Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A classification of coverings yielding Heun-to-hypergeometric reductions   Pull-back transformations between Heun and Gauss hypergeometric equations
give useful expressions of Heun functions in terms of better understood
hypergeometric functions. This article classifies, up to Mobius automorphisms,
the coverings P1-to-P1 that yield pull-back transformations from hypergeometric
to Heun equations with at least one free parameter (excluding the cases when
the involved hypergeometric equation has cyclic or dihedral monodromy). In all,
61 parametric hypergeometric-to-Heun transformations are found, of maximal
degree 12. Among them, 28 pull-backs are compositions of smaller degree
transformations between hypergeometric and Heun functions. The 61
transformations are realized by 48 different Belyi coverings (though 2
coverings should be counted twice as their moduli field is quadratic). The same
Belyi coverings appear in several other contexts. For example, 38 of the
coverings appear in Herfutner's list of elliptic surfaces over P1 with four
singular fibers, as their j-invariants. In passing, we demonstrate an elegant
way to show that there are no coverings P1-to-P1 with some branching patterns.

28
The categories of this paper are: alg-geom
29
The categories of this paper are: funct-an
34
The categories of this paper are: chem-ph
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
20
The categories of this paper are: acc-phys
33
The categories of this paper are: plasm-ph
16
The categories of this paper are: stat
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
12
The categories of this paper are: nucl-th