P2, C =
4(PA − Q)
P.
Define αi by∏2
i=0(X − αi ) = X 3 + AX 2 + BX + C . Then thegenus 2 curve
Y 2 = D
2∏
i=0
(X 2 − αiX + Pα2i + Qαi + 4P)
is defined over K and has RM by Q(√
2), also defined over K .Hence if K = Q, these curves are modular.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example 2 (P. Bending) Let K ⊂ C be a field. LetA,P 6= 0,Q,D 6= 0 in K . Define
B =Q(PA − Q) + 4P2 + 1
P2, C =
4(PA − Q)
P.
Define αi by∏2
i=0(X − αi ) = X 3 + AX 2 + BX + C . Then thegenus 2 curve
Y 2 = D
2∏
i=0
(X 2 − αiX + Pα2i + Qαi + 4P)
is defined over K and has RM by Q(√
2), also defined over K .Hence if K = Q, these curves are modular.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example 2 (P. Bending) Let K ⊂ C be a field. LetA,P 6= 0,Q,D 6= 0 in K . Define
B =Q(PA − Q) + 4P2 + 1
P2, C =
4(PA − Q)
P.
Define αi by∏2
i=0(X − αi ) = X 3 + AX 2 + BX + C . Then thegenus 2 curve
Y 2 = D
2∏
i=0
(X 2 − αiX + Pα2i + Qαi + 4P)
is defined over K and has RM by Q(√
2), also defined over K .Hence if K = Q, these curves are modular.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example 3 (Bending/Hashimoto/Murabayashii) Let K ⊂ C be afield. Let N,D in K ∗ with N2 6= −1.Let C be the curve
Y 2 = D[(N−1)X 6−6NX 5+3(N+1)X 4−8N2X 3+3(N−1)X 2+6N+N+1]
Then End0K(Jac(C )) contains the division quaternion algebra B6
of discriminant 6, namely(
2,−3Q
)
. The action of Q(√
2) ⊂ B6 is
defined over K and the action of Q(√−3) ⊂ B6 is defined over
K (√−3). Hence if N,D ∈ Q these are modular.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example 3 (Bending/Hashimoto/Murabayashii) Let K ⊂ C be afield. Let N,D in K ∗ with N2 6= −1.Let C be the curve
Y 2 = D[(N−1)X 6−6NX 5+3(N+1)X 4−8N2X 3+3(N−1)X 2+6N+N+1]
Then End0K(Jac(C )) contains the division quaternion algebra B6
of discriminant 6, namely(
2,−3Q
)
. The action of Q(√
2) ⊂ B6 is
defined over K and the action of Q(√−3) ⊂ B6 is defined over
K (√−3). Hence if N,D ∈ Q these are modular.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example 3 (Bending/Hashimoto/Murabayashii) Let K ⊂ C be afield. Let N,D in K ∗ with N2 6= −1.Let C be the curve
Y 2 = D[(N−1)X 6−6NX 5+3(N+1)X 4−8N2X 3+3(N−1)X 2+6N+N+1]
Then End0K(Jac(C )) contains the division quaternion algebra B6
of discriminant 6, namely(
2,−3Q
)
. The action of Q(√
2) ⊂ B6 is
defined over K and the action of Q(√−3) ⊂ B6 is defined over
K (√−3). Hence if N,D ∈ Q these are modular.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example 3 (Bending/Hashimoto/Murabayashii) Let K ⊂ C be afield. Let N,D in K ∗ with N2 6= −1.Let C be the curve
Y 2 = D[(N−1)X 6−6NX 5+3(N+1)X 4−8N2X 3+3(N−1)X 2+6N+N+1]
Then End0K(Jac(C )) contains the division quaternion algebra B6
of discriminant 6, namely(
2,−3Q
)
. The action of Q(√
2) ⊂ B6 is
defined over K and the action of Q(√−3) ⊂ B6 is defined over
K (√−3). Hence if N,D ∈ Q these are modular.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example 4 (QM by B10) Let
C : y2 = x6 − 16x5 + 40x4 + 140x3 + 80x2 − 64x + 64.
Then Jac(C ) has multiplications by the quaternion division ring ofdiscriminant 10. We have
det(X − ρ(Frobp)) = gp(X )gp(X )
for gp(X ) ∈ Q(√
5)[X ].
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
p gp(X )
3 X 2 +√
5X + 3
11 X 2 + 3X + 11
13 X 2 + 3√
5X + 13
17 X 2 +√
5X + 17
19 X 2 + 19
23 X 2 + 23
29 X 2 + 9X + 29
31 X 2 + 31
37 X 2 + 37
41 X 2 + 41
43 X 2 + 43
47 X 2 + 5√
5X + 47
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example 5 (RM by Q(√
2)) Let
C : y2 = 21x5 + 50x4 + 5x3 − 20x2 + 4x .
Then Jac(C ) has multiplications by Q(√
2), but theendomorphisms are defined over Q(
√2). The ℓ-adic representation
here is not obviously automorphic. The conductor is 2?3272. This isspecial case of a 1-parameter family of Hashimoto/Sakai.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
p det(X − ρ(Frobp))5 (X 2 − 5)2
11 (X 2 + 4√
2X + 11)(X 2 − 4√
2X + 11)13 X 4 + 6X 2 + 132
17 (X 2 + (4√
2 − 2)X + 17)(X 2 + (−4√
2 − 2)X + 17)19 X 4 − 10X 2 + 192
23 (X 2 + 4√
2X + 23)(X 2 − 4√
2X + 23)
29 (X 2 + 6√
2X + 29)(X 2 − 6√
2X + 29)31 (X 2 + 31)2
37 (X 2 + 10X + 37)(X 2 − 10X + 37)
41 (X 2 + (4√
2 − 2)X + 41)(X 2 + (−4√
2 − 2)X + 41)43 X 4 + 3X 2 + 432
47 (X 2 − 8X + 47)2
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Example 6 (RM by Q(√
2)) Let
C : y2 = 9x5 − 210x4 + 165x3 + 16740x2 − 74844x
Then Jac(C ) has multiplications by Q(√
2), but the endomorphismsare defined over Q(
√−2). The ℓ-adic representation here is not
obviously automorphic. The conductor is 2?3?7?11?. This is specialcase of a 1-parameter family of Hashimoto/Sakai.
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
p det(X − ρ(Frobp))5 (X 2 − 5)2
13 X 4 − 6X 2 + 132
17 (X 2 + (4√
2 − 2)X + 17)(X 2 + (−4√
2 − 2)X + 17)
19 (X 2 + 4√
2X + 19)(X 2 − 4√
2X + 19)
23 (X 2 + 4√
2X + 23)(X 2 − 4√
2X + 23)29 SX 4 − 10X 2 + 29
2
31 (X 2 + 4√
2X + 31)(X 2 − 4√
2X + 31)37 (X 2 + 6X + 37)(X 2 − 6X + 37)
41 (X 2 + (4√
2 + 6)X + 41)(X 2 + (−4√
2 + 6)X + 41)43 (X 2 + 8X + 43)2
47 (X 2 + 8X + 47)(X 2 − 8X + 47)
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
Galois representations of GL2-type Explicit moduli for genus two curves Humbert surfaces Examples
Thanks to Ramin Takloo-Bighash andWinnie Li!
Jerome William Hoffman Atkin Memorial Conference Louisiana State University,
Algebraic curves of GL2-type
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