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 $\genfrac{}{}{0pt}{}{{\sum }_{k=0}^{p^a-1}(p^a-1}{\genfrac{}{}{0pt}{}{k\left)\right(2k}{{k)(-1)}^k\equiv 3^{p-1}(p^a/3)\left(modp²\right)}}$.

We have ${\sum }_{K-G\le k_j\le K+G}{\alpha }_j{H}_j^3\left(\frac{1}{2}\right){\ll }_{ϵ}G{K}^{1+ϵ}$ for $K^{ϵ}\le G\le K,\text{where}{\alpha }_j={\left|{\rho }_j\left(1\right)\right|}^2{(cosh\pi k_j)}^{-1},\text{and}{\rho }_j\left(1\right)$ is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue ${\lambda }_j=k_j^2+\frac{1}{4}$ to which the Hecke series $H_j\left(s\right)$ is attached. This result yields the new bound $H_j(\frac{1}{2}{\ll }_{ϵ}{k}_j^{\frac{1}{3}+ϵ}.$

The pairs (k,m) are studied such that for every positive integer n we have $1^k+2^k+\cdots +n^k|1^{km}+2^{km}+\cdots +n^{km}$.

We address the question of when an integer in a totally real number field can be written as the sum of three squared integers from the field and more generally whether it can be represented by a positive definite integral ternary quadratic form over the field. In recent work with Piatetski-Shapiro and Sarnak we have shown that every sufficiently large totally positive square free integer is globally integrally represented if and only if it is so locally at all places, thus essentially resolving...