## 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)

### Citations:

Zbl 0792.11015; Zbl 0617.14001; Zbl 0675.10025; Zbl 0713.11082
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