A remark on Serre's example of \(p\)-adic Eisenstein series
Let \(p>3, p\equiv 3 \pmod 4\) be a prime number. Building on ideas of Serre, the author evaluates the \(p\)-adic limit of the Siegel-Eisenstein series of arbitrary (fixed) degree \(n\) and weight \(1+{p-1\over 2}p^m, m\to\infty,\) and expresses the constant term as half a class number and the other coefficients by arithmetic expressions. The main input is information on Bernoulli numbers and the Fourier coefficients of the Siegel-Eisenstein series. In degree 2, the author also derives a formula for the mod \(p\) Fourier coefficients of Siegel-Eisenstein series of weight \({p+1\over 2}\). As a last result, it is shown that the \(p\)-adic Eisenstein series under consideration coincides with a genus theta series, and, up to a factor, also with a certain weight-1-Eisenstein series.
scientific article; zbMATH DE number 1546901