Heights of vector bundles and the fundamental group scheme of a curve
\textit{A. Weil} proved in [J. Math. Pures Appl. (9) 17, 47-87 (1938; Zbl 0018.06302)] that every irreducible vector bundle of degree 0 over a Riemann surface \(X\) comes from a representation of the fundamental group of \(X\). Furthermore, he also proved that a vector bundle \(E\) over \(X\) becomes isomorphic to a direct sum of the trivial bundle over an étale Galois cover of \(X\) if and only if there exist \(f,g\in\mathbb{N}[x]\) such that \(f(E)\simeq g(E)\). Vector bundles satisfying this property are called finite vector bundles. More recently, \textit{M. V. Nori} proved [Compos. Math. 33, 29-41 (1976; Zbl 0337.14016)] that if \(X\) is a projective variety defined over a perfect field \(k\), then the finite bundles form an Abelian category that when endowed with the tensor and the ``restriction to the fiber over a rational point'' functor becomes a Tannaka category. Therefore one associates with it a profinite group scheme \(\pi(X/k.x_0)\) which generalizes the ``usual'' fundamental group. In the paper under review the author starts by giving an alternative approach to the fundamental group scheme which works for reduced schemes over Dedekind schemes. Then he analyzes two situations. In the first one he considers the fundamental group scheme of a smooth projective curve defined over a \(p\)-adic field. In the second one the geometric object is an arithmetic surface. In the first setup, if \(K\) denotes the \(p\)-adic field and \(\pi(X_K/K,x_0)^{[p]}\) the prime to \(p\) part of \(\pi(X_K/K,x_0)\), under the hypothesis that \(X_K\) has good reduction, he proves that the set of isomorphism classes of rational representations \(\rho:\pi(X_K/K,x_0)^{[p]}\to\text{GL}_{N,K}\) is finite. In the second half of the paper he considers representations of the fundamental group scheme of an arithmetic surface \(X\). They are related to a special kind of vector bundles which play a role analogue to that of the finite vector bundles in the geometric (function field) case. The set of the ``new'' rank \(N\) finite vector bundles over \(X\) is denoted \(\text{FVect}_N(X)\). Moreover, the author refines a construction of \textit{C. Gasbarri} [Forum Math. 12, 135-153 (2000; Zbl 0955.14018)], obtaining an intrinsic height on the moduli space of semistable vector bundles over a smooth projective curve defined over a number field \(K\). This relies on the connection between geometric invariant theory and Arakelov geometry (C. Gasbarri, loc. cit.). This construction is similar to that of intrinsic heights on projective varieties by \textit{J.-B. Bost} [Duke Math. J. 82, 21-70 (1996; Zbl 0867.14010)]. Let \(N\) and \(d\) be fixed positive integers (\(d\) sufficiently large), \(\mathcal{E}\) a semistable vector bundle over \(X_K\) of rank \(N\) and degree \(d\), \(\mathcal{W}_E\) the set of Hermitian \(\mathcal{O}_K\)-modules \(H\) such that \(H_K=H^0(X_K,\mathcal{E})\), \(x\in X_K(K)\) and \(H_x=\alpha_x(H)\), where \(\alpha_x:H^0(X_K,\mathcal{E})\to\mathcal{E}_{|x}\) is the surjective restriction map. The following result is proved. There exist \(M\) points \(x_1,\cdots,x_M\in X_K(\overline{K})\) depending only on \(N\), \(d\), the genus of \(X\) and a constant \(A=A(d,n,X)\) such that for every semistable vector bundle \(\mathcal{E}\) of rank \(N\) and degree \(d\) over \(X\) the height of \(\mathcal{E}\) satisfies the inequality \[ h(\mathcal{E}):=\inf_{\mathcal{W}_E}\left(\frac 1{[K:\mathbb{Q}]}\left(\sum_{i=1}^M\widehat{\deg}(H_{x_i})-NM\frac{\widehat{\deg}(H)} {\text{rk}(H)}\right)\right)\geq A. \] Moreover, if \(B\geq A\) is a real number the set of isomorphism classes of stable vector bundles \(\mathcal{E}\) of rank \(N\) and degree \(d\) defined over \(K\) and such that \(h(\mathcal{E})\leq B\) is finite. So the function \(h(\mathcal{E})\) defines an intrinsic height on the moduli space. Fix an ample Hermitian line bundle \(\overline{L}\) on \(X\) (of sufficiently high degree) and a metric on the relative dualizing sheaf \(\omega_{X/\mathcal{O}_K}\). The author also proves that there exists a constant \(C=C(X,N,\overline{L},\omega_{X/\mathcal{O}_K})\) such that if \(E\in\text{FVect}_N(X)\) then \(h(E_K\otimes L_K)\leq C\). In particular, \(\text{FVect}_N(X)\) is finite. As a consequence finite bundles have bounded height (as torsion points on the Jacobian), and thus (by Theorem 4.1 in the paper) there are only finitely many \(K\)-isomorphism classes of representations of the fundamental group scheme of \(X\).
smooth projective curve over a \(p\)-adic field
scientific article; zbMATH DE number 1933166
scientific article