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Siegel’s theorem for Drinfeld modules. (English) Zbl 1170.11012
The authors obtain analogues for Drinfeld modules of both Siegel’s Theorem on integral points on algebraic curves and a theorem of J. H. Silverman on non-constant rational maps of $${\mathbb P}^1$$ over a number field [see Duke Math. J. 71, No. 3, 793–829 (1993; Zbl 0811.11052)]. The authors’ results are based on a Drinfeld type linear forms in logarithm estimate proved by V. Bosser [J. Number Theory 75, No. 2, 279–323 (1999; Zbl 0922.11062)], but also on another type of linear forms estimate (their Statement 3.2) which is quite plausible but which they state without proof.
More precisely, let $$q$$ be any prime power, $$A=\mathbb F_q [t]$$, $$K$$ a finite extension of $$\mathbb F_q (t)$$ and $$\bar K$$ an algebraic closure of $$K$$. Denote by $$M_K$$ the set of places of $$K$$ and choose the absolute values $$|\cdot |_v$$ for $$v\in M_K$$ in such a way that they satisfy the product formula $$\prod_{v\in M_K} |x|_v=1$$ for $$x\in K^*$$. Call a place $$v$$ of $$K$$ infinite if $$|t|_v>1$$. Each absolute value $$|\cdot |_v$$ is extended to $$\bar K$$. Given $$\alpha\in\bar K$$, and a finite set of places $$S$$ of $$K$$, an element $$\beta$$ of $$\bar K$$ is said to be $$S$$-integral with respect to $$\alpha$$ if $$|\sigma_1 (\alpha )-\sigma_2(\beta )|_v= \max (1,|\sigma_1(\alpha )|_v)\max (1,|\sigma_2(\beta )|_v)$$ for every $$v\in M_K\setminus S$$ and every pair of $$K$$-automorphisms $$\sigma_1,\sigma_2$$ of $$\bar K$$.
In the usual Drinfeld module setting, let $$\tau$$ denote the Frobenius map $$x\mapsto x^q$$, extended to $$\bar K$$, and $$K\{\tau\}$$ the ring of polynomial expressions in $$\tau$$ with coefficients $$K$$, endowed with usual addition and composition as multiplication. Let $$A=\mathbb F_q [t]$$ and consider Drinfeld modules $$\phi :\, A\to K\{\tau\}$$, where $$\phi_a :=\phi (a) =a\tau^0+$$(higher powers of $$\tau$$). Further, let $$\Gamma$$ be a finitely generated $$\phi$$-submodule of the additive group $${\mathbb G}_a(K)$$, i.e., $$\phi_a(x)\in\Gamma$$ for every $$x\in\Gamma$$, $$a\in A$$. We call $$\beta\in\Gamma$$ a torsion point if there is a non-zero $$Q\in A$$ such that $$\phi_Q(\beta )=0$$.
Inspired by the analogy between elliptic curves and Drinfeld modules of rank 2, the authors prove (modulo the unproved linear forms in logarithms estimate mentioned above) the following finiteness result, which may thus be viewed as an analogue of Siegel’s Theorem: Let $$K$$, $$\phi$$, $$\Gamma$$ be as above, but assume that $$K$$ has only one infinite place, and let $$S$$ be a finite set of places in $$K$$ and $$\alpha\in K$$. Then there are only finitely many $$\gamma\in\Gamma$$ which are $$S$$-integral with respect to $$\alpha$$.
The authors’ analogue of Silverman’s result, also depending on the unproved linear forms in logarithms estimate is as follows: Let $$K$$, $$\phi$$, $$\Gamma$$, $$S$$, $$\alpha$$ be as above, but drop the assumption that $$K$$ have only one infinite place. Further, let $$\beta\in\Gamma$$ be a non-torsion point. Then there are only finitely many $$Q\in A$$ such that $$\phi_Q(\beta )$$ is integral with respect to $$\alpha$$.

