Equations for Modular Curves - Magma

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Definition Modularity Finiteness Results Computationality C /Q is a new modular curve of level N if Then Departure point Generalization Newness J1(N) J(C) J1(N) new X1(N) π C π ∗ H 0 (C,Ω 1 new dq ) ↩→ S2(N) q Enrique González Jiménez Equations for Modular Curves
Modularity Finiteness Results Computationality Departure point Generalization Newness Let C be a new modular curve of level N then ∃f1,...,fk ∈ NewN := {normalized newforms on S2(N)} such that Then J1(N) new Q ∼ ∏ Af Q J(C) ∼ f∈NewN/GQ π ∗ (H 0 (C,Ω 1 )) = k i=1 dq S2(Afi ) q = k i=1 Enrique González Jiménez Equations for Modular Curves k ∏Afi i=1 , 〈 σ dq fi(q) | σ ∈ Gal(Efi /Q)〉 q
Page 1 and 2: Modularity Finiteness Results Compu
Page 5 and 6: Theorem Modularity Finiteness Resul
Page 7 and 8: Theorem Modularity Finiteness Resul
Page 9 and 10: Generalization Modularity Finitenes
Page 17 and 18: Definition Modularity Finiteness Re
Page 27: Definition Modularity Finiteness Re
Page 45 and 46: Fact Modularity Finiteness Results
Page 47 and 48: Fact Modularity Finiteness Results
Page 49 and 50: Notation Modularity Finiteness Resu
Page 51 and 52: g = 0 M C 0 = M C new 0 #M C 0 = #M
Page 61 and 62: g = 2 Theorem #M C new 2 < ∞ Modu
Page 63 and 64: Theorem #M C new 2 < ∞ Moreover,
Page 65 and 66: Theorem #M C new 2 < ∞ Moreover,
Page 67 and 68: Theorem #M C new 2 = 213 Moreover,
Page 73 and 74: C : y 2 = F(x) Modularity Finitenes
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C : y 2 = F(x) Modularity Finitenes
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g ≥ 2 Theorem Modularity Finitene
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Notation Modularity Finiteness Resu
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Conjecture 1 Modularity Finiteness
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Hyperelliptic Curves Definition Mod
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Algorithm Modularity Finiteness Res
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Algorithm (cont.) Modularity Finite
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Modularity Finiteness Results Compu
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C A 21A {0,2} C : y 2 = F(x) Modula
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C A 21A {0,2} C : y 2 = F(x) Modula
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C A 21A {0,2} C : y 2 = F(x) Modula
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C A 21A {0,2} C : y 2 = F(x) Modula
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C A 21A {0,2} C : y 2 = F(x) Modula
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Conjecture #M C new (2) = 300 Modul
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Example Modularity Finiteness Resul
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C X1(17) : ⎧ ⎨ ⎩ Modularity F
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Final Remark k a field, char(k) = 0
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Equations for Modular Curves - Magma

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