Exponential and Weierstrass equations

Transcription

1 Exponential and Weierstrass equations Jonathan Kirby 8th April

2 Complex functions Exponentiation is a group homomorphism C exp C. For each elliptic curve E there is a -function which gives a group homomorphism C (, ) E C 2. The graphs are analytic subgroups of algebraic groups Γ exp C C Γ (, ) C E Fibres: right fibres are trivial, left fibres are cosets of the kernel. The kernels interpret arithmetic causes complications. 2

3 Differential equations d exp(x) dx = exp(x) d (x) dx = 4 (x) 3 g 2 (x) g 3 Look at these equations in a differential field F ; +,, D, C. Dy Dx = y Dy Dx = 4y 3 g 2 y g 3 which can be rewritten as Dx = Dy y Dx = Dy 4y 3 g 2 y g 3 3

4 Theorem 1. Group structure The solution sets form subgroups of the algebraic groups E F F W F E The fibres of the projections are cosets of the subgroups C F and E C E, so are strongly minimal and non-orthogonal to the constants. Proof. Exponentiation easy. Weierstrass equation a little work using differential Galois theory, see [1]. This gives a complete description of the relations E and W as groups. We want to relate these groups to the field structure. 4

5 Schanuel s conjecture Schanuel s conjecture for the complex exponential states: For any x 1,..., x n C, td Q (x 1, e x 1,..., x n, e xn ) ldim Q (x 1,..., x n ) 0 A conjecture in transcendental number theory wide open. There is a similar conjecture for -functions. 5

6 Schanuel conditions Theorem 2a (Ax, 1971) If (x i, y i ) E for i = 1,..., n and not all x i, y i C then td C (x 1, y 1,..., x n, y n ) ldim Q mod C (x 1,..., x n ) 1 Theorem 2b (In [1], using Brownawell & Kubota 1977) If (x i, y i ) W for i = 1,..., n and not all x i, y i C then td C (x 1, y 1,..., x n, y n ) ldim CM(E) mod C (x 1,..., x n ) 1 ldim K mod C X is the dimension of the K-vector space generated by C and X, quotiented by C. Using a compactness argument we can rewrite these statements in a uniform, first order way. 6

7 Uniform Schanuel condition Theorem 2b If (x 1, y 1,..., x n, y n ) W n V c for some algebraic variety V c of dimension n defined over parameters c C k then there is m CM(E) n {0} such that n i=1 m i x i C. Furthermore, this m may be chosen from a finite set H V independent of the parameters c and of the point ( x, ȳ). which is Proof. This uses the compactness theorem and the fact that the set of parameters c for which dim V c n is definable. 7

8 Universal theory of the equations Now consider reducts of algebraically closed differential fields in the language L = +,, W, C where W F E F 3. They satisfy the following theory T 0. ACF 0 C is an algebraically closed subfield, containing the parameters used to define the elliptic curve E. W is a subgroup of F E, whose fibres are cosets of C and E C (Theorem 1). (SC) The uniform, first order version of the Schanuel condition (Theorem 2). 8

9 Amalgamation This is a theory with a predimension function given by the Schanuel condition, so by the general argument of Hrushovski constructions, after checking some conditions, we get the following. Theorem 3. The following existentially closed condition (EC) is first order, and T W = T 0 + (EC) is a complete, consistent, ω-stable, first order theory. (EC) Any system of equations which is consistent with the group structure and the Schanuel condition has a solution. Exponentiation can be dealt with in the same way to get a theory T E. 9

10 Differential fields again Theorem 4. Let F be a differentially closed field and consider the reducts F ; +,, E, C and F ;, +,, W, C defined as before with E(x, y) Dx = Dy y W (x, (y, z)) Dx = Dy z. Then F ; +,, E, C = (EC) E and F ; +,, W, C = (EC) W these reducts have theories T E and T W respectively. and thus For exponentiation this result is due to Cecily Crampin. for Weierstrass equations is based on her method. The proof 10

11 Back to complex functions Think of E as Γ exp blurred by the constants. Claim 5. Let C C; +,, exp be a proper elementary substructure such that (*) Every analytic component of every set definable with parameters in C is realised in C. Define E(x, y) iff there is c C such that y = c exp(x). Then C; +,, E, C = T E. T 0 including (SC) proven, (EC) probable. Similarly for -functions. 11

12 Pseudo-exponentiation Boris Zilber has constructed K; +,, ex conjecturally isomorphic to complex exponentiation. We can take an elementary substructure C of this and define E as before. Claim 6. K; +,, E, C = T E. Again, (SC) holds but I have yet to check (EC). Assuming these claims, we have shown that, outside elementary substructures, pseudo-exponentiation and exponentiation agree. 12

13 Future directions Exponential and Weierstrass equations together Parametric Weierstrass equations More than one derivation Higher dimensional (semi-)abelian varieties Equations governed by non-commutative differential Galois groups [Speculation] Positive characteristic 13