We determine the distribution over square-free integers $n$ of the pair $({dim}_{{𝔽}_2}{\mathrm{Sel}}^{\Phi }(E_n/\mathbb{Q}),{dim}_{{𝔽}_2}{\mathrm{Sel}}^{\widehat{\Phi }}(E_n^{\text{'}}/\mathbb{Q}))$, where $E_n$ is a curve in the congruent number curve family, $E_n^{\text{'}}:y^2=x^3+4n^{2}x$ is the image of isogeny $\Phi :E_n\to E_n^{\text{'}}$, $\Phi (x,y)=(y^2/x^2,y(n^2-x^2)/x^2)$, and $\widehat{\Phi }$ is the isogeny dual to $\Phi$.

Jordan pairs of quadratic forms are generalized so that they have forms with values in invertible modules. The role of such pairs turns out to be natural in describing 'big cells', a kind of open charts around unit sections, of Clifford and orthogonal groups as group schemes. Group germ structures on big cells are particularly interested in and related also to Cayley-Lipschitz transforms.

In this article we study the interplay between algebro-geometric notions related to $\pi$-points and structural features of the stable Auslander-Reiten quiver of a finite group scheme. We show that $\pi$-points give rise to a number of new invariants of the AR-quiver on one hand, and exploit combinatorial properties of AR-components to obtain information on $\pi$-points on the other. Special attention is given to components containing Carlson modules, constantly supported modules, and endo-trivial modules.

We prove that among counterexamples to the Jacobian Conjecture, if there are any, we can find one of lowest degree, the coordinates of which have the form $x^m{y}^n$ + terms of degree < m+n.