Siegel's lemma for function fields
This interesting paper presents a function field analogue for the version of Thue-Siegel's lemma due to \textit{E. Bombieri} and \textit{J. Vaaler} [Invent. Math. 73, 11-32 (1983; Zbl 0533.10030)]. It is worthwhile to compare both the results and the methods. First assume that \(K\) is a number field of degree \(d\) over \(\mathbb{Q}\) with \(r_2\) distinct complex embeddings and discriminant \(\Delta_K\), and let \(M\) be an \(m\times n\) matrix of rank \(m\) with coefficients in \(K\). Bombieri and Vaaler show the existence of a basis \(x_1, \dots, x_{n- m}\) of the kernel of \(M\) in \(K^n\) such that \[ H(x_1) \dots H(x_{n-m})\leq \Biggl( {{2^{r_2} \sqrt {|\Delta_K |}} \over {\pi^{r_2}}} \Biggr)^{(n- m)/d} H' (M), \tag{1} \] where \(H(x)\) denotes the absolute multiplicative Weil height of a point \(x\in K^n\) computed using the maximum norm at each place, and where \(H' (M)\) denotes the absolute multplicative Weil height of the \({n\choose m}\)- tuple of minors of order \(m\) of \(M\) computed using the Euclidean norm at each archimedean place. Now assume that \(K\) is a function field in one variable over its constant field \(K_0\). More precisely, assume that \(K\) is an algebraic extension of \(K_0 (T)\) of degree \(d\) where \(T\in K\) is transcendental over \(K_0\). Let \(g\) be the genus of \(K\) and let \(M(K)\) be the set of all its valuations that are trivial on \(K_0\) and have value group \(\mathbb{Z}\). For each \(v\in M(K)\), let \(K_v\) denote the completion of \(K\) at \(v\) and, for each non-zero \(x\in (K_v )^n\), let \(\text{ord}_v (x)\) denote the minimum of the order at \(v\) of the \(n\) coordinates of \(x\). The author defines the divisor of a non-zero \(x\in K^n\) as \(\text{div} (x)= \sum_v \text{ord}_v (x)\) and the (logarithmic) height of \(x\) as \(h(x)= -(1/ d)\deg (\text{div} (x))\). For a matrix \(M\) as above, he shows the existence of a basis \(x_1, \dots, x_{n-m}\) of the kernel of \(M\) in \(K^n\) such that \[ h(x_1)+ \cdots+ h(x_{n -m})\leq (g-1+ d)(n- m)/d+ h(M), \tag{2} \] where \(h(M)\) denotes the height of the \({n \choose m}\)-tuple of minors of order \(m\) of \(M\). In [\textit{D. Roy} and \textit{J. the author}, A note on Siegel's lemma over number fields, Monatsh. Math. (to appear)], it is shown that some positive power of \(|\Delta_K |\) is needed in the right hand side of (1) when \(n> m+2\). In the present paper, the author shows that some positive multiple of \(g\) is also needed in the right hand side of (2) when \(K\) is separable over \(K_0 (T)\) and when \(n> m+2\). While the proof of the result of Bombieri and Vaaler rested on a version of Minkowski's second convex bodies theorem in the context of adeles, the proof of the result of the author uses a version of the Riemann-Roch theorem for subspaces of \(K^n\). This, in fact, allows him to work with the twisted height \(h_A\) attached to an element \(A\) of \(\text{GL}_n (K_\mathbb{A})\) where \(K_\mathbb{A}\) stands for the adele ring of \(K\). His main result is formulated in these terms and (2) follows as a corollary.
scientific article; zbMATH DE number 784589