Periods Relations for Riemann Surfaces with Many Automorphisms

Christopher Marks

California State University, Chico

West Coast Number Theory CSU, Chico December 17, 2018

Christopher Marks Period Relations Let ∆ = ∆(p, q, r) be a hyperbolic triangle group in PSL2(R). 1 1 1 In other words, p, q, r ∈ N such that p + q + r < 1 and there is a presentation

p q r ∆ = hδp, δq, δr | δp = δq = δr = δpδqδr = 1i.

∆ acts on the complex upper half-plane H = {x + iy ∈ C | y > 0} by linear fractional transformations,

a b aτ + b , τ ∈ Γ × 7→ ∈ . c d H cτ + d H

If G is any ﬁnite index subgroup of ∆, then the orbit space X (G) = G\H for this action carries the structure of a compact Riemann surface, called the automorphic curve associated to G.

Triangle groups and automorphic curves

Christopher Marks Period Relations 1 1 1 In other words, p, q, r ∈ N such that p + q + r < 1 and there is a presentation

p q r ∆ = hδp, δq, δr | δp = δq = δr = δpδqδr = 1i.

∆ acts on the complex upper half-plane H = {x + iy ∈ C | y > 0} by linear fractional transformations,

a b aτ + b , τ ∈ Γ × 7→ ∈ . c d H cτ + d H

If G is any ﬁnite index subgroup of ∆, then the orbit space X (G) = G\H for this action carries the structure of a compact Riemann surface, called the automorphic curve associated to G.

Triangle groups and automorphic curves

Let ∆ = ∆(p, q, r) be a hyperbolic triangle group in PSL2(R).

Christopher Marks Period Relations ∆ acts on the complex upper half-plane H = {x + iy ∈ C | y > 0} by linear fractional transformations,

a b aτ + b , τ ∈ Γ × 7→ ∈ . c d H cτ + d H

If G is any ﬁnite index subgroup of ∆, then the orbit space X (G) = G\H for this action carries the structure of a compact Riemann surface, called the automorphic curve associated to G.

Triangle groups and automorphic curves

Let ∆ = ∆(p, q, r) be a hyperbolic triangle group in PSL2(R). 1 1 1 In other words, p, q, r ∈ N such that p + q + r < 1 and there is a presentation

p q r ∆ = hδp, δq, δr | δp = δq = δr = δpδqδr = 1i.

Christopher Marks Period Relations If G is any ﬁnite index subgroup of ∆, then the orbit space X (G) = G\H for this action carries the structure of a compact Riemann surface, called the automorphic curve associated to G.

Triangle groups and automorphic curves

Let ∆ = ∆(p, q, r) be a hyperbolic triangle group in PSL2(R). 1 1 1 In other words, p, q, r ∈ N such that p + q + r < 1 and there is a presentation

p q r ∆ = hδp, δq, δr | δp = δq = δr = δpδqδr = 1i.

∆ acts on the complex upper half-plane H = {x + iy ∈ C | y > 0} by linear fractional transformations,

a b aτ + b , τ ∈ Γ × 7→ ∈ . c d H cτ + d H

Christopher Marks Period Relations Triangle groups and automorphic curves

Let ∆ = ∆(p, q, r) be a hyperbolic triangle group in PSL2(R). 1 1 1 In other words, p, q, r ∈ N such that p + q + r < 1 and there is a presentation

p q r ∆ = hδp, δq, δr | δp = δq = δr = δpδqδr = 1i.

∆ acts on the complex upper half-plane H = {x + iy ∈ C | y > 0} by linear fractional transformations,

a b aτ + b , τ ∈ Γ × 7→ ∈ . c d H cτ + d H

If G is any ﬁnite index subgroup of ∆, then the orbit space X (G) = G\H for this action carries the structure of a compact Riemann surface, called the automorphic curve associated to G.

Christopher Marks Period Relations Let N be a ﬁnite index, normal subgroup of ∆ = ∆(p, q, r), and assume that the genus of X (N) is g ≥ 1.

There is a covering X (N) → X (∆) of compact Riemann surfaces and, following Wolfart, we call X (N) a curve with many automorphisms.

∆/N is the automorphism group for X (N), and also acts linearly on the space Ω1(X (N)) of holomorphic 1-forms for X (N).

1 Identifying Ω (X (N)) with the space M2(N) of weight two automorphic forms for N, this action is given by f 7→ f |2γ where

a b aτ + b f | (τ) = f (cτ + d)−2. 2 c d cτ + d

This yields a g-dimensional complex representation ρN : ∆ → GL(M2(N)), called the canonical representation of the cover X (N) → X (∆).

