Endomorphism rings of elliptic curves over finite fields by David Kohel

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CHAPTER 3. COMPLEX MULTIPLICATION 30These three cases correspond to Aut(E) having 2, 4, and 6 elements, respectively.The weights of the numerators and denominators of the Weber functions above areeach 12, in the sense that the map (z; Λ) ↦→ (λz; λΛ) multiplies both numerator anddenominator by λ −12 . Thus up to some change of variable z ↦→ λz, the Weber functionis an invariant of the homothety class of Λ.Before stating the next results, we define ray class fields and ring class fields overK. For any integral ideal m of O = End(E) we define E[m] = {P ∈ E(C) : ϕ(P) =O for all ϕ ∈ m}. Let Λ be a lattice such that, as before, C/Λ ∼ = E(C). The torsionpoints E[m] corresponds to m −1 Λ/Λ ⊆ C/Λ. Thus in terms of our Weber functionson E and on C, we have h(E[m]) = h(m −1 Λ; Λ). For α ∈ K ∗ by α ≡ 1 mod*m wemean that v p (α −1) ≥ r for every prime power p r dividing m with positive exponent.We defineP 1 (m) = {(α) ∈ P(m) : α ≡ 1 mod*m} and,P Z (m) = {(α) ∈ P(m) : α/n ≡ 1 mod*m for some n ∈ Z}.Note that (α) ∈ P 1 (m) does not imply that α ≡ 1 mod*m, only that there exists aunit µ ∈ OK ∗ such that µα ≡ 1 mod*m. Also note that if m is the largest integer suchthat m is contained in (m), then P Z (mO K ) = P Z (m). Thus we assume m = (m) andwrite P Z (m) for P Z (mO K ).The ray class field modulo m, denoted K m , is defined to be the largest unramifiedabelian extension L of K such that the Artin map [ · , L/K] : I(m) → Gal(L/K)contains P 1 (m) in its kernel.The ring class field of conductor m is defined to be the largest abelian extension Lof K such that the Artin map [ · , L/K] : I(m) → Gal(L/K) contains P Z (m) in itskernel. To justify the nomenclature for the definition of the ring class field, we firstrecall that an order O in K has the form Z+mO K , for a unique positive integer m, theconductor of O. The ideal class group Cl(O) of projective ideals of O is isomorphicto I(m)/P Z (m). We call the ring class field of conductor m the ring class field for Oand denote it by K O .We can now state the main theorems of this section.Theorem 18 Let K/Q be a quadratic imaginary extension of Q, let m be an integralideal of O K , and let a be any fractional ideal for O K . Thenis the ray class field modulo m.K m = K(j(a), h(m −1 a; a))Proof. Silverman [30, Theorem II.5.6] or Lang [16, Chapter 10, §1, Theorem 2].
CHAPTER 3. COMPLEX MULTIPLICATION 31Theorem 19 Let K/Q be a quadratic imaginary extension of Q and let O be an orderof conductor m in K. Then j(O) is an algebraic integer which generates the ring classfield for O over K. The Galois conjugates for j(O) are j(a i ), where {a i } is a completeset of coset representatives for the projective ideal classes of O. The Artin map definesan isomorphism of Cl(O) with Gal(K O /K) such that [ pO K , K O /K](j(a)) = j(p −1 a),where p is a prime ideal of O not dividing m.Proof. Lang [16, Chapter 10, §3, Theorem 5]As an application we can now define the class polynomial H D (X). Let O be anorder of discriminant D in an imaginary quadratic extension of Q, and let {a i } be acomplete set of coset representatives of the h(O) projective ideal classes of O. Theabove theorem implies thatis an irreducible polynomial in Z[X].h(O)∏H D (X) = (X − j(a i ))For example, if we take D = −71, the class polynomial H −71 (X) isi=1X 7 + 313645809715X 6 − 3091990138604570X 5 + 98394038810047812049302X 4− 823534263439730779968091389X 3 + 5138800366453976780323726329446X 2− 425319473946139603274605151187659X + 11 9 · 17 6 · 23 3 · 41 3 · 47 3 · 53 3 .As with the modular equation, the coefficients grow rapidly with the size of thediscriminant. And as with the modular equations, one can try to deduce simplerexpressions for the class polynomial using different modular functions. For instance,Yui and Zagier [37] use special values of certain classical Weber functions to find areduced class equationW −71 (t) = t 7 − t 6 − t 5 + t 4 − t 3 − t 2 + 2t + 1,for the discriminant −71, where t and X satisfy the relation (t 24 − 16) 3 = t 24 X.The following commutative diagram of exact sequences summarizes the ideal class
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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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