Formal groups and elliptic functions

Transcription

1 Steklov Mathematical Institute, Russian Academy of Sciences Magadan conference 7 December 2015

2 References V. M. Buchstaber,, Manifolds of solutions of Hirzebruch functional equations, Tr. Mat. Inst. Steklova, 290, (2015) V. M. Buchstaber,, The universal formal group that denes the elliptic function of level 3, Chebyshevskii Sb., 16:2 (2015) V. M. Buchstaber, A. V. Ustinov,, Rings of coecients for Tate formal groups, determining Krichever genera, Tr. Mat. Inst. Steklova, (2016)

3 Recall classical denitions An elliptic function is a meromorphic function in C that is doubly periodic: f (z + 2ω) = f (z), f (z + 2ω ) = f (z), Im ω ω > 0. Properties For any nonconstant elliptic functions f (z), g(z), there exists a polynomial P(x 1, x 2 ), such that P(f (z), g(z)) 0, any elliptic function h(z) is a rational function of f (z) and f (z). The Weierstrass function (z; g 2, g 3 ) is the unique elliptic function with periods 2ω, 2ω and poles only in lattice points such that lim It denes the uniformization of an elliptic curve V (z) 2 = 4 (z) 3 g 2 (z) g 3. ( ) (z) 1 z 0 z = 0. 2

4 Relations between parameters and periods The elliptic curve in standard Weierstrass form a V = {(x, y) y 2 = 4x 3 g 2 x g 3 }. The parameters ω, ω, η, η are determined by the relations dx 2ω = y, dx 2ω = y, 2η = xdx y, xdx 2η = y, where dx y b xdx and y are a set of holomorphic dierentials on V, a and b are basis cycles on the curve such that ηω ωη = πi 2. Vice versa, g 2 = (n,m) (0,0) 60 (2mω + 2nω ) 4, g 3 = a (n,m) (0,0) b 140 (2mω + 2nω ) 6.

5 Sigma function The Weierstrass ζ-function is dened by ζ(z; g 2, g 3 ) = (z; g 2, g 3 ), We have the relation η = ζ(ω; g 2, g 3 ). lim (zζ(z)) = 1. z 0 The Weierstrass σ-function is dened by ( ln σ(z; g2, g 3 ) ) ( ) σ(z) = ζ(z; g2, g 3 ), lim = 1. z 0 z It is an entire odd quasiperiodic function homogeneous with respect to the grading deg z =1, deg g k = 2k. We have σ(z; g 2, g 3 ) = z g 2z 5 2 5! 6g 3z 7 g 2 2z9 7! 4 8! 18g 2g 3 z 11 + (z 13 ) 11!

6 Elliptic function of level N For a lattice L in C consider the elliptic function g(x) with divisor N 0 N z. We demand that g(x) = x N Set f (x) = g(x) 1/N where f (x) = x Periodic properties f (x) = σ(x)σ(z) exp (αx ζ(z)x). σ(z x) f (x + 2ω k ) = f (x) exp (2η k z + 2ω k (α ζ(z))). For z = 2n N ω 1 + 2m N ω 2, α = 2n N η 1 2m N η 2 + ζ(z), we get elliptic functions of level N. The function f (x) is elliptic with respect to a sublattice L of L.

7 Let the bundle CP(ξ) B with ber CP(2) be the projectivization of a 3-dimensional complex vector bundle ξ B. A Hirzebruch genus L f : Ω U R is called CP(2)-multiplicative, if we have L f [CP(ξ)] = L f [CP(2)]L f [B]. Theorem (V.M. Buchstaber, E.Yu. Bunkova 2014) Let L f be a CP(2)-multiplicative genus. If L f [CP(2)] 0, then L f is the two-parametric Todd genus, and f (x) = eαx e βx αe αx βe βx, If L f [CP(2)] = 0, then L f is level 3 elliptic genus and 2 (x) + a2 2 f (x) =. (x) a (x) + b a3 4 Here g 2 = 1 4 (8b 3a3 )a, g 3 = 1 24 (8b2 12a 3 b + 3a 6 ).