Curves with many automorphisms and the canonical representation

Christopher Marks Period Relations There is a covering X (N) → X (∆) of compact Riemann surfaces and, following Wolfart, we call X (N) a curve with many automorphisms.

∆/N is the automorphism group for X (N), and also acts linearly on the space Ω1(X (N)) of holomorphic 1-forms for X (N).

1 Identifying Ω (X (N)) with the space M2(N) of weight two automorphic forms for N, this action is given by f 7→ f |2γ where

a b aτ + b f | (τ) = f (cτ + d)−2. 2 c d cτ + d

This yields a g-dimensional complex representation ρN : ∆ → GL(M2(N)), called the canonical representation of the cover X (N) → X (∆).

Curves with many automorphisms and the canonical representation

Let N be a ﬁnite index, normal subgroup of ∆ = ∆(p, q, r), and assume that the genus of X (N) is g ≥ 1.

Christopher Marks Period Relations ∆/N is the automorphism group for X (N), and also acts linearly on the space Ω1(X (N)) of holomorphic 1-forms for X (N).

1 Identifying Ω (X (N)) with the space M2(N) of weight two automorphic forms for N, this action is given by f 7→ f |2γ where

a b aτ + b f | (τ) = f (cτ + d)−2. 2 c d cτ + d

This yields a g-dimensional complex representation ρN : ∆ → GL(M2(N)), called the canonical representation of the cover X (N) → X (∆).

Curves with many automorphisms and the canonical representation

Let N be a ﬁnite index, normal subgroup of ∆ = ∆(p, q, r), and assume that the genus of X (N) is g ≥ 1.

There is a covering X (N) → X (∆) of compact Riemann surfaces and, following Wolfart, we call X (N) a curve with many automorphisms.

Christopher Marks Period Relations 1 Identifying Ω (X (N)) with the space M2(N) of weight two automorphic forms for N, this action is given by f 7→ f |2γ where

a b aτ + b f | (τ) = f (cτ + d)−2. 2 c d cτ + d

This yields a g-dimensional complex representation ρN : ∆ → GL(M2(N)), called the canonical representation of the cover X (N) → X (∆).

Curves with many automorphisms and the canonical representation

Let N be a ﬁnite index, normal subgroup of ∆ = ∆(p, q, r), and assume that the genus of X (N) is g ≥ 1.

There is a covering X (N) → X (∆) of compact Riemann surfaces and, following Wolfart, we call X (N) a curve with many automorphisms.

∆/N is the automorphism group for X (N), and also acts linearly on the space Ω1(X (N)) of holomorphic 1-forms for X (N).

Christopher Marks Period Relations This yields a g-dimensional complex representation ρN : ∆ → GL(M2(N)), called the canonical representation of the cover X (N) → X (∆).

Curves with many automorphisms and the canonical representation

Let N be a ﬁnite index, normal subgroup of ∆ = ∆(p, q, r), and assume that the genus of X (N) is g ≥ 1.

There is a covering X (N) → X (∆) of compact Riemann surfaces and, following Wolfart, we call X (N) a curve with many automorphisms.

∆/N is the automorphism group for X (N), and also acts linearly on the space Ω1(X (N)) of holomorphic 1-forms for X (N).

1 Identifying Ω (X (N)) with the space M2(N) of weight two automorphic forms for N, this action is given by f 7→ f |2γ where

a b aτ + b f | (τ) = f (cτ + d)−2. 2 c d cτ + d

Christopher Marks Period Relations Curves with many automorphisms and the canonical representation

Let N be a ﬁnite index, normal subgroup of ∆ = ∆(p, q, r), and assume that the genus of X (N) is g ≥ 1.

There is a covering X (N) → X (∆) of compact Riemann surfaces and, following Wolfart, we call X (N) a curve with many automorphisms.

∆/N is the automorphism group for X (N), and also acts linearly on the space Ω1(X (N)) of holomorphic 1-forms for X (N).

1 Identifying Ω (X (N)) with the space M2(N) of weight two automorphic forms for N, this action is given by f 7→ f |2γ where

a b aτ + b f | (τ) = f (cτ + d)−2. 2 c d cτ + d

This yields a g-dimensional complex representation ρN : ∆ → GL(M2(N)), called the canonical representation of the cover X (N) → X (∆).

