The algebraic groups leading to the Roth inequalities
A general ROTH system \(S=(\ell_1,\dots,\ell_n,c(1),\dots,c(n))\) is given by \(n\) linearly independent linear forms \((\ell_1,\dots,\ell_n)\) in \(n\) variables with real algebraic coefficients and \(n\) real numbers \(c(1),\dots,c(n)\) such that \(c(1)+\dots+c(n)=0\) and such that, for each \(\delta>0\), there are only finitely many integral solutions to the system of inequalities \[ |\ell_q|<Q^{-c(q)-\delta}, \quad Q>1, \; 1\leq q\leq n. \] The special case \(n=2\), \(\ell_1(x_1,x_2)=x_1\), \(\ell_2(x_1,x_2)=\alpha x_1-x_2\) where \(\alpha\) is an irrational real algebraic number and \(c(1)=-1\), \(c(2)=1\) is called a classical ROTH system. To a general ROTH system \(S\) is attached a filtration \(F_S^\cdot V\) over \(\overline{\mathbb Q}\) of the \(\mathbb Q\)-vector space \(V=\mathbb Q x_1+\dots + \mathbb Q x_n\), defined as \[ F_S^{i}V=\sum_{c(q)\geq i}\overline{\mathbb Q} \ell_q \quad (i\in\mathbb R). \] In the case of a classical ROTH system attached to an irrational algebraic number \(\alpha\), this filtered space is denoted by \((\check{V}, F_\alpha^\cdot \check{V})\). The slope of a filtered vector space \((V, F^\cdot V)\) is \[ \mu(V)=\frac{1}{\dim_\mathbb Q V} \sum_{w\in \mathbb R} w \dim_{\overline{\mathbb Q}} {\mathrm{gr}}^w(F^{\cdot} V) \] where \( {\mathrm{gr}}^w(F^{\cdot} V)=F^wV/F^{w+}V\), \(F^{w+}V=\bigcup _{j>w}F^jV\). The filtration is semi stable if \(\mu(W)=\mu(V)\) for any nonzero \(\mathbb Q\)-vector subspace \(W\) of \(V\). The category of finite dimensional vector spaces over \(\mathbb Q\) with semistable filtration of slope zero is denoted by \(C_0^{\mathrm{ss}}(\mathbb Q,\overline{\mathbb Q})\). See Chap.~VI, Theorem 2B of [\textit{W. M. Schmidt}, Diophantine approximation. Berlin etc.: Springer-Verlag (1980; Zbl 0421.10019)]. See also [\textit{G. Faltings}, Proc. ICM '94. Vol. I. Basel: Birkhäuser, 648--655 (1995; Zbl 0871.14010)] and [\textit{B. Totaro}, Duke Math. J. 83, No. 1, 79--104 (1996; Zbl 0873.14019)]. Here is the main result of the paper under review. Let \(\alpha\) be an irrational real algebraic number. If \(\alpha\) is not quadratic, then there exists a fully faithful tensor functor \(\iota\) of the category \({\mathrm{Rep}}_\mathbb Q {\mathrm{SL}}_2\) of finite dimensional representations over \(\mathbb Q\) of the special linear group \({\mathrm{SL}}_2\) of degree \(2\) into the tensor category \(C_0^{\mathrm{ss}}(\mathbb Q,\overline{\mathbb Q})\) such that the functor \(\iota\) commutes with the forgetful tensor functor to the tensor category \({\mathrm{Vec}}_\mathbb Q\) of finite dimensional vector spaces over \(\mathbb Q\) and such that the image of \(\iota\) contains the filtered vector space \((\check{V}, F_\alpha^\cdot \check{V})\) derived from a classical ROTH System. On the other hand, if \(\alpha\) is quadratic, then there exists a fully faithful tensor functor \(\iota\) of the category \({\mathrm{Rep}}_\mathbb Q T_\alpha\) of finite dimensional representations over \(\mathbb Q\) of a one dimensional anisotropic torus \(T_\alpha\) over \(\mathbb Q\) into the tensor category \(C_0^{\mathrm{ss}}(\mathbb Q,\overline{\mathbb Q})\) such that the group \(T_\alpha(\mathbb Q)\) is isomorphic to the kernel of the norm map of the quadratic number field \(\mathbb Q(\alpha)\) over \(\mathbb Q\), such that the functor \(\iota\) is compatible with the forgetful tensor functor to \({\mathrm{Vec}}_\mathbb Q\) and such that the image of \(\iota\) contains the filtered vector space \((\check{V}, F_\alpha^\cdot \check{V})\).
scientific article; zbMATH DE number 6099144
scientific article