We study effective versions of unlikely intersections of images of torsion points of elliptic curves on the projective line.
The paper is devoted to the description of family of scalene triangles for which the triangle formed by the intersection points of bisectors with opposite sides is isosceles. We call them Sharygin triangles. It turns out that they are parametrized by an open subset of an elliptic curve. Also we prove that there are infinitely many non-similar integer Sharygin triangles. © 2017 Korean Mathematial Soiety.
A one-dimensional generalization of the Riemann–Hilbert problem from the Riemann sphere to an elliptic curve is considered. A criterion for its positive solvability is obtained and the explicit form of all possible solutions is found. As in the spherical case, the solutions turn out to be isomonodromic.
This is a brief textbook on complex analysis intended for the students of upper undergraduate or beginning graduate level. The author stresses the aspects of complex analysis that are most important for the student planning to study algebraic geometry and related topics. The exposition is rigorous but elementary: abstract notions are introduced only if they are really indispensable. This approach provides a motivation for the reader to digest more abstract definitions (e.g., those of sheaves or line bundles, which are not mentioned in the book) when he/she is ready for that level of abstraction indeed. In the chapter on Riemann surfaces, several key results on compact Riemann surfaces are stated and proved in the first nontrivial case, i.e. that of elliptic curves.
We apply Orlov's equivalence to derive formulas for the Betti numbers of maximal Cohen-Macaulay modules over the cone an elliptic curve $(E,x)$ embedded into $\mathbb{P}^{n-1}$, by the full linear system $|\mathcal{O}(nx)|$, for $n>3$. The answers are given in terms of recursive sequences. These results are applied to give a criterion of (Co-)Koszulity.
In the last two sections of the paper we apply our methods to study the cases $n=1,2$\,. Geometrically these correspond to the embedding of an elliptic curve into a weighted projective space. The singularities of the corresponding cones are called minimal elliptic. They were studied by K.Saito \cite{Saito74}, where he introduced the notation $\widetilde{E_8}$ for $n=1$, $\widetilde{E_7}$ for $n=2$ and $\widetilde{E_6}$ for the cone over a smooth cubic, that is, for the case $n=3$\,. For the singularities $\widetilde{E_7}$ and $\widetilde{E_8}$ we obtain formulas for the Betti numbers and the numerical invariants of MCM modules analogous to the case of a plane cubic.