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Camilo Sanabria Malagón

Publications and source records attributed to Camilo Sanabria Malagón.

6 recordsLinked to original sources
arXiv2026

Schwarz maps for modular curves

We solve a classical problem posed by F. Klein and studied by A. Hurwitz concerning the construction of linear ordinary differential equations associated with modular transformations of fixed degree. For every odd integer $N\ge 3$ (respectively, even integer $N\ge 4$), we construct a canonical invariant model of the modular curve $X(N)=\mathbb{H}/Γ(N)$ (respectively, $X_H(N)=\mathbb{H}/H(N)$ where $H(N)=Γ(N)\capΓ_0^0(2N)$), together with a linear ordinary differential equation with rational coefficients whose Schwarz map parametrizes this model and whose projective monodromy group is the finite quotient $PSL_2(\mathbb{Z})/\tildeΓ(N)$ (respectively, $PSL_2(\mathbb{Z})/\tilde{H}(N)$). The construction is expressed in terms of invariant projective geometry and Picard-Vessiot theory and yields equations that are canonical up to projective equivalence. In this framework, Hurwitz's classical equation for degree $7$ appears as a special case of a general mechanism. The results place Klein's question within the modern theory of algebraic linear ordinary differential equations and provide a uniform geometric realization of modular transformation groups as projective differential Galois groups. As an application, we construct an explicit example of a linear ordinary differential equation associated with $X(9)$.

math.NT↗
arXiv2024

A note on the inverse problem for finite differential Galois groups

In this paper we revisit the following inverse problem: given a curve invariant under an irreducible finite linear algebraic group, can we construct an ordinary linear differential equation whose Schwarz map parametrizes it? We present an algorithmic solution to this problem under the assumption that we are given the function field of the quotient curve. The result provides a generalization and an efficient implementation of the solution to the inverse problem exposed by M. van der Put, C. Sanabria and J.Top [19]. As an application, we show that there is no hypergeometric equation with differential Galois group isomorphic to $H_{72}^{SL_3}$, thus completing Beuker and Heckman's answer [4] to the question of which irreducible finite subgroup of $SL_3(\mathbb{C})$ are the monodromy of a hypergeometric equation.

math.AG↗
arXiv2021

Solutions of algebraic linear ordinary differential equations

A classical result of F.Klein states that, given a finite primitive group $G\subseteq SL_2(\mathbb{C})$, there exists a hypergeometric equation such that any second order LODE whose differential Galois group is isomorphic to $G$ is projectively equivalent to the pullback by a rational map of this hypergeometric equation. In this paper, we generalize this result. We show that, given a finite primitive group $G\subseteq SL_n(\mathbb{C})$, there exist a positive integer $d=d(G)$ and a standard equation such that any LODE whose differential Galois group is isomorphic to $G$ is gauge equivalent, over a field extension $F$ of degree $d$, to an equation projectively equivalent to the pullback by a map in $F$ of this standard equation. For $n=3$, these standard equations can be chosen to be hypergeometric.

math.AG↗
arXiv2017

An elementary proof of the symplectic spectral theorem

The classical spectral theorem completely describes self-adjoint operators on finite dimensional inner product vector spaces as linear combinations of orthogonal projections onto pairwise orthogonal subspaces. We prove a similar theorem for self-adjoint operators on finite dimensional symplectic vector spaces over perfect fields. We show that these operators decompose according to a polarization, i.e. as the product of an operator on a lagrangian subspace and its dual on a complementary lagrangian. Moreover, if all the eigenvalues of the operator are in the base field, then there exist a Darboux basis such that the matrix representation of the operator is two-by-two blocks block-diagonal, where the first block is in Jordan normal form and the second block is the transpose of the first one.

math.RA↗
arXiv2016

Schwarz maps of Algebraic Linear Ordinary Differential Equations

A linear ordinary differential equation is called algebraic if all its solution are algebraic over its field of definition. In this paper we solve the problem of finding closed form solution to algebraic linear ordinary differential equations in terms of standard equations. Furthermore, we obtain a method to compute all algebraic linear ordinary differential equations with rational coefficients by studying their associated Schwarz map through the Picard-Vessiot Theory.

math.AG↗