Class invariants for quartic CM-fields

Transcription

1 Number Theory Seminar Oxford 2 June 2011

2 Elliptic curves An elliptic curve E/k (char(k) 2) is a smooth projective curve y 2 = x 3 + ax 2 + bx + c. Q P E is a commutative algebraic group P Q

3 Endomorphisms End(E) = (ring of algebraic group morphisms E E) (φ + ψ)(p) = φ(p) + ψ(p) (φψ)(p) = φ(ψ(p)) Examples: Z End(E) with n : P P + + P. For most E s in characteristic 0, have End(E) = Z. If E : y 2 = x 3 + x and i 2 = 1 in k, then we have i : (x, y) ( x, iy), and Z[i] End(E). If #k = q, we have Frob : (x, y) (x q, y q ).

4 The j-invariant is The Hilbert class polynomial j(e) = b b c 2 for E : y 2 = x 3 + bx + c. j(e) = j(f ) E = k F Definition Let K be an imaginary quadratic number field. Its Hilbert class polynomial is E/C End(E) =O K H K = ( X j(e) ) Z[X ]. Application 1: roots generate the Hilbert class field of K over K. Application 2: make elliptic curves with prescribed order over F p.

5 Curves with prescribed order If p = ππ for π O K, then (H K mod p) splits into linear factors. Let j 0 F p be a root and let E 0 /F p have j(e 0 ) = j 0. Then a twist E of E 0 has Frob = π. Reason: E0 is the reduction of some Ẽ with End(Ẽ) = O K CM theory = Frob OK and N(Frob) = p. We get #E(F p ) = N(π 1) = p + 1 tr(π).

6 Computing Hilbert class polynomials (1) Any E is complex analytically C/Λ for a lattice Λ Endomorphisms induce C-linear maps α : C C with α(λ) Λ If End(E) = O K, then Λ = ca for an ideal a O K and c C. We get Cl K {E/C : End(E) = O K } = [a] C/a.

7 Computing Hilbert class polynomials (2) Write a = τz + Z and let q = exp(2πiτ). Let θ 0 = n=1 q 1 2 n2 and θ 1 = n=1 ( 1)n q 1 2 n2. Then j(z) = 256 (θ8 0 θ4 0 θ4 1 + θ8 1 )3 θ 8 0 θ8 1 (θ4 0 θ4 1 )2 Compute H K = Other algorithms: [a] CL K (X j(c/a)) Z[X ]. p-adic, [Couveignes-Henocq 2002, Bröker 2006] Chinese remainder theorem. [Chao-Nakamura-Sobataka-Tsujii 1998, Agashe-Lauter-Venkatesan 2004]

8 Performance The Hilbert class polynomial is huge: the degree h K grows like D 1 2, as do the logarithms of the coefficients. Example: for K = Q( 101), get H K =X X X X \ X \ X \ X X X X X \ 320X \ X \ X \ Under GRH or heuristics, all three quasi-linear O( D 1+ɛ ).

9 Curves of genus 2 Definition A curve of genus 2 is a smooth geometrically irreducible curve of which the genus is 2. Definition (char. 2) A curve of genus 2 is a smooth projective curve that has an affine model y 2 = f (x), deg(f ) {5, 6}, where f has no double roots.

10 The group law on the Jacobian The Jacobian: group of equivalence classes of pairs of points. More precisely, divisor class group Pic 0 (C) {P 1, P 2 } [P 1 + P 2 D ] P1 P2 Q2 Q1 {P 1, P 2 } + {Q 1, Q 2 } =?

11 The group law on the Jacobian The Jacobian: group of equivalence classes of pairs of points. More precisely, divisor class group Pic 0 (C) {P 1, P 2 } [P 1 + P 2 D ] R2 P1 P2 Q2 Q1 S1 R1 S2 {P 1, P 2 } + {Q 1, Q 2 } = {S 1, S 2 }

12 Complex multiplication Elliptic curves E have CM if End(E) is an order in an imaginary quadratic field K = Q( r) with r Q negative. Curves C of genus 2 have CM if End(J(C)) is an order in a CM field K of degree 4, i.e. K = K 0 ( r) with K 0 real quadratic and r K 0 totally negative. Assume K contains no imaginary quadratic field.

