Related papers: A theory of generalized Lam\'e curves

The computation of the topology of a real algebraic plane curve is greatly simplified if there are no more than one critical point in each vertical line: the general position condition. When this condition is not satisfied, then a finite…

We describe degenerations of projective plane curves to curves containing a fixed line $l$ as a component, and show that $H^1({\overline V}_{n,d,m}, {\Cal O} (r))=0, r \in{\Bbb Z}$, where $V_{n,d,m}\subset {\Bbb P}^N (N = n(n+3)/2)$ is the…

We construct relative Gromov--Witten theory with expanded degenerations in the normal crossings setting and establish a degeneration formula for the resulting invariants. Given a simple normal crossings pair $(X,D)$, we show that there…

Consider genus g curves that admit degree d covers to an elliptic curve simply branched at 2g-2 points. Vary a branch point and the locus of such covers forms a one-parameter family W. We investigate the geometry of W by using admissible…

Let $p \geq 5$ be a prime number. We find all the possible subgroups $G$ of ${\rm GL}_2 ( \mathbb{Z} / p \mathbb{Z} )$ such that there exists a number field $k$ and an elliptic curve ${\mathcal{E}}$ defined over $k$ such that the ${\rm Gal}…

Consider genus $g$ curves that admit degree $d$ covers to elliptic curves only branched at one point with a fixed ramification type. The locus of such covers forms a one parameter family $Y$ that naturally maps into the moduli space of…

We provide an angular parametrization of the special unitary group $\textrm{SU}(2^{n})$ generalizing Euler angles for $\textrm{SU}(2)$ by successively applying the KAK decomposition. We then determine constraint equations for the parametric…

We carry out the generalized symmetry classification of polylinear autonomous discrete equations defined on the square, which belong to a twelve-parametric class. The direct result of this classification is a list of equations containing no…

A stable generalized complex structure is one that is generically symplectic but degenerates along a real codimension two submanifold, where it defines a generalized Calabi-Yau structure. We introduce a Lie algebroid which allows us to view…

We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total $\delta$ invariant is preserved. These are also known as equi-normalizable or equi-generic…

We prove that, on average, elliptic curves over Q have finitely many primes p for which they possess a p-adic point of order p. We include a discussion of applications to companion forms and the deformation theory of Galois representations.

Motivated by the finite-gap structure of the classical Lam\'{e} equation (1.2) and its central role in mathematical physics, generalized Lam\'{e}-type equations (1.12) are investigated. For the fundamental case $n=1$, a monodromy…

We revisit the results on admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements…

We give more or less explicit equations for all two-dimensional cusp singularities of embedding dimension at least 4. They are closely related to Felix Klein's equations for universal curves with level n structure. The main technical result…

We introduce a new method to study mixed characteristic deformation of line bundles. In particular, for sufficiently large smooth projective families $f : \mathscr{X} \to \mathscr{S}$ defined over the ring of $N$-integers…

We formulate a refined theory of linear systems, using the methods of a previous paper, "A Theory of Branches for Algebraic Curves", and use it to give a geometric interpretation of the genus of an algebraic curve. Using principles of…

We study the Generalized Fermat Equation $x^2 + y^3 = z^p$, to be solved in coprime integers, where $p \ge 7$ is prime. Using modularity and level lowering techniques, the problem can be reduced to the determination of the sets of rational…

In this paper, we classify the algebraic isomonodromic deformations that can be obtained through restriction to generic lines of logarithmic flat connections on the complex projective plane $\mathbb{P}^2_\mathbb{C}$ whose singular locus is…

The logarithmic connections studied in the paper are direct images of regular connections on line bundles over genus-2 double covers of the elliptic curve. We give an explicit parametrization of all such connections, determine their…

\[ \Delta u+g(u)=f(x) \s \mbox{for $x \in \Omega$}, \s u=0 \s \mbox{on $\partial \Omega$} \] decompose $f(x)=\mu _1 \p _1+e(x)$, where $\p _1$ is the principal eigenfunction of the Laplacian with zero boundary conditions, and $e(x) \perp \p…