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Periods Relations for Riemann Surfaces with Many Automorphisms 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.
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