Eta Products and Models for Modular Curves

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Lemma: (C ∗ p/q d , 〈ζq〉) ∼ (C ∗ p/q d 1 , 〈ζq 1〉) if and only if q gcd(d,N/d) = q gcd(d,N/d) 1 . Pf/ Let λ be a primitive N th root of unity. So ζ = λ jN/r for some j with (j, r) = 1. Then the two elliptic curves are isomorphic if and only if q 1 = (λ) kN/d q for some k. First suppose 〈λ jN/r q〉 = 〈λ jN/r λ kN/d q〉. Then jN/r = (jN/r + kN/d)(1 + ld) + mN 0 = kN/d + ldjN/r + ldkN/d + mN. Thus, we see that d ∣ ∣ (kN/d). So d gcd(d,N/d) ∣ k. Therefore q gcd(d,N/d) 1 = λ kN gcd(d,N/d)/d q d = q d . The other direction is similar.
The Tate Curve C ∗ p/q d with Some of its N-torsion C p !. 3 ! q. . 2 . q . 2 . q d . q 3 ! . 1
Page 1 and 2: Eta Products and Models for Modular
Page 3 and 4: Cusps, as Families of Tate Curves D
Page 5 and 6: Cusps, as Families of Tate Curves D
Page 7: Lemma: (C ∗ p/q d , 〈ζq〉)
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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