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Analytic homomorphisms into Drinfeld modules. (English) Zbl 0881.11055

Starting more than 10 years ago, the author has developed a remarkable theory of transcendence on Drinfeld modules (see for instance [Jing Yu, Transcendence in finite characteristic, The arithmetic of function fields, de Gruyter, Berlin, 253-264 (1992; Zbl 0792.11015)]). The paper under review represents one of the most important steps in this theory.
The author proves an analog of Philippon’s zero estimate [P. Philippon, Bull. Soc. Math. Fr. 114, No. 3, 355-383 (1986; Zbl 0617.14001)] and deduces an analog of Wüstolz’s analytic subgroup theorem [G. Wüstholz, Ann. Math., II. Ser. 129, No. 3, 501-517 (1989; Zbl 0675.10025)].
Denote by \(A\) the ring \({\mathbb{F}}_q[T]\) of polynomials over a finite field \({\mathbb{F}}_q\), by \(k\) the rational function field \({\mathbb{F}}_q(t)\) and by \(k_\infty\) its completion \({\mathbb{F}}_q((t))\). Let \(\overline k_\infty\) be an algebraic closure of \(k_\infty\) and \(\overline k\) be the algebraic closure of \(k\) in \(\overline k_\infty\). Let \(G=({\mathcal G}_{\text{a}}^m,\phi)\) be an \(m\)-dimensional \(t\)-module and \(\Phi:\overline k_\infty^n\rightarrow {\mathcal G}_{\text{a}}^m(\overline k_\infty)\) an \(n\)-parameters analytic submodule of \(G\). For each \(t\)-submodule \(H\) of \(G\), the non negative integer \(r(\Phi,H)\) is defined as the codimension in \(\overline k_\infty^n\) of \(d\Phi^{-1}(\text{Im } d\Phi\cap \text{Lie } H)\). For \(a\in k\) let \(d_\infty(a)\) denote the degree in \(t\) of \(a\). Given elements \(\gamma_1,\ldots,\gamma_\ell\) in \(G(\overline k_\infty)\) and a positive integer \(S\), define \[ \Gamma(S)=\bigl\{a_1\gamma_1+\cdots+a_\ell\gamma_\ell;\;a_i\in A,\;d_\infty(a_i)<S\bigr\}. \] Here is the zero estimate: Let \(Q\) be a polynomial in \(m\) variables which is of total degree \(\leq D\) and vanishes along \(\Phi\) to order \(\geq mT+1\) at all points of \(\Gamma(S)\). Then there exists a proper \(t\)-submodule \(H\subset G\) such that \[ {T+r(\Phi,H)\choose r(\Phi,H)} \text{Card }\bigl(\Gamma(S-m+1)+H/H\bigr)\deg H \leq C(G) D^{\dim G/H}, \] where \(C(G)\) is a constant depending only on the \(t\)-module \(G\). A consequence of this zero estimate is the following analytic subgroup Theorem for Drinfeld modules: Assume that the regular \(t\)-module \(G=({\mathcal G}_{\text{a}}^m,\phi)\) is defined over \(\overline k\). Let \(u\) be a point in \(\text{Lie } G(\overline k_\infty)\) such that \(\exp_G(u)\) is in \(G(\overline k)\). Then the smallest vector subspace in \(\text{Lie }G(\overline k_\infty)\), defined over \(\overline k\), which is invariant under \(d\phi(t)\) and contains \(u\), is the tangent space at the origin of a \(t\)-submodule of \(G\).
This deep transcendence result has far reaching consequences. For instance, using G. W. Anderson and D. S. Thakur [Ann. Math., II. Ser. 132, No. 1, 159-191 (1990; Zbl 0713.11082)], the author deduces that the only linear relations over \(\overline k\) between the values of the Carlitz zeta function \[ \zeta_C(n)=\sum_{\substack{ a\in A\\ a\text{ monic}}} a^{-n},\quad n=1,2,\dots \] and the powers \(\widetilde\pi^m\), (\(m=0,1,\dots\)) of the fundamental period \(\widetilde\pi\) of the Carlitz module \(C\) are the known ones between \(\zeta_C(n)\) and \(\widetilde\pi^n\) when \(n\) is divisible by \(q-1\). Namely, if \(m\) and \(n\) are positive integers, then the dimension of the \(\overline k\)-vector space spanned by the \(n+m+1\) elements \[ \zeta_C(1),\ldots,\zeta_C(n),1,\widetilde\pi,\ldots,\widetilde\pi^m \] is \(1+\max\{n,m\}+\kappa\), where \(\kappa\) is the number of positive integers \(\leq\min\{n,m\}\) which are not divisible by \(q-1\).

MSC:

11G09 Drinfel’d modules; higher-dimensional motives, etc.
11J81 Transcendence (general theory)
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