### 14-term arithmetic progressions on quartic elliptic curves.

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We explicitly perform some steps of a 3-descent algorithm for the curves ${y}^{2}={x}^{3}+a$, $a$ a nonzero integer. In general this will enable us to bound the order of the 3-Selmer group of such curves.

Let $K$ be a number field. We consider a local-global principle for elliptic curves $E/K$ that admit (or do not admit) a rational isogeny of prime degree $\ell $. For suitable $K$ (including $K=\mathbb{Q}$), we prove that this principle holds for all $\ell \equiv 1\phantom{\rule{0.277778em}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}4$, and for $\ell \<7$, but find a counterexample when $\ell =7$ for an elliptic curve with $j$-invariant $2268945/128$. For $K=\mathbb{Q}$ we show that, up to isomorphism, this is the only counterexample.

We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have nondegenerate solutions. Each of these cases gives rise to a polynomial system of equations, the first being solved by Elkies in 1988 using the resultant methods of Macsyma, with there being a unique rational nondegenerate solution. For the second case...

We show that the number of squares in an arithmetic progression of length $N$ is at most ${c}_{1}{N}^{3/5}{\left(\mathrm{log}N\right)}^{{c}_{2}}$, for certain absolute positive constants ${c}_{1}$, ${c}_{2}$. This improves the previous result of Bombieri, Granville and Pintz [1], where one had the exponent $\frac{2}{3}$ in place of our $\frac{3}{5}$. The proof uses the same ideas as in [1], but introduces a substantial simplification by working only with elliptic curves rather than curves of genus $5$ as in [1].

Let $E/K$ be an elliptic curve defined over a number field, and let $P\in E\left(K\right)$ be a point of infinite order. It is natural to ask how many integers $n\ge 1$ fail to occur as the order of $P$ modulo a prime of $K$. For $K=\mathbb{Q}$, $E$ a quadratic twist of ${y}^{2}={x}^{3}-x$, and $P\in E\left(\mathbb{Q}\right)$ as above, we show that there is at most one such $n\ge 3$.