Bounds for arithmetic multiplicities
This is a conference report on recent progress and open questions for several notoriously difficult problems concerning arithmetic multiplicities. Recent methods to find upper bounds go under the headings ``approximate orthogonality'' and ``rigidity of arithmetic coefficients''. Among others, the following problems are discussed. 1. For the dimension of the space of cusp forms of weight 1, level \(N\), and character \(\chi\), find a good upper bound depending on \(N\). 2. For given \(n\) and \(D\), find an upper bound for the number \(M_n(D)\) of number fields of degree \(n\) and discriminant \(D\). Find an estimate for the average of \(M_n (D)\), taken over all \(| D|\) less than some given \(x\). This is related to problems on bounds for class numbers. 3. For the number of elliptic curves over \(\mathbb{Q}\) with conductor \(N\), find a bound depending on \(N\). Standard conjectures about \(L\)-functions of elliptic curves will imply good answers.
scientific article; zbMATH DE number 1184273