13 Rosenhain form Over algebraically closed fields, we can write any elliptic curve in Legendre form E : y 2 = x(x 1)(x λ). and any curve of genus 2 in Rosenhain form C : y 2 = x(x 1)(x λ 1 )(x λ 2 )(x λ 3 ). The moduli space of elliptic curves is one-dimensional, that of curves of genus 2 is three-dimensional.

14 Igusa invariants Igusa gave a genus-2 analogue of the j-invariant (i.e., a model for the moduli space) Mestre s algorithm (available in Magma) constructs an equation for the curve from its invariants. This may need an extension of degree 2 over the field of moduli. The Igusa class polynomials of a quartic CM field K are a set of polynomials of which the roots are the Igusa invariants of curves C of genus 2 with CM by O K.

15 Applications Roots generate class fields. not of K, but of its reflex field (no problem) not the full Hilbert class field (but we know which field) useful? efficient? (work in progress!) If p = ππ in O K, construct curve C with #J(C)(F p ) = N(π 1) and #C(F p ) = p + 1 tr(π).

16 Algorithms 1. Complex analytic [Spallek 1994, Van Wamelen 1999] 2. p-adic [Gaudry-Houtmann-Kohel-Ritzenthaler-Weng 2002, Carls-Kohel-Lubicz 2008] 3. Chinese remainder theorem [Eisenträger-Lauter 2005] None of these had running time bounds: denominators (now bounded by Goren and Lauter) not known how to bound i n (C). algorithms not explicit enough no rounding error analysis for alg. 1 (not even for genus 1!!)

17 Complex CM surfaces K R = C 2 as R-algebra let Φ be an isomorphism and a K an O K -ideal Then C 2 /Φ(a) has CM by O K (also need a polarization ξ, which we ll ignore) Can enumerate triples (Φ, a, ξ) up to isomorphism. By choosing the right kind of basis, get C 2 /(τz 2 + Z) with τ Mat 2 (C) symmetric with positive definite imaginary part.

18 Theta constants Thomae s formula [1870] gives a model for C, given τ. For c 1, c 2 Q g, let θ[c 1, c 2 ](τ) = v Z g exp(πi(v + c 1 )τ(v + c 1 ) t + 2πi(v + c 1 )c t 2). For c 1 = 1 2 (a, b), c 2 = 1 2 (c, d), write θ a+2b+4c+8d = θ[c 1, c 2 ]. Let λ 1 = θ2 0 θ2 1 θ 2 2 θ2 3, λ 2 = θ2 1 θ2 12 θ 2 2 θ2 15, λ 3 = θ2 0 θ2 12 θ3 2. θ2 15 Then the polarized ab. surface corresponding to τ is Jac(C) with C : y 2 = x(x 1)(x λ 1 )(x λ 2 )(x λ 3 ).

19 Igusa invariants in terms of theta Write out, get [Bolza 1887, Igusa 1967] i k (τ) = pol. in θ s θ 4, e.g. i 2 (τ) = ( θ 8 ) 5 θ 4, sums and products taken over θ[c 1, c 2 ] with c i {0, 1 2 }2 and 2c t 1 c 2 Z. Evaluate Igusa class polynomials numerically. Bound absolute values by bounding entries of τ.