8 2-nd Hirzebruch functional equation 1 f (x 1 x 2 )f (x 1 x 3 ) + 1 f (x 2 x 1 )f (x 2 x 3 ) + 1 f (x 3 x 1 )f (x 3 x 2 ) = C. Solutions are formal series f (x) such that f (0) = 0, f (0) = 1. If C 0, then f (x) = eαx e βx αe αx βe βx. If C = 0, then f (x) is elliptic function of level 3 and 2 (x) + a2 2 f (x) =. (x) a (x) + b a3 4 Here g 2 = 1 4 (8b 3a3 )a, g 3 = 1 24 (8b2 12a 3 b + 3a 6 ).

9 n-th Hirzebruch functional equation n+1 j=1 i j 1 f (x i x j ) = C Solutions are formal series f (x) such that f (0) = 0, f (0) = 1. The elliptic genus of level N is strictly multiplicative in bre bundles with a manifold M d with c 1 (M d ) 0 mod N as bre and a compact connected Lie group G of automorphisms of M d as structure group. (F. Hirzebruch comparing a paper by P.S. Landweber with a paper by S. Ochanine) => Elliptic function of level n + 1 is a (two-parametric) solution to n-th Hirzebruch equation (with C = 0).

10 Let the bundle CP(ξ) B with ber CP(2) be the projectivization of a 3-dimensional complex vector bundle ξ B. A Hirzebruch genus L f : Ω U R is called CP(2)-multiplicative, if we have L f [CP(ξ)] = L f [CP(2)]L f [B]. Theorem (V.M. Buchstaber, E.Yu. Bunkova 2014) Let L f be a CP(2)-multiplicative genus. If L f [CP(2)] 0, then L f is the two-parametric Todd genus, and f (x) = eαx e βx αe αx βe βx, If L f [CP(2)] = 0, then L f is level 3 elliptic genus and 2 (x) + a2 2 f (x) =. (x) a (x) + b a3 4 Here g 2 = 1 4 (8b 3a3 )a, g 3 = 1 24 (8b2 12a 3 b + 3a 6 ).

11 Application to isogenies of elliptic curves Theorem Let and J be the discriminant and J-invariant of an elliptic curve with generators (2ω 1, 2ω 2 ). Then the discriminant and J-invariant of elliptic curves with generators (2ˆω 1 = 2ω 1, 2ˆω 2 = 2 3 ω 2) and (2ˆω 1 = 2 3 ω 1, 2ˆω 2 = 2ω 2 ) can be expressed as follows: Let us introduce formal parameters p, s such that = 1 27 p3 s, J = 1 (p s)(p 9s) 3 64 p 3, s then ˆ = 27s 3 p, Ĵ = 1 (s p)(s 9p) 3 64 s 3. p

12 1-st Hirzebruch functional equation 1 f (x 1 x 2 ) + 1 f (x 2 x 1 ) = C. Elliptic function of level 2 is Jacobi's elliptic sine sn(x). sn (x) 2 = 1 2δsn(x) 2 + εsn(x) 4. sn(x) is the exponential of universal formal group of the form F (u, v) = Thus B(u) 2 = 1 2δu 2 + εu 4. u 2 v 2 ub(v) vb(u).

13 Formal group Let R be a commutative ring with unity 1. A commutative one-dimensional formal group over R is a formal series F (u, v) = u + v + a i,j u i v j, a i,j R, with conditions i,j>0 F (v, u) = F (u, v), F (u, F (v, w)) = F (F (u, v), w). It's exponential f (x) R Q[[x]], f (0) = 0, f (0) = 1, is determined by the addition law f (x + y) = F (f (x), f (y)).

14 From the addition law we get the relations F (u, v) v = w(u), v=0 F (f (x), f (y)) = f (x + y) 2 F (u, v) v 2 = w(u)(w (u) w 1 ), v=0 3 F (u, v) v 3 = w(u) 2 w (u) 2w(u)w 2 + v=0 + w(u)(w (u) w 1 )(w (u) 2w 1 ).

15 1-st Hirzebruch functional equation 1 f (x 1 x 2 ) + 1 f (x 2 x 1 ) = C. Elliptic function of level 2 is Jacobi's elliptic sine sn(x). sn (x) 2 = 1 2δsn(x) 2 + εsn(x) 4. sn(x) is the exponential of universal formal group of the form F (u, v) = Thus B(u) 2 = 1 2δu 2 + εu 4. u 2 v 2 ub(v) vb(u).

