Endomorphism rings of elliptic curves over finite fields by David Kohel

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CHAPTER 3. COMPLEX MULTIPLICATION 26whereσ ∗ 1 (n) = {σ1 (n) if n ≢ 0 mod Nσ 1 (n/N) otherwiseis a modular form. The modular interpretation for E2(τ) ∗ stems from the followingformula [28]:NE ∗ 2 (τ) = −1 2N−1∑i=1℘(iτ/N; τ).Thus it gives the first symmetric function in the x-values of the points of a cyclicsubgroup of order N on E Λ (C), where x(P) and x(−P) are counted once and x(Q)is counted with multiplicity 1/2 for all Q in E Λ [2].Next we define η(τ) = q 1/24 ∏ (1−q n ), a 24-th root of ∆(τ). Then η(τ) is holomorphicon H and transforms as follows [30, Theorem I.8.3] under the generators for SL 2 (Z):η(−1/τ) = √ −iτ η(τ), andη(τ + 1) = e 2πi/24 η(τ),where √ is a branch of the square root which is positive on the positive real axis.While η(τ) is not a modular form we use η(τ) to construct modular forms. Forinstance, set( ) 2 η(13τ)u = 13 .η(τ)Then u is a modular form for Γ 0 (13) and the Atkin-Lehner operator acts in a particularlysimple fashion on u.( ) 2 η(τ)u| W13 = = 13η(13τ) u .Theta functionsTheta functions associated to positive definite quadratic forms over Z provide anabundant source of modular forms. This will be particularly useful when appliedwith the binary and quaternary quadratic forms associated to ideal classes of ordersin complex imaginary extensions of Q and of orders in quaternion algebras over Q.Let q : V → Q be a positive definite quadratic form of even dimension n = 2k overQ with integral lattice Λ of determinant det(Λ). Then we can form a holomorphicfunction on Hθ Λ (τ) = ∑ ω∈Λq q(ω) ,where q = e 2πiτ , and the transformation of θ Λ (τ) under elements of the modular groupSL 2 (Z) is well understood (see Chapter IX of Schoeneberg [26]). In the special casethat n = 4 and det(Λ) = N 2 then θ Λ (τ) is a modular form of weight 2 for Γ 0 (N).
CHAPTER 3. COMPLEX MULTIPLICATION 27Models for modular curvesWe can now make use of the above constructions for modular forms to producemodels for modular curves, in particular for X 0 (N). Classically one uses the functionsj = j(τ) and j N = j(Nτ) to construct the field of modular functions on X 0 (N). Bythe following theorem, this gives us all functions on X 0 (N).Theorem 16 The field of modular functions for Γ 0 (N) is C(j, j N ).The modular functions j and j N satisfy the classical modular equation Φ N (j, j N ) =0, where Φ N (X, Y ) ∈ Z[X, Y ]. While this gives an aesthetically pleasing relationbetween the j-invariant of a curve E and the j-invariants of the curves, Φ N (X, Y ) isa singular model for X 0 (N) and has many singularities over Spec(Z). As a result,the coefficients of Φ N (X, Y ) can be quite large. For instance for the first few valuesof N, we haveΦ 2 (X, Y ) = (X + Y ) 3 − X 2 Y 2 + 1485XY (X + Y ) − 162000(X + Y ) 2+ 41097375XY + 8748000000(X + Y ) − 157464000000000,Φ 3 (X, Y ) = (X + Y ) 4 − X 3 Y 3 + 2232X 2 Y 2 (X + Y ) + 36864000(X + Y ) 3− 1069960XY (X + Y ) 2 + 2590058000X 2 Y 2+ 8900112384000XY (X + Y ) + 452984832000000(X + Y ) 2− 771751936000000000XY + 1855425871872000000000(X + Y ),Φ 4 (X, Y ) = (X + Y ) 6 − X 4 Y 4 (X + Y ) + 1488X 4 Y 4 + 2976X 3 Y 3 (X + Y ) 2− 2533680X 2 Y 2 (X + Y ) 3 + 561444603XY (X + Y ) 4− 8507430000(X + Y ) 5 + 80975207520X 3 Y 3 (X + Y )− 120497741069824X 3 Y 3 + 1425218210971653X 2 Y 2 (X + Y ) 2+ 1194227286647130000XY (X + Y ) 3+ 24125474716854750000(X + Y ) 4− 917945232480970290000X 2 Y 2 (X + Y )+ 1362750357225997008000000X 2 Y 2+ 12519709864947556179750000XY (X + Y ) 2− 22805180351548032195000000000(X + Y ) 3+ 257072180519642551869287109375XY(X + Y )+ 158010236947953767724187500000000(X + Y ) 2− 410287056959130938575699218750000XY− 364936327796757658404375000000000000(X + Y )+ 280949374722195372109640625000000000000,and the modular polynomial of level 13 is
Page 1 and 2: Endomorphism rings of elliptic curv
Page 3: Endomorphism rings of elliptic curv
Page 6 and 7: ivContents1 Introduction 12 Ellipti
Page 8 and 9: viAcknowledgementsThis work would n
Page 11 and 12: CHAPTER 1. INTRODUCTION 3ring of E
Page 13 and 14: CHAPTER 2. ELLIPTIC CURVES AND ISOG
Page 27 and 28: 19Chapter 3Complex multiplication3.
Page 29 and 30: CHAPTER 3. COMPLEX MULTIPLICATION 2
Page 48 and 49: CHAPTER 4. THE ORDINARY CASE 40In p
Page 50 and 51: CHAPTER 4. THE ORDINARY CASE 42with
Page 52 and 53: CHAPTER 4. THE ORDINARY CASE 44of n
Page 54 and 55: CHAPTER 4. THE ORDINARY CASE 46Proo
Page 56 and 57: CHAPTER 4. THE ORDINARY CASE 48Inde
Page 58 and 59: CHAPTER 4. THE ORDINARY CASE 50by t
Page 60 and 61: CHAPTER 4. THE ORDINARY CASE 52Proo
Page 62 and 63: CHAPTER 4. THE ORDINARY CASE 54havi
Page 64 and 65: CHAPTER 4. THE ORDINARY CASE 56and
Page 66 and 67: 58Chapter 5Arithmetic of quaternion
Page 68 and 69: CHAPTER 5. ARITHMETIC OF QUATERNION
Page 78 and 79: 70Chapter 6Quadratic spacesThe equi
Page 80 and 81: CHAPTER 6. QUADRATIC SPACES 72mapΦ
Page 82 and 83: CHAPTER 6. QUADRATIC SPACES 74Propo
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CHAPTER 6. QUADRATIC SPACES 76that
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CHAPTER 6. QUADRATIC SPACES 78of th
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CHAPTER 6. QUADRATIC SPACES 82Proof
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CHAPTER 6. QUADRATIC SPACES 84which
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CHAPTER 6. QUADRATIC SPACES 86In th
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CHAPTER 7. SUPERSINGULAR ELLIPTIC C
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94Bibliography[1] A. O. L. Atkin an
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BIBLIOGRAPHY 96[32] J. Tate. Global
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Endomorphism rings of elliptic curves over finite fields by David Kohel

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