Théories de Galois différentielles et transcendance (Q962063)

Let \(S\) be a smooth algebraic curve over \(\mathbb C\), \(K=\mathbb C(S)\) its function field, and \(\bar{K}\) the algebraic closure of \(K\). Let \(\partial\) be a derivation of \(K\) with field of constants \(\mathbb C\) and let \(\hat{K}\) be the differential closure of \(K\). Let \(\mathcal G\) be a group scheme over \(S\) whose generic fibre \(G\) is a connected commutative algebraic group defined over \(K\), and let \(L\mathcal G\) and \(LG\) be their Lie algebras. In the complex analytic category, there is an exponential map \( \text{exp}_G: L\mathcal G \to \mathcal G\). Let \(x\) be a local analytic section of \(L\mathcal G\) and let \(y=\text{exp}_G(x)\). The functional Schanuel problem is to bound the transcendence degree of \(K(x,y)\) over \(K\). In this exposition, the author provides such a result under some special cases of \(\mathcal G\), \(x\), and \(y\), in particular where \(\mathcal G\) has a differential structure. The proofs of the results have both a differential Galois and an algebraic geometry part. The differential structure in question is an extension of the derivation \(\partial\) of \(K\) to a derivation \(D_\partial\) of the structure sheaf of \(G\), which respects the group structure of \(G\). In this case, there also exists a logarithmic derivative \( \partial \ell n_G: G \to LG\) and its derivative at the origin \(\partial_{LG}: LG \to LG\). The groups considered are the following: let \(A\) be an abelian variety over \(K\) and \(\tilde{A}\) its universal extension by a vector group (the latter being the dual of \(H^1(A, \mathcal O_A)\)). There is a unique differential structure on such a group \(\tilde{A}\). The groups \(G\) are then of the form \(T \times \tilde{A}\), with \(T\) a torus. The results are as follows: Theorem (L): Suppose the element \(y \in G(K)\) does not lie in \(H+G_0(\mathbb C)\) for any proper \(K\) algebraic subgroup of \(G\) (\(G_0 = T \times \tilde{A}_0\), where \(\tilde{A}_0\) is the \(\bar{K}/\mathbb C\) trace of \(\tilde{A}\).) Let \(x\) in \(LG(\hat{K})\) be a solution of the equation \(\partial\ell n_G(x)=\partial_{LG}(y)\). Then the transcendence degree of \(K(x)\) over \(K\) is the dimension of \(G\). Theorem (E) Let \(G\) and \(G_0\) be as above, and suppose the element \(x\) of \(LG(K)\) does not lie in \(LH+LG_0(\mathbb C)\) for any proper \(K\) algebraic subgroup of \(G\). Let \(y\) in \(G(\hat{K})\) be a solution of the equation \(\partial\ell n_G(y)=\partial_{LG}(x)\). Then the transcendence degree of \(K(y)\) over \(K\) is the dimension of \(G\). The part of the proof that uses Galois theory considers the solutions \(y\) in \(G(\hat{K})\) of the equations \(\partial\ell n_G(y)=\beta\), where \(\beta \in LG(K)\) for case (E) and the solutions \(x\) in \(LG(\hat{K})\) of the equations \(\partial\ell n_G(y)=\alpha\), where \(\alpha \in LG(\bar{K})\) for case (L).