Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Generalized Lam\'e equation with finite monodromy   In this paper, we study the algebraic form of the symmetric generalized
Lam\'e equations which have finite projective monodromy groups. In particular,
we consider equations with $3$ regular singular points on a flat torus $T$
which takes the form \begin{equation*} \begin{split} \frac{d^2
y}{dz^2}-\left[n_1(n_1+1)(\wp(z+a) +\wp(z-a))\right.\left. +A_1(\zeta(z+a) -
\zeta(z-a))+n_0(n_0+1) \wp(z)+B\right]y=0, \end{split} \end{equation*} where
$n_1, n_0 \in \Bbb R$, $A_1,B \in \Bbb C$, and $\wp$ is the Weierstrass
elliptic function.
  We give a complete list of all the group types that occur as the finite
projective monodromy groups on the algebraic form and give the corresponding
parameters $n_0$ and $n_1$. For equations with only $1$ or $2$ regular singular
points, we further determine their monodromy group types.
  The main tool used is the Grothendieck correspondence which gives a bijection
between Belyi pairs and dessin d'enfants. By Klein's theorem, we may regard
generalized Lam\'e equations with finite monodromy as pullbacks of the
hypergeometric equations. In this paper, we will restrict our cases with the
assumption that the pullback maps are Belyi functions. Under this setup, our
main results consist of a systematic construction on the required dessin. In
particular a gluing procedure of dessin will be developed to enable inductive
constructions of the dessin.

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The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an
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The categories of this paper are: astro-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-th
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The categories of this paper are: eess
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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: q-bio
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The categories of this paper are: chao-dyn
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The categories of this paper are: physics