20 Thesis result Theorem Algorithm computes the Igusa class polynomials of K in time less than cst.(d 7/2 1 D 11/2 0 ) 1+ɛ, where D 0 = disc K 0 and D 1 D0 2 = disc K. The bit size of the output is between Bottlenecks: cst.(d 1/2 1 D 1/2 0 ) 1 ɛ and cst.(d 2 1D 3 0) 1+ɛ. 1. quasi-quadratic time theta evaluation (quasi-linear method not proven [Dupont 2006]) 2. denominator bounds not optimal (special cases/conjectures [Bruinier-Yang 2006, Yang (preprint)])

21 A cube root of j with K = Q( 101) H K =X X X X \ X \ X \ X X X X X \ 320X \ X \ X \ But 3 j H K, with minimal polynomial x x x x x x x x x x x x x x

22 The Hilbert class field The Hilbert class field H K of K is the largest unramified abelian extension of K. CM fact: HK = K[X ]/H K There is a natural isomorphism Cl K Gal(H K /K). CM fact: [b] j(c/a) = j(c/b 1 a). A class invariant is a value f (τ) H K of a modular function f in a point τ K. Example: j(ω) with O K = Z + τz Example: on previous slide, also j(ω) 1/3 Weber (around 1900) studied smaller class invariants.

23 For A = ( a c Modular functions b d ) GL 2(Q), let Aτ = aτ+b cτ+d. A modular form of weight k and level n is a holomorphic map f : H C satisfying f (Aτ) = (cτ + d) k f (τ) for all A Γ(n) = ker(sl 2 (Z) SL 2 (Z/nZ)), and a convergence condition at the cusps. Example: linear combination of products of 2k theta s Has a Fourier expansion f (τ) = k=0 a kq k/n with q = e 2πiτ. Let F n = { f g : f, g of weight k, level n, coefficients in Q(ζ n)} SL 2 (Z) acts on F n by f A (τ) := f (Aτ), induces action of SL 2 (Z/nZ).

24 Modular functions Let F n = { f g : f, g of weight k, level n, coefficients in Q(ζ n)} SL 2 (Z) acts on F n by f A (τ) := f (Aτ), induces action of SL 2 (Z/nZ). (Z/nZ) = Gal(Q(ζ n )/Q) acts on F n by acting on coeffs. 0 v ). Let (Z/nZ) GL 2 (Z/nZ) via v ( 1 0 Fact: F 1 = Q(j), Gal(F n /F 1 ) = GL 2 (Z/nZ)/{±1}

25 Shimura s reciprocity law Let O K = ωz + Z. The values f (ω) for f F n generate the ray class field H n K of K of conductor n. Gal(H n K /H K ) = (O K /no K ). Shimura s reciprocity law gives a map g = g ω : (O K /no K ) GL 2 (Z/nZ) such that f (ω) x = f g(x) (ω). g(x) t is the matrix of multiplication by x with respect to the basis ω, 1 of O K.

26 Shimura s reciprocity law Gal(H n K /H K ) = (O K /no K ). Shimura s reciprocity law gives a map g = g ω : (O K /no K ) GL 2 (Z/nZ) such that f (ω) x = f g(x) (ω). g(x) t is the matrix of multiplication by x wrt ω, 1. If f is fixed under g ω ((O K /no K ) ), then f (ω) H K, that is, f (ω) is a class invariant Any example gives a whole family of number fields that are examples! A more general version of g gives the orbit of f (ω) under Gal(H K /K).

27 Shimura reciprocity for dimension > 1 Shimura s reciprocity law for dimension > 1 can be made as explicit as on the previous slide (just more complicated). The action of Sp 2g (Z) on the θ s is also explicit. Example: λ 1 = θ2 0 θ2 1 in K Example: θ 2 2 θ2 3 is a class invariant if 2 splits completely e 5πi/4 θ 2θ 3 θ 4 θ 8 θ 9 θ 15 is a class invariant for a certain τ for K = Q(α) with α α = 0.

28 Example The Hilbert class field of K = Q(α) with α α = 0 is given by But also by 10201X 7 + ( w )X 6 +( w )X ( w )X 4 +( w )X ( w )X 2 +( w )X ( w ) w = 13 = 1 4 (α2 + 27) Q(w)[X ] 32724y 7 + ( 2565α α α ) y 6 + ( α α α ) y 5 + ( 86247α α α ) y 4 + ( α α α ) y 3 + ( α α α ) y 2 + ( α α α ) y α α α