Derivatives of p-adic Siegel Eisenstein series and p-adic degrees of arithmetic cycles

Abstract

Kudla has given for each $n\ge 1$ a genus $n$ weight $\tfrac{n + 1}{2}$ Siegel Eisenstein series with odd functional equation whose central derivative he speculates to have arithmetic content. Specifically, these {\it incoherent} Eisenstein series vanish at $s = 0$ and their derivatives are nonholomorphic modular forms whose Fourier coefficients seem be degrees of $0$-cycles on certain Shimura varieties. When $n$ is odd, we search for evidence of a $p$-adic analogue which relates the derivative of a $p$-adic Siegel Eisenstein series to $p$-adic degrees of $0$-cycles. Indeed, when $n = 1$ or $3$, we construct an analogous $p$-adic Siegel Eisenstein series, compute the Fourier expansion of its derivative, and relate the resulting Fourier coefficients to $p$-adic degrees of the same $0$-cycles studied by Kudla.