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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 \(\alpha\).

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
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