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A Lindemann-Weierstrass theorem for semi-abelian varieties over function fields. (English) Zbl 1276.12003

The classical Lindemann-Weierstrass Theorem on the algebraic independence of exponentials of algebraic numbers can be stated by saying that if \(G\) is an algebraic torus over the algebraic closure \(\mathbb Q^{\mathrm {alg}}\) of \(\mathbb Q\) and \(x\) a point in the Lie group \(LG\) of \(G\) which is defined over \(\mathbb Q^{\mathrm {alg}}\) and which does not belong to the Lie group of a proper algebraic subgroup of \(G\) over \(\mathbb Q^{\mathrm {alg}}\), then the transcendence degree of the field of definition of \(y=\exp_G(x)\) over \(\mathbb Q^{\mathrm {alg}}\) is equal to the dimension of \(G\). In the paper under review, the authors study the analogous problem where \(\mathbb Q^{\mathrm {alg}}\) is replaced by the algebraic closure \(K^{\mathrm {alg}}\) of the function field \(K=\mathbb C(S)\) of a smooth algebraic curve \(S\) over \(\mathbb C\) and where the torus is replaced by a commutative algebraic group over \(K^{\mathrm {alg}}\). A simple example of the general result is the following, which is deduced by considering the very special case where \(G\) is an abelian variety which is a power of an elliptic curve. Let \(\wp\) and \(\zeta\) be the associated Weierstrass elliptic function and zeta function respectively. Let \(x_1(z),\dots,x_n(z)\) be algebraic functions which are \(\mathbb Z\)-linearly independent. If the invariant \(j\in\mathbb C(z)\) is not constant, then the \(2n\) functions \(\wp\bigl(x_1(z)\bigr),\dots,\wp\bigl(x_n(z)\bigr)\), \(\zeta\bigl(x_1(z)\bigr),\dots,\zeta\bigl(x_n(z)\bigr)\) are algebraically independent over \(\mathbb C(z)\). If \(j\) is constant, the same conclusion holds only under the assumption that \(x_1(z),\dots,x_n(z)\) are linearly independent over the ring of multipliers of \(\wp\). Previous results in this direction were known only under the assumption that the abelian variety is constant, by means of a method of J. Ax [Am. J. Math. 94, 1195–1204 (1972; Zbl 0258.14014)].
The framework of this article involves differential fields. Let \(\partial\) be a non trivial derivation on \(K\) and let \(K^{\mathrm {diff}}\) be a differential closure of \(K\) in some universal differential extension \(\mathcal U\) of \(K\). Let \(G\) be a connected commutative algebraic group over \(K\) equipped with an extension to \(\mathcal O_G\) of the derivation \(\partial\) which respects the group structure of \(G\). Let \(\partial ln_G:G\rightarrow LG\) be the logarithmic derivative on \(G\) and \(\partial_{LG}: LG \rightarrow LG\) the canonical connection, contracted with \(\partial\), which \(\partial ln_G\) induces on \(LG\). The basic object of study of the paper under review is the differential relation \[ \partial ln_G(y)=\partial_{LG}(x) \] where \((x,y)\in (LG\times G)(\mathcal U)\). The authors compute the transcendence degree \(K(y)\) over \(K\) under the assumption that \(x\) is \(K\)-rational. The proof uses differential algebraic geometry (Ritt and Kolchin) and model theory, a variant of the “socle theorem” of E. Hrushovski [J. Am. Math. Soc. 9, No. 3, 667–690 (1996; Zbl 0864.03026)], the Manin-Coleman-Chai theorem of the kernel [C.-L. Chai, Am. J. Math. 113, No. 3, 387–389 (1991; Zbl 0759.14017)], the Griffiths-Schmid-Deligne theorem of the fixed part and P. Deligne’s semi-simplicity theorem [Publ. Math., Inst. Hautes Étud. Sci. 40, 5–57 (1971; Zbl 0219.14007); Publ. Math., Inst. Hautes Étud. Sci. 44, 5–77 (1974; Zbl 0237.14003)].
An extensive appendix on exponentials on algebraic \(D\)-groups clarifies a number of issues, including a discussion of the theorem of the kernel.

MSC:

12H05 Differential algebra
11J95 Results involving abelian varieties
14K05 Algebraic theory of abelian varieties
34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
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References:

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