We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $$\left\{\begin{array}{c}n=a_1+a_2+\cdots +a_{s-1},\hfill \\ a_1{a}_2\cdots a_{s-1}(a_1+a_2+\cdots +a_{s-1})=b^s\hfill \end{array}\right.$$ has positive integer or rational solutions $n$, $b$, $a_i$, $i=1,2,\cdots ,s-1$, $s\ge 3.$ Using the theory of elliptic curves, we prove that it has no positive integer solution for $s=3$, but there are infinitely many positive integers $n$ such that it has a positive integer solution for $s\ge 4$. As a corollary, for $s\ge 4$ and any positive integer $n$, the above Diophantine system has a positive rational solution. Meanwhile, we give conditions such that...

The famous problem of determining all perfect powers in the Fibonacci sequence ${\left(F_n\right)}_{n\ge 0}$ and in the Lucas sequence ${\left(L_n\right)}_{n\ge 0}$ has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations $L_n=q^a{y}^p$, with $a\>0$ and $p\ge 2$, for all primes $q\<1087$ and indeed for all but $13$ primes $q\<{10}^6$. Here the strategy of [10] is not sufficient due to the sizes of...

We provide a lower bound for the number of distinct zeros of a sum $1+u+v$ for two rational functions $u,v$, in term of the degree of $u,v$, which is sharp whenever $u,v$ have few distinct zeros and poles compared to their degree. This sharpens the “$abcd$-theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface $x^a+y^a+z^c=1$ contains only finitely many rational or elliptic curves,...

Let $E/Q$ be a modular elliptic curve, and let $K$ be an imaginary quadratic field. We show that the $p$-Selmer group of $E$ over certain finite anticyclotomic extensions of $K$, modulo the universal norms, is annihilated by the «characteristic ideal» of the universal norms modulo the Heegner points. We also extend this result to the anticyclotomic ${\mathbb{Z}}_p$-extension of $K$. This refines in the current contest a result of [1].

We prove that for any $t\in 𝐐$, the curve$$5x^3+9y^3+10z^3+12{\left(\frac{t^2+82}{t^2+22}\right)}^3{(x+y+z)}^3=0$$in ${𝐏}^2$ is a genus $1$ curve violating the Hasse principle. An explicit Weierstrass model for its jacobian $E_t$ is given. The Shafarevich-Tate group of each $E_t$ contains a subgroup isomorphic to $𝐙/3\times 𝐙/3$.

We prove that for any prime p there is a constant Cₚ > 0 such that for any n > 0 and for any p-power q there is a smooth, projective, absolutely irreducible curve over ${}_q$ of genus g ≤ Cₚqⁿ without points of degree smaller than n.

We study the Iwasawa theory of a CM elliptic curve $E$ in the anticyclotomic ${\mathbf{Z}}_p$-extension of the CM field, where $p$ is a prime of good, ordinary reduction for $E$. When the complex $L$-function of $E$ vanishes to even order, Rubin’s proof of the two variable main conjecture of Iwasawa theory implies that the Pontryagin dual of the $p$-power Selmer group over the anticyclotomic extension is a torsion Iwasawa module. When the order of vanishing is odd, work of Greenberg show that it is not a torsion module. In...

We give a survey of methods used to connect the study of ternary diophantine equations to modern techniques coming from the theory of modular forms.