##### MSC:
 11G09 Drinfel’d modules; higher-dimensional motives, etc. 11J68 Approximation to algebraic numbers 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 11G50 Heights
##### Keywords:
Drinfeld modules; Siegel’s Theorem
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##### References:
  Baker A. (1975). Transcendental Number Theory. Cambridge University Press, Cambridge · Zbl 0297.10013  Baker, M., Ih, S.I., Rumely, R.: A finiteness property of torsion points. preprint, Available at arxiv: math.NT/0509485, pp. 30 (2005)  Bosser V. (1999). Minorations de formes linéaires de logarithmes pour les modules de Drinfeld. J. Number Theory 75(2): 279–323 · Zbl 0922.11062  Breuer F. (2005). The André-Oort conjecture for products of Drinfeld modular curves. J. Reine Angew. Math. 579: 115–144 · Zbl 1064.11045  David S. (1995). Minorations de formes linéaires de logarithmes elliptiques. Mem. Soc. Math. Fr. 62: 143  Denis, L.: Géométrie diophantienne sur les modules deDrinfel’ d. The arithmetic of function fields (Columbus,1991). Ohio State Univ. Math. Res. Inst. Publ., vol. 2, pp. 285–302. de Gruyter, Berlin (1992)  Denis L. (1992). Hauteurs canoniques et modules de Drinfel’ d. Math. Ann. 294(2): 213–223 · Zbl 0764.11027  Edixhoven, B., Yafaev, A.: Subvarieties of Shimura type. Ann. of Math. (2) 157(2), 621–645 (2003) · Zbl 1053.14023  Ghioca D. (2005). The Mordell-Lang theorem for Drinfeld modules. Int. Math. Res. Not. 53: 3273–3307 · Zbl 1158.11030  Ghioca D. (2006). Equidistribution for torsion points of a Drinfeld module. Math. Ann. 336(4): 841–865 · Zbl 1171.11038  Ghioca, D.: Towards the full Mordell-Lang conjecture for Drinfeld modules, 6p (2006)(in press) · Zbl 1219.11086  Ghioca D. (2007). The Lehmer inequality and the Mordell-Weil theorem for Drinfeld modules. J. Number Theory 122(1): 37–68 · Zbl 1125.11033  Ghioca D. (2007). The local Lehmer inequality for Drinfeld modules. J. Number Theory 123(2): 426–455 · Zbl 1173.11035  Goss, D.: Basic structures of function field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [results in mathematics and related areas (3)], vol. 35, Springer, Berlin (1996)  Ghioca, D., Tucker, T.J.: Equidistribution and integral points for Drinfeld modules. Trans AM Math Soc, pp. 29 (2006) · Zbl 1178.11046  Ghioca, D., Tucker, T.J.: A dynamical version of the Mordell-Lang conjecture, pp. 14 (2007)(in press) · Zbl 1169.11024  Poonen B. (1995). Local height functions and the Mordell-Weil theorem for Drinfel’ d modules. Composit. Math. 97(3): 349–368 · Zbl 0839.11024  Scanlon T. (2002). Diophantine geometry of the torsion of a Drinfeld module. J. Number Theory 97(1): 10–25 · Zbl 1055.11037  Serre, J.-P.: Lectures on the Mordell-Weil theorem, 3 edn. Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, With a foreword by Brown and Serre (1997)  Siegel, C.L.: Über einige anwendungen diophantisher approximationen. Abh. Preuss. Akad. Wiss. Phys. Math. Kl. 41–69 (1929) · JFM 56.0180.05  Silverman J.H. (1993). Integer points, Diophantine approximation and iteration of rational maps. Duke Math. J. 71(3): 793–829 · Zbl 0811.11052  Szpiro L., Ullmo E., Zhang S. (1997). Equirépartition des petits points. Invent. Math. 127: 337–347 · Zbl 0991.11035  Taguchi Y. (1993). Semi-simplicity of the Galois representations attached to Drinfel’ d modules over fields of ”infinite characteristics. J. Number Theory 44(3): 292–314 · Zbl 0781.11024  Yafaev A. (2006). A conjecture of Yves André’s. Duke Math. J. 132(3): 393–407 · Zbl 1097.11032  Zhang, S.: Equidistribution of small points on abelian varieties. Ann. Math. (2) 147(1), 159–165 (1998) · Zbl 0991.11034
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