Eta Products and Models for Modular Curves

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Cusps, as Families of Tate Curves Definition: X 0 (N) is the projective curve whose points over C p correspond to pairs (E, C) where E is a generalized elliptic curve and C ⊆ E is a cyclic subgroup of order N which meets every component of E. Every cusp is surrounded by a family of Tate curves (with level structure) which degenerate into the corresponding Néron n-gon. Definition: For d|N, the canonical families of width d on X 0 (N) are given by (C ∗ p/q d , 〈ζq〉), where ζ is any primitive r th root of unity such that lcm(r, d) = N.
Cusps, as Families of Tate Curves Definition: X 0 (N) is the projective curve whose points over C p correspond to pairs (E, C) where E is a generalized elliptic curve and C ⊆ E is a cyclic subgroup of order N which meets every component of E. Every cusp is surrounded by a family of Tate curves (with level structure) which degenerate into the corresponding Néron n-gon. Definition: For d|N, the canonical families of width d on X 0 (N) are given by (C ∗ p/q d , 〈ζq〉), where ζ is any primitive r th root of unity such that lcm(r, d) = N. Each family defines a rigid map from the disk |q| < ɛ into X 0 (N) which takes q = 0 to one of the cusps (z → z gcd(d,N/d) followed by an injection). Note: it is easy to check when two families represent the same cusp (there are φ(gcd(d, N/d)) different ones of width d).
Page 1 and 2: Eta Products and Models for Modular
Page 3: Cusps, as Families of Tate Curves D
Page 7 and 8: Lemma: (C ∗ p/q d , 〈ζq〉)
Page 9 and 10: The Tate Curve C ∗ p/q d with Som
Page 11 and 12: Definition of Modular Forms Definit
Page 13 and 14: Pullbacks by Level-Lowering Maps Su
Page 15 and 16: Pullbacks by Level-Lowering Maps Su
Page 17 and 18: Theorem 1: Suppose r, d, M, and l a
Page 19 and 20: Theorem 1: Suppose r, d, M, and l a
Page 21 and 22: Although we will only use the theor
Page 23 and 24: Ligozat’s Criteria Definition: Le
Page 25 and 26: q-expansions of ∆ pullbacks Theor
Page 27 and 28: Corollary 3.1: The leading term of
Page 29 and 30: Corollary 3.1: The leading term of
Page 31 and 32: Eta products on X 0 (18) Applying t
Page 33 and 34: Moreover, the values of f 24 at the
Page 39 and 40: What if there aren’t enough eta p
Page 41 and 42: A Nice Example Involving X 0 (11) D
Page 43 and 44: Remark: This reduces mod 11 to t 2
Page 45 and 46: Remark: This reduces mod 11 to t 2
Page 47 and 48: Conclusions: (1) It’s fairly stra
Page 49 and 50: Conclusions: (1) It’s fairly stra
Page 51 and 52: Eta Product Package Wish List Easy:
Page 53 and 54: Eta Product Package Wish List Easy:
Page 55 and 56:
Basis Calculation for X 0 (18) Ligo
Page 57 and 58:
Basis Calculation for X 0 (18) Ligo
Page 59 and 60:
Using eta products to find an equat
Page 61 and 62:
Using eta products to find an equat
Page 63 and 64:
Since u = 13 at the cusp 0, we let
Page 65:
Since u = 13 at the cusp 0, we let
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Eta Products and Models for Modular Curves

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