Christopher Marks Period Relations The theory of modular symbols, due to Manin and (independently) Birch, implies that there are hyperbolic matrices γj ∈ G, 1 ≤ j ≤ 2g, such that H1(X (N), Z) is generated by the closed paths {τ0, γj τ0}.

(Recall that γ ∈ SL2(R) is hyperbolic iﬀ |tr(γ)| > 2.)

Let {f1 ··· , fg } be a basis for M2(N). The periods of X (N) are generated by the complex numbers

Z γj τ0 ωjk = fk (τ) dτ, 1 ≤ j ≤ 2g, 1 ≤ k ≤ g. τ0

These numbers span a full rank lattice Λ in Cg (i.e. a free Z-module of rank 2g), and Cg /Λ is an abelian variety called the Jacobian of X (N).

Periods of X (N)

Fix a base point τ0 ∈ H and consider the homology group H1(X (N), Z) relative to τ0.

Christopher Marks Period Relations (Recall that γ ∈ SL2(R) is hyperbolic iﬀ |tr(γ)| > 2.)

Let {f1 ··· , fg } be a basis for M2(N). The periods of X (N) are generated by the complex numbers

Z γj τ0 ωjk = fk (τ) dτ, 1 ≤ j ≤ 2g, 1 ≤ k ≤ g. τ0

These numbers span a full rank lattice Λ in Cg (i.e. a free Z-module of rank 2g), and Cg /Λ is an abelian variety called the Jacobian of X (N).

Periods of X (N)

Fix a base point τ0 ∈ H and consider the homology group H1(X (N), Z) relative to τ0. The theory of modular symbols, due to Manin and (independently) Birch, implies that there are hyperbolic matrices γj ∈ G, 1 ≤ j ≤ 2g, such that H1(X (N), Z) is generated by the closed paths {τ0, γj τ0}.

Christopher Marks Period Relations Let {f1 ··· , fg } be a basis for M2(N). The periods of X (N) are generated by the complex numbers

Z γj τ0 ωjk = fk (τ) dτ, 1 ≤ j ≤ 2g, 1 ≤ k ≤ g. τ0

These numbers span a full rank lattice Λ in Cg (i.e. a free Z-module of rank 2g), and Cg /Λ is an abelian variety called the Jacobian of X (N).

Periods of X (N)

(Recall that γ ∈ SL2(R) is hyperbolic iﬀ |tr(γ)| > 2.)

Christopher Marks Period Relations The periods of X (N) are generated by the complex numbers

Z γj τ0 ωjk = fk (τ) dτ, 1 ≤ j ≤ 2g, 1 ≤ k ≤ g. τ0

These numbers span a full rank lattice Λ in Cg (i.e. a free Z-module of rank 2g), and Cg /Λ is an abelian variety called the Jacobian of X (N).

Periods of X (N)

(Recall that γ ∈ SL2(R) is hyperbolic iﬀ |tr(γ)| > 2.)

Let {f1 ··· , fg } be a basis for M2(N).

Christopher Marks Period Relations These numbers span a full rank lattice Λ in Cg (i.e. a free Z-module of rank 2g), and Cg /Λ is an abelian variety called the Jacobian of X (N).

Periods of X (N)

(Recall that γ ∈ SL2(R) is hyperbolic iﬀ |tr(γ)| > 2.)

Let {f1 ··· , fg } be a basis for M2(N). The periods of X (N) are generated by the complex numbers

Z γj τ0 ωjk = fk (τ) dτ, 1 ≤ j ≤ 2g, 1 ≤ k ≤ g. τ0

Christopher Marks Period Relations Periods of X (N)

(Recall that γ ∈ SL2(R) is hyperbolic iﬀ |tr(γ)| > 2.)

Let {f1 ··· , fg } be a basis for M2(N). The periods of X (N) are generated by the complex numbers

Z γj τ0 ωjk = fk (τ) dτ, 1 ≤ j ≤ 2g, 1 ≤ k ≤ g. τ0

These numbers span a full rank lattice Λ in Cg (i.e. a free Z-module of rank 2g), and Cg /Λ is an abelian variety called the Jacobian of X (N).

Christopher Marks Period Relations 2 Trivially one sees that dim VN ≤ 2g . Q

Using the fact that X (N) → X (∆) is a Galois cover and ∆/N acts transitively on H1(X (N), ), Wolfart proved in 2002 that dim VN ≤ g. Z Q

This was, until very recently, the best known bound on dim VN . Q

Although not particularly strong, it already implies nontrivial arithmetic results.

Period relations for X (N)

Let VN = span {ωjk | 1 ≤ j ≤ 2g, 1 ≤ k ≤ g} be the -span of the Q Q periods of X (N), viewed as a subspace of C.

Christopher Marks Period Relations Using the fact that X (N) → X (∆) is a Galois cover and ∆/N acts transitively on H1(X (N), ), Wolfart proved in 2002 that dim VN ≤ g. Z Q

This was, until very recently, the best known bound on dim VN . Q

Although not particularly strong, it already implies nontrivial arithmetic results.

Period relations for X (N)

Let VN = span {ωjk | 1 ≤ j ≤ 2g, 1 ≤ k ≤ g} be the -span of the Q Q periods of X (N), viewed as a subspace of C.

2 Trivially one sees that dim VN ≤ 2g . Q

Christopher Marks Period Relations This was, until very recently, the best known bound on dim VN . Q

Although not particularly strong, it already implies nontrivial arithmetic results.

Period relations for X (N)

Let VN = span {ωjk | 1 ≤ j ≤ 2g, 1 ≤ k ≤ g} be the -span of the Q Q periods of X (N), viewed as a subspace of C.

2 Trivially one sees that dim VN ≤ 2g . Q

Using the fact that X (N) → X (∆) is a Galois cover and ∆/N acts transitively on H1(X (N), ), Wolfart proved in 2002 that dim VN ≤ g. Z Q

Christopher Marks Period Relations Although not particularly strong, it already implies nontrivial arithmetic results.

Period relations for X (N)

Let VN = span {ωjk | 1 ≤ j ≤ 2g, 1 ≤ k ≤ g} be the -span of the Q Q periods of X (N), viewed as a subspace of C.

2 Trivially one sees that dim VN ≤ 2g . Q

Using the fact that X (N) → X (∆) is a Galois cover and ∆/N acts transitively on H1(X (N), ), Wolfart proved in 2002 that dim VN ≤ g. Z Q

This was, until very recently, the best known bound on dim VN . Q

Christopher Marks Period Relations Period relations for X (N)

Let VN = span {ωjk | 1 ≤ j ≤ 2g, 1 ≤ k ≤ g} be the -span of the Q Q periods of X (N), viewed as a subspace of C.

2 Trivially one sees that dim VN ≤ 2g . Q

Using the fact that X (N) → X (∆) is a Galois cover and ∆/N acts transitively on H1(X (N), ), Wolfart proved in 2002 that dim VN ≤ g. Z Q

This was, until very recently, the best known bound on dim VN . Q

Although not particularly strong, it already implies nontrivial arithmetic results.

Christopher Marks Period Relations ω1 By Wolfart’s bound we have dim VN = g = 1, so ∈ . Q ω2 Q

In other words, X (N) is an elliptic curve with complex multiplication and, as is well-known the above period ratio lies in a quadratic imaginary extension of Q.

Example: Elliptic curves with many automorphisms

Suppose the genus of X (N) is 1, so that X (N) deﬁnes an elliptic curve C/Λ, where Λ = Zω1 ⊕ Zω2 denotes the period lattice of X (N).

Christopher Marks Period Relations In other words, X (N) is an elliptic curve with complex multiplication and, as is well-known the above period ratio lies in a quadratic imaginary extension of Q.

Example: Elliptic curves with many automorphisms

Suppose the genus of X (N) is 1, so that X (N) deﬁnes an elliptic curve C/Λ, where Λ = Zω1 ⊕ Zω2 denotes the period lattice of X (N).

ω1 By Wolfart’s bound we have dim VN = g = 1, so ∈ . Q ω2 Q

Christopher Marks Period Relations Example: Elliptic curves with many automorphisms

Suppose the genus of X (N) is 1, so that X (N) deﬁnes an elliptic curve C/Λ, where Λ = Zω1 ⊕ Zω2 denotes the period lattice of X (N).

ω1 By Wolfart’s bound we have dim VN = g = 1, so ∈ . Q ω2 Q

In other words, X (N) is an elliptic curve with complex multiplication and, as is well-known the above period ratio lies in a quadratic imaginary extension of Q.

Christopher Marks Period Relations 1 By studying the cohomology group H (∆, ρN ) for the canonical representation of the covering X (N) → X (∆), one may obtain the following result: Theorem For j = p, q, r let dj denote the dimension of the eigenspace of ρN (δj ) associated to the eigenvalue 1. Then

dim VN ≤ g − dp − dq − dr . Q

Note that, on average, this gives a bound of

g g g 1 1 1 g − − − = g 1 − + + g. p q r p q r

A new bound on dim V Q N Recent joint work with Luca Candelori (Wayne State University) and two Chico State undergraduates (Jack Fogliasso and Skip Moses) has yielded a much improved bound for the number of period relations over Q.

Christopher Marks Period Relations Theorem For j = p, q, r let dj denote the dimension of the eigenspace of ρN (δj ) associated to the eigenvalue 1. Then

dim VN ≤ g − dp − dq − dr . Q

Note that, on average, this gives a bound of

g g g 1 1 1 g − − − = g 1 − + + g. p q r p q r

1 By studying the cohomology group H (∆, ρN ) for the canonical representation of the covering X (N) → X (∆), one may obtain the following result:

Christopher Marks Period Relations Note that, on average, this gives a bound of

g g g 1 1 1 g − − − = g 1 − + + g. p q r p q r

1 By studying the cohomology group H (∆, ρN ) for the canonical representation of the covering X (N) → X (∆), one may obtain the following result: Theorem For j = p, q, r let dj denote the dimension of the eigenspace of ρN (δj ) associated to the eigenvalue 1. Then

dim VN ≤ g − dp − dq − dr . Q

Christopher Marks Period Relations A new bound on dim V Q N Recent joint work with Luca Candelori (Wayne State University) and two Chico State undergraduates (Jack Fogliasso and Skip Moses) has yielded a much improved bound for the number of period relations over Q.

dim VN ≤ g − dp − dq − dr . Q

Note that, on average, this gives a bound of

g g g 1 1 1 g − − − = g 1 − + + g. p q r p q r

Christopher Marks Period Relations The canonical representation ρN for the covering X (N) → X (∆) is irreducible, and this may be combined with a theorem of Shiga and Wolfart to show that the Jacobian of X (N) factors as E 2 for an elliptic curve E with complex multiplication.

A genus two example

The largest possible automorphism group for a genus two algebraic curve is GL(2,3), and this is realized by a normal subgroup N / ∆ = ∆(2, 3, 8) of index 48. X (N) is called a Bolza surface.

Christopher Marks Period Relations A genus two example

The largest possible automorphism group for a genus two algebraic curve is GL(2,3), and this is realized by a normal subgroup N / ∆ = ∆(2, 3, 8) of index 48. X (N) is called a Bolza surface.

The canonical representation ρN for the covering X (N) → X (∆) is irreducible, and this may be combined with a theorem of Shiga and Wolfart to show that the Jacobian of X (N) factors as E 2 for an elliptic curve E with complex multiplication.

Christopher Marks Period Relations For all examples computed by the authors, it was observed that in fact

1 dim VN = H (∆, ρN ). Q Might this be true for all curves with many automorphisms?

If so, then this would provide a powerful method for determining when Jacobians for curves with many automorphisms have complex multiplication.

Don’t call it a conjecture

The proof of the above bounding theorem shows that when X (N) has many automorphisms then

1 dim VN ≤ H (∆, ρN ). Q

Christopher Marks Period Relations Might this be true for all curves with many automorphisms?

If so, then this would provide a powerful method for determining when Jacobians for curves with many automorphisms have complex multiplication.

Don’t call it a conjecture

The proof of the above bounding theorem shows that when X (N) has many automorphisms then

1 dim VN ≤ H (∆, ρN ). Q For all examples computed by the authors, it was observed that in fact

1 dim VN = H (∆, ρN ). Q

Christopher Marks Period Relations If so, then this would provide a powerful method for determining when Jacobians for curves with many automorphisms have complex multiplication.

Don’t call it a conjecture

The proof of the above bounding theorem shows that when X (N) has many automorphisms then

1 dim VN ≤ H (∆, ρN ). Q For all examples computed by the authors, it was observed that in fact

1 dim VN = H (∆, ρN ). Q Might this be true for all curves with many automorphisms?

Christopher Marks Period Relations Don’t call it a conjecture

The proof of the above bounding theorem shows that when X (N) has many automorphisms then

1 dim VN ≤ H (∆, ρN ). Q For all examples computed by the authors, it was observed that in fact

1 dim VN = H (∆, ρN ). Q Might this be true for all curves with many automorphisms?

If so, then this would provide a powerful method for determining when Jacobians for curves with many automorphisms have complex multiplication.

Christopher Marks Period Relations Thanks very much!

Christopher Marks Period Relations