Group representations and lattices. (English) Zbl 0745.11035

Various authors have constructed and classified \(G\)-stable lattices in higher-dimensional Euclidean space using sufficiently restrictive hypotheses on the action of a finite group \(G\). The present paper contains a particularly deep application of this principle.
The author considers lattices \(L\) for representations \(V\) of \(G\) over \(\mathbb{Q}\) which satisfy the following conditions of “global irreducibility”: \(V\otimes\mathbb{R}\) is irreducible, there is a maximal order \(R\) in \(K=\text{End}_ G V\) such that \(L\) is an \(RG\)-lattice, and for any maximal two-sided ideal \({\mathfrak p}\) of \(R\) the reduction \(V_{\mathfrak p}=L/{\mathfrak p}L\) is irreducible. These conditions ensure that, after some normalization of the inner product, the class group of \(R\) acts simply transitively on the \(\mathbb{Z} G\)-isometry classes of the lattices (so this is the classification, although not yet over \(\mathbb{Z}\)). To find all admissible pairs \((G,V)\) remains an open problem. For the case \(K=\mathbb{Q}\), studied earlier by J. G. Thompson, there are almost no examples (and no new ones) known. But when \(K\) is imaginary quadratic or a definite quaternion algebra over \(\mathbb{Q}\), many examples are given here. Some of the pairs \((G,L)\) have been known before, e.g., for the Weil representation of \(G=\text{Sp}_{2n}(q)\) by R. Gow (his lattices should have been referred too).
It is an important aspect of the paper that in several cases the lattices arising from representation theory are also described as a Mordell-Weil group modulo torsion for some elliptic curve \(E\) over a global function field \(F\). Such Mordell-Weil lattices have been studied by T. Shioda and N. Elkies; the author especially refers to Elkies’ (unpublished) work. In the last example, \(E\) is defined over the field \(k\) with \(q^ 2\) elements such that \(E(k)\) has order \((q+1)^ 2\), \(F\) is the function field of the Fermat curve with exponent \(q+1\) over \(k\), and \(G=PU_ 3(q^ 2)\). Here the global irreducibility condition is satisfied only when \(q=p\) (a prime) or \(q=p^ 2\), but for higher \(q\) the author calculates part of the Tate-Shafarevich group which gives an upper bound for the determinant of the Mordell-Weil lattice.


11H56 Automorphism groups of lattices
20C15 Ordinary representations and characters
11G05 Elliptic curves over global fields
Full Text: DOI


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