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 2 N such that p + q + r < 1 and there is a presentation p q r ∆ = hδp; δq; δr j δp = δq = δr = δpδqδr = 1i: ∆ acts on the complex upper half-plane H = fx + iy 2 C j y > 0g by linear fractional transformations, a b aτ + b ; τ 2 Γ × 7! 2 : c d H cτ + d H If G is any finite index subgroup of ∆, then the orbit space X (G) = GnH 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 2 N such that p + q + r < 1 and there is a presentation p q r ∆ = hδp; δq; δr j δp = δq = δr = δpδqδr = 1i: ∆ acts on the complex upper half-plane H = fx + iy 2 C j y > 0g by linear fractional transformations, a b aτ + b ; τ 2 Γ × 7! 2 : c d H cτ + d H If G is any finite index subgroup of ∆, then the orbit space X (G) = GnH 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 = fx + iy 2 C j y > 0g by linear fractional transformations, a b aτ + b ; τ 2 Γ × 7! 2 : c d H cτ + d H If G is any finite index subgroup of ∆, then the orbit space X (G) = GnH 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 2 N such that p + q + r < 1 and there is a presentation p q r ∆ = hδp; δq; δr j δp = δq = δr = δpδqδr = 1i: Christopher Marks Period Relations If G is any finite index subgroup of ∆, then the orbit space X (G) = GnH 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 2 N such that p + q + r < 1 and there is a presentation p q r ∆ = hδp; δq; δr j δp = δq = δr = δpδqδr = 1i: ∆ acts on the complex upper half-plane H = fx + iy 2 C j y > 0g by linear fractional transformations, a b aτ + b ; τ 2 Γ × 7! 2 : 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 2 N such that p + q + r < 1 and there is a presentation p q r ∆ = hδp; δq; δr j δp = δq = δr = δpδqδr = 1i: ∆ acts on the complex upper half-plane H = fx + iy 2 C j y > 0g by linear fractional transformations, a b aτ + b ; τ 2 Γ × 7! 2 : c d H cτ + d H If G is any finite index subgroup of ∆, then the orbit space X (G) = GnH 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 j2γ where a b aτ + b f j (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 j2γ where a b aτ + b f j (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 j2γ where a b aτ + b f j (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 j2γ where a b aτ + b f j (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 j2γ where a b aτ + b f j (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 j2γ where a b aτ + b f j (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 2 G, 1 ≤ j ≤ 2g, such that H1(X (N); Z) is generated by the closed paths fτ0; γj τ0g. (Recall that γ 2 SL2(R) is hyperbolic iff jtr(γ)j > 2.) Let ff1 ··· ; fg g be a basis for M2(N). The periods of X (N) are generated by the complex numbers Z γj τ0 !jk = fk (τ) dτ; 1 ≤ j ≤ 2g; 1 ≤ k ≤ g: τ0 These numbers span a full rank lattice Λ in Cg (i.e.