16 Theorem (V.M. Buchstaber, E.Yu. Bunkova 2015) The exponential of the universal formal group of the form is the elliptic function of level 3. F (u, v) = u2 C(v) v 2 C(u) uc(v) 2 vc(u) 2

17 Krichever genus F B (u, v) = u2 A(v) v 2 A(u), A(0) = B(0) = 1. (1) ub(v) vb(u) Theorem (V.M. Buchstaber, 1990) The exponential of the universal formal group of the form (1) is f (x) = exp(αx) Φ(x) = σ(x)σ(z) σ(z x) eαx ζ(z)x.

18 Two-parametric Todd genus, elliptic genera of level 2 and 3 u 2 A(v) v 2 A(u) ua(v) va(u), u 2 v 2 ub(v) vb(u), u 2 C(v) v 2 C(u) uc(v) 2 vc(u) 2, Z[µ 1, µ 2 ], Z[µ 2, µ 4 ], Z[µ 1, µ 3, µ 6 ]/{µ 2 3 = µ 6 }, e αx e βx αe αx βe βx, sn(x), σ(x)σ(z) exp (αx ζ(z)x), σ(z x) 2( (x) λ) (x) µ 1 (x) + µ 1 λ, 2( (x) δ 3 ), (x) 2( (x) + a2 4 ). (x) a (x) + b a3 4

19 Tate elliptic formal group General Weierstrass model of elliptic curve in Tate coordinates: s = u 3 + µ 1 us + µ 2 u 2 s + µ 3 s 2 + µ 4 us 2 + µ 6 s 3. F T (u, v) = ( (u+v)(1 µ 3 k µ 6 k 2 ) uv(µ 1 +µ 3 m+µ 4 k +2µ 6 mk) ) m = s(u) s(v), k = u v f (x) = 2 (1 + µ 2 m + µ 4 m 2 + µ 6 m 3 ) (1 + µ 2 n + µ 4 n 2 + µ 6 n 3 )(1 µ 3 k µ 6 k 2 ) 2 us(v) vs(u) u v, n = m+uv (1 + µ 2m + µ 4 m 2 + µ 6 m 3 ) (1 µ 3 k µ 6 k 2. ) F T (u, v) Z[µ 1, µ 2, µ 3, µ 4, µ 6 ][[u, v]]. (x) 1 12 (µ µ 2) (x) µ 1 (x) µ 1(µ µ. 2) µ 3

20 Two-parametric Todd genus, elliptic genera of level 2 and 3 u 2 A(v) v 2 A(u) ua(v) va(u), u 2 v 2 ub(v) vb(u), u 2 C(v) v 2 C(u) uc(v) 2 vc(u) 2, Z[µ 1, µ 2 ], Z[µ 2, µ 4 ], Z[µ 1, µ 3, µ 6 ]/{µ 2 3 = µ 6 }, e αx e βx αe αx βe βx, sn(x), σ(x)σ(z) exp (αx ζ(z)x), σ(z x) 2( (x) λ) (x) µ 1 (x) + µ 1 λ, 2( (x) δ 3 ), (x) 2( (x) + a2 4 ). (x) a (x) + b a3 4

21 Rings of coecients F B (u, v) = u2 A(v) v 2 A(u) ub(v) vb(u), Z[a 1, b 2, a 3, b 3,...]/J a F T (u, v) =..., Z[µ 1, µ 2, µ 3, µ 4, µ 6 ].

22 Tate and Buchstaber formal groups F 1 (u, v) = u + v µ 1uv, Z[µ 1, µ 2 ] 1 + µ 2 uv F 2 (u, v) = u 2 v 2 u (1 µ 2 v 2 ) 2 4µ 4 v 4 v (1 µ 2 u 2 ) 2 4µ 4 u 4, Z[µ 2, µ 4 ] F 3 (u, v) = u2 (1 µ 1 v µ 3 s(v)) v 2 (1 µ 1 u µ 3 s(u)) u(1 µ 1 v µ 3 s(v)) 2 v(1 µ 1 u µ 3 s(u)) 2, Z[µ 1, µ 3, µ 6 ]/{3µ 6 = µ 2 3} F 4 (u, v) = u 2 (1 + µ 6 s(v) 2 ) v 2 (1 + µ 6 s(u) 2 ) u(1 + µ 2 v 2 + µ 6 s(v) 2 ) v(1 + µ 2 u 2 + µ 6 s(u) 2 ), F 2[µ 2, µ 4, µ 6 ]

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