2 Classical algebraic solutions of irregular Garnier systems

In this section, we describe one of the two methods we use to produce algebraic isomonodromic deformations, namely those that are characterized by their Galois group. It is convenient here to work with a holomorphic vector bundle $E$ equipped with a linear meromorphic connection $\unicode[STIX]{x1D6FB}$ instead of dealing with systems, as these latter ones can be viewed as connections on the trivial bundle, and some arguments need to deal with modified bundles. Also, most of the arguments and results remain valid for arbitrary curves $C$ instead of $\mathbb{P}^{1}$.

2.1 Classical solutions: the Galois group is $C_{\infty }$ or $D_{\infty }$

The rough idea is as follows. Since the differential Galois group of a linear differential equation can be determined from its coefficients by algebraic operations, it follows that iso-Galois deformations are of algebraic nature: we have an algebraic stratification of each moduli space ${\mathcal{S}}ys^{\unicode[STIX]{x1D6F4}}$ where strata are defined in terms of the Galois group. In fact, this is not exactly true since there might be infinitely many strata corresponding to finite groups in the dihedral case. However, the locus of each finite group is algebraic and coincides with a finite number of isomonodromic leaves. It follows that the leaf associated to a finite linear group is algebraic. This is the approach followed by Hitchin in [Reference HitchinHit95b] and Boalch in [Reference BoalchBoa06a, Reference BoalchBoa07a], where they considered deformations of Fuchsian systems with finite linear group, and therefore finite Galois group. Recall that in the Fuchsian case, the differential Galois group of the system is the Zariski closure of the monodromy group in $\text{SL}_{2}(\mathbb{C})$. However, in the irregular case, there are additional entries. The local monodromy at an irregular pole admits a decomposition

(5)$$\begin{eqnarray}\displaystyle & & \displaystyle \underbrace{\left(\begin{array}{@{}cc@{}}e^{i\unicode[STIX]{x1D70B}\unicode[STIX]{x1D703}} & 0\\ 0 & e^{-i\unicode[STIX]{x1D70B}\unicode[STIX]{x1D703}}\end{array}\right)}_{\text{formal monodromy}}\underbrace{\left(\begin{array}{@{}cc@{}}1 & s_{1}\\ 0 & 1\end{array}\right)\left(\begin{array}{@{}cc@{}}1 & 0\\ t_{1} & 1\end{array}\right)\cdots \left(\begin{array}{@{}cc@{}}1 & s_{k}\\ 0 & 1\end{array}\right)\left(\begin{array}{@{}cc@{}}1 & 0\\ t_{k} & 1\end{array}\right)}_{\text{Stokes matrices}}\quad (\text{unramified case}),\end{eqnarray}$$

(6)$$\begin{eqnarray}\displaystyle & & \displaystyle \underbrace{\left(\begin{array}{@{}cc@{}}0 & 1\\ -1 & 0\end{array}\right)}_{\text{formal monodromy}}\underbrace{\left(\begin{array}{@{}cc@{}}1 & s_{1}\\ 0 & 1\end{array}\right)\left(\begin{array}{@{}cc@{}}1 & 0\\ t_{1} & 1\end{array}\right)\cdots \left(\begin{array}{@{}cc@{}}1 & s_{k}\\ 0 & 1\end{array}\right)}_{\text{Stokes matrices}}\quad (\text{ramified case}),\end{eqnarray}$$

tracing back the local analytic structure of the pole. Then the differential Galois group contains all Stokes matrices as well as the ‘exponential torus’, which is the diagonal group in the normalization (5) and (6) (see [Reference Martinet and RamisMR90]). These observations give us the following possible differential Galois groups in the irregular case:

(7)$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}C_{\infty }=\bigg\{\!\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D706} & 0\\ 0 & \unicode[STIX]{x1D706}^{-1}\end{array}\right);\unicode[STIX]{x1D706}\in \mathbb{C}^{\ast }\bigg\},\\ D_{\infty }=\bigg\{\!\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D706} & 0\\ 0 & \unicode[STIX]{x1D706}^{-1}\end{array}\right),\left(\begin{array}{@{}cc@{}}0 & \unicode[STIX]{x1D706}\\ -\unicode[STIX]{x1D706}^{-1} & 0\end{array}\right);\unicode[STIX]{x1D706}\in \mathbb{C}^{\ast }\bigg\}=\langle C_{\infty },\left(\begin{array}{@{}cc@{}}0 & 1\\ -1 & 0\end{array}\right)\rangle ,\\ P_{\infty }=\bigg\{\!\left(\begin{array}{@{}cc@{}}1 & \unicode[STIX]{x1D707}\\ 0 & 1\end{array}\right);\unicode[STIX]{x1D707}\in \mathbb{C}\bigg\},\\ T_{\infty }=\bigg\{\!\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D706} & \unicode[STIX]{x1D707}\\ 0 & \unicode[STIX]{x1D706}^{-1}\end{array}\right);\unicode[STIX]{x1D706}\in \mathbb{C}^{\ast },\unicode[STIX]{x1D707}\in \mathbb{C}\bigg\},\\ \text{SL}_{2}(\mathbb{C}).\end{array}\right.\end{eqnarray}$$

As we shall see, Galois groups $P_{\infty }$ and $T_{\infty }$ do not occur in algebraic solutions of Garnier systems (see § 3). However, the locus of $C_{\infty }$ and $D_{\infty }$ is algebraic, and is a finite union of isomonodromy leaves in many cases when $C=\mathbb{P}^{1}$.

Let us first explain the diagonal case $C_{\infty }$. The poles can only be of types $\mathbf{Log}$ and $\mathbf{Irr}^{\mathbf{un}}$ and, in this latter case, the Galois group $C_{\infty }$ coincides with all local exponential torii at irregular singular points and all Stokes matrices are trivial. The two eigendirections of the Galois group correspond to two $\unicode[STIX]{x1D6FB}$-invariant line bundles $L,L^{-1}\subset E$ of the vector bundle for the normalized equation, and we have $E=L\oplus L^{-1}$. The connection $\unicode[STIX]{x1D6FB}$ restricts as meromorphic connections on $(L,\unicode[STIX]{x1D6FB}|_{L})$ and $(L,\unicode[STIX]{x1D6FB}|_{L})^{\otimes (-1)}$ and, at each pole $t_{i}$, the corresponding residues are $\pm \unicode[STIX]{x1D703}_{i}/2$: they are opposite for $L$ and $L^{-1}$. The Fuchs relation yields

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{i=1}^{n}\unicode[STIX]{x1D716}_{k}\unicode[STIX]{x1D703}_{k}\in \mathbb{Z},\quad \unicode[STIX]{x1D716}_{i}=\pm 1. & & \displaystyle \nonumber\end{eqnarray}$$

There are finitely many such relations for each formal data and, given one relation, the connection $(L,\unicode[STIX]{x1D6FB}|_{L})$ (and therefore $(E,\unicode[STIX]{x1D6FB})$) can be uniquely determined by Mittag–Leffler’s theorem from the data of the irregular curve ($C=\mathbb{P}^{1}$ + principal parts).

Something similar occurs when the Galois group is $D_{\infty }$. The poles must be of type $\mathbf{Log}$, $\mathbf{Irr}^{\mathbf{un}}$ or $\mathbf{Irr}^{\mathbf{ram}}$ and the normal subgroup $C_{\infty }\subset D_{\infty }$ must coincide with all local exponential torii at irregular singular points. In that case, Stokes matrices are trivial and, if $C=\mathbb{P}^{1}$, the global monodromy group is generated by matrices $M_{i}\in D_{\infty }$ (local monodromy at $t_{i}$) satisfying

$$\begin{eqnarray}\displaystyle M_{1}\cdots M_{n}=I. & & \displaystyle \nonumber\end{eqnarray}$$

Precisely, for a normalized equation, we have

$$\begin{eqnarray}\displaystyle \begin{array}{@{}lcl@{}}M_{i}~\text{diagonal} & \Rightarrow & \unicode[STIX]{x1D705}_{i}\in \mathbb{Z}_{{\geqslant}0},\\ M_{i}~\text{anti-diagonal} & \Rightarrow & \left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D705}_{i}\\ \unicode[STIX]{x1D703}_{i}\end{array}\right)=\left(\begin{array}{@{}c@{}}0\\ \frac{1}{2}\end{array}\right)~\text{or}~\left(\begin{array}{@{}c@{}}\frac{1}{2}+\unicode[STIX]{x1D708}\\ 0\end{array}\right),\quad \unicode[STIX]{x1D708}\geqslant 0.\end{array} & & \displaystyle \nonumber\end{eqnarray}$$

The number of anti-diagonal matrices among $M_{1},\ldots ,M_{n}$ is even.

Remark 5. One can check