Euler noted the relation $6^{3}\,=\,3^{3}+4^{3}+5^{3}$ and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular, Stroeker determined all squares that can be written as a sum of at most 50 consecutive cubes. We generalize Stroeker’s work by determining all perfect powers that are sums of at most 50 consecutive cubes. Our methods include descent, linear forms in two logarithms and Frey–Hellegouarch curves.

For a given cusp form $\phi $ of even integral weight satisfying certain hypotheses, Waldspurger’s theorem relates the critical value of the $\mathrm{L} $-function of the $n\mathrm{th} $ quadratic twist of $\phi $ to the $n\mathrm{th} $ coefficient of a certain modular form of half-integral weight. Waldspurger’s recipes for these modular forms of half-integral weight are far from being explicit. In particular, they are expressed in the language of automorphic representations and Hecke characters. We translate these recipes into congruence conditions involving easily computable values of Dirichlet characters. We illustrate the practicality of our ‘simplified Waldspurger’ by giving several examples.

Let $C$ be an elliptic curve defined over $\mathbb{Q}$. We can associate two formal groups with $C$: the formal group $\^{C}(X,Y)$ determined by the formal completion of the Néron model of $C$ over $\mathbb{Z}$ along the zero section, and the formal group $F_L(X,Y)$ of the L-series attached to $l$-adic representations on $C$ of the absolute Galois group of $\mathbb{Q}$. Honda shows that $F_L(X, Y)$ is defined over $\mathbb{Z}$, and it is strongly isomorphic over $\mathbb{Z}$ to $\^{C}(X,Y)$. In this paper we give a generalization of the result of Honda to building blocks over finite abelian extensions of $\mathbb{Q}$. The difficulty is to define new matrix L-series of building blocks. Our generalization contains the generalization of Deninger and Nart to abelian varieties of $\rm{GL}_2$-type. It also contains the generalization of our previous paper to $\mathbb{Q}$-curves over quadratic fields.

Here, a classical observation of Siegel is generalized by determining all the linear relations among the initial Fourier coefficients of a modular form on $\SL_2(\ZZ)$. As a consequence, spaces $M_k$ are identified, in which there are universal $p$-divisibility properties for certain $p$-power coefficients. As a corollary, let $f(z)=\sum_{n=1}^{\infty}a_f(n)q^n \in S_k\cap O_{L}[[q]]$ be a normalized Hecke eigenform (note that $q:=e^{2\pi i z}$), and let $k\equiv \delta(k)\pmod{12}$, where $\delta(k)\in \{4, 6, 8, 10, 14\}$. Reproducing earlier results of Hatada and Hida, if $p$ is a prime for which $k\equiv \delta(k)\pmod{p-1}$, and $\mathfrak{p}\subset O_L$ is a prime ideal above $p$, a proof is given to show that $a_f(p)\equiv 0\pmod{\mathfrak{p}}$.