On the Mumford-Tate group of an abelian variety with complex multiplication
Let K be a CM field, and (K,\(\phi)\) a primitive CM type. Suppose \([K:{\mathbb{Q}}]=2n\), and view \(\phi\) as an element of the free \({\mathbb{Z}}\)- module on the 2n embeddings of K into \({\mathbb{C}}\). Define rank(\(\phi)\) to be the rank of the \({\mathbb{Z}}\)-module spanned by all translates of \(\phi\) under the Galois group of the Galois closure of K over \({\mathbb{Q}}\). The number rank(\(\phi)\) is of interest because it coincides with the dimension of the Mumford-Tate group of an abelian variety A of CM type \((K,\phi)\), and controls the properties of class fields constructed using A. - Now let n be fixed, let K vary over all CM fields of degree 2n, and let S(n) be the set of all numbers rank(\(\phi)\) for primitive CM types \((K,\phi)\) as above. Let \(B(n)\) be the ``sharp lower bound for the rank'' in dimension \(n,\) the least element of \(S(n)\). This paper contains many partial results describing the set S(n) and the number \(B(n)\). As might be expected, they vary strongly with the factorization of n and the types of groups which can occur as Galois groups for CM fields of degree n. - The first section is primarily concerned with B(n). The kinds of results obtained are: If \(n=2^ j\), then \(B(n)=j+2\). If n is not a power of 2, then \(\log_ 2n+2<B(n)\leq n+1\). If \(n=p\), a prime, then \(B(n)=p+1\). If \(n=p^ 2\), then \(B(n)=2p\). If \(n=2^{(p-1)/2}pt\) with t dividing (p-1)/2, then \(B(n)=p+1\). These results follow from specific examples of CM types and from a ``nondegeneracy theorem'' of Ribet. - The second section studies \(S(n)\) when n is odd, showing that many of its elements are accounted for by CM types coming from fields K whose Galois group satisfies a relative solvability criterion. These elements are called ``computable''. - The third section contains a partial classification of simple groups which have a permutation representation in degree \(n.\) - The final section treats \(n=9\) as an example, showing that \(S(9)=\{6,8,10\}\), and describing the CM-types that account for each of these ranks.
scientific article; zbMATH DE number 4037997
scientific article