We investigate the average behavior of the $n$th normalized Fourier coefficients of the $j$th ($j\ge 2$ be any fixed integer) symmetric power $L$-function (i.e., $L(s,{\mathrm{sym}}^{j}f)$), attached to a primitive holomorphic cusp form $f$ of weight $k$ for the full modular group $SL(2,\mathbb{Z})$ over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum $$S_j^{*}:=\sum _{a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2\le x(a_1,a_2,a_3,a_4,a_5,a_6)\in {\mathbb{Z}}^6}{\lambda }_{{\mathrm{sym}}^{j}f}^2(a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2),$$ where $x$ is sufficiently large, and $$L(s,{\mathrm{sym}}^{j}f):=\sum _{n=1}^{\infty }\frac{{\lambda }_{{\mathrm{sym}}^{j}f}\left(n\right)}{n^s}.$$ When $j=2$, the error term which we obtain improves the earlier known result.

Let $E/F$ be a modular elliptic curve defined over a totally real number field $F$ and let $\phi$ be its associated eigenform. This paper presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of $E$ over suitable quadratic imaginary extensions $K/F$. In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachëv, that is, when $[F:\mathbb{Q}]$ is even and $\phi$ not new at any prime.

We show that the coefficients of the characteristic power series of Atkin’s U operator acting on overconvergent $p$-adic modular forms of weight $k$ vary $p$-adically continuously as functions of $k$. Are they in fact Iwasawa functions of $k$ ?