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Strongly walk regular graphs (SWRGs or $s$-SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length~2 are replaced by paths of length~$s$. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible parameters of these codes in the binary and ternary case for medium size code lengths. For the binary case, the divisibility of the weights of these codes is investigated and several general results are shown. It is known that an $s$-SWRG has at most 4 distinct eigenvalues $k > \theta_1 > \theta_2 > \theta_3$, and that the triple $(\theta_1, \theta_2, \theta_3)$ satisfies a certain homogeneous polynomial equation of degree $s - 2$ (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we use methods from algorithmic arithmetic geometry to show that for $s = 5$ and $s = 7$, there are only the obvious solutions, and we conjecture this to remain true for all (odd) $s \ge 9$., Comment: 42 pages

ZusammenfassungDie Forschung widmete sich der Frage, wie Einzelcoaching dazu beiträgt, dass Coachees eine situationsadäquate Selbstregulation von Dominanz und Unterordnung gelingt. Zunächst wird die Relevanz des Themas für die Coachingpraxis und das Spannungsfeld von Dominanz und Unterordnung erörtert. Das Instrument der operationalisierten psychodynamischen Diagnostik diente als Grundlage für das Forschungsdesign. Anschließend wird die verwendete Methode beschrieben, und die Ergebnisse der Forschung werden komprimiert dargelegt. Schließlich werden einzelne inhaltliche Kernaspekte diskutiert und Implikationen für die Beratungspraxis angeführt.

We study the generalized Fermat equation $x^{2}+y^{3}=z^{p}$, to be solved in coprime integers, where $p\geqslant 7$ is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve $X(p)$. We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic $p$-torsion modules. Using these criteria we produce the minimal list of twists of $X(p)$ that have to be considered, based on local information at 2 and 3; this list depends on $p\hspace{0.2em}{\rm mod}\hspace{0.2em}24$. We solve the equation completely when $p=11$, which previously was the smallest unresolved $p$. One new ingredient is the use of the ‘Selmer group Chabauty’ method introduced by the third author, applied in an elliptic curve Chabauty context, to determine relevant points on $X_{0}(11)$ defined over certain number fields of degree 12. This result is conditional on the generalized Riemann hypothesis, which is needed to show correctness of the computation of the class groups of five specific number fields of degree 36. We also give some partial results for the case $p=13$. The source code for the various computations is supplied as supplementary material with the online version of this article.

We show that there is essentially a unique elliptic curve $E$ defined over a cubic Galois extension $K$ of $\mathbb Q$ with a $K$-rational point of order 13 and such that $E$ is not defined over $\mathbb Q$.

We show that there is a bound depending only on g and [K : Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its Jacobian is at most g − 3. If K = Q, an explicit bound is 8(r + 4)(g − 1) + max{1, 4r} · g. The proof is based on Chabauty’s method; the new ingredient is an estimate for the number of zeros of a logarithm in a p-adic ‘annulus’ on the curve, which generalizes the standard bound on disks. The key observation is that for a p-adic field k, the set of k-points on C can be covered by a collection of disks and annuli whose number is bounded in terms of g (and k).