In this paper we study certain moduli spaces of Barsotti-Tate groups constructed by Rapoport and Zink as local analogues of Shimura varieties. More precisely, given an isogeny class of Barsotti-Tate groups with unramified additional structures, we investigate how the associated (non-basic) moduli spaces compare to the (basic) moduli spaces associated with its isoclinic constituents. This aspect of the geometry of the Rapoport-Zink spaces is closely related to Kottwitz’s prediction that their $l$-adic...

Let $D(s,f,g)$ be the Rankin product $L$-function for two Hilbert cusp forms $f$ and $g$. This $L$-function is in fact the standard $L$-function of an automorphic representation of the algebraic group $GL\left(2\right)\times GL\left(2\right)$ defined over a totally real field. Under the ordinarity assumption at a given prime $p$ for $f$ and $g$, we shall construct a $p$-adic analytic function of several variables which interpolates the algebraic part of $D(m,f,g)$ for critical integers $m$, regarding all the ingredients $m$, $f$ and $g$ as variables.

Let $(K,\nu )$ be a valued field, where $\nu$ is a rank one discrete valuation. Let $R$ be its ring of valuation, $𝔪$ its maximal ideal, and $L$ an extension of $K$, defined by a monic irreducible polynomial $F\left(X\right)\in R\left[X\right]$. Assume that $\overline{F}\left(X\right)$ factors as a product of $r$ distinct powers of monic irreducible polynomials. In this paper a condition which guarantees the existence of exactly $r$ distinct valuations of $K$ extending $\nu$ is given, in such a way that it generalizes the results given in the paper “Prolongations of valuations to finite...

Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum $\genfrac{}{}{0pt}{}{{\sum }_{k=0}^{p^a-1}(h{p}^a-1}{\genfrac{}{}{0pt}{}{k\left)\right(2k}{k)/m^k\left(modp²\right)}}$, where h and m are p-adic integers with m ≢ 0 (mod p). For example, we show that if h ≢ 0 (mod p) and $p^a>3$, then $\genfrac{}{}{0pt}{}{{\sum }_{k=0}^{p^a-1}(h{p}^a-1}{\genfrac{}{}{0pt}{}{k\left)\right(2k}{{k)(-h/2)}^k\equiv ((1-2h)/\left(p^a\right))(1+h({(4-2/h)}^{p-1}-1))\left(modp²\right)}}$, where (·/·) denotes the Jacobi symbol. Here is another remarkable congruence: If $p^a>3$ then $\genfrac{}{}{0pt}{}{{\sum }_{k=0}^{p^a-1}(p^a-1}{\genfrac{}{}{0pt}{}{k\left)\right(2k}{{k)(-1)}^k\equiv 3^{p-1}(p^a/3)\left(modp²\right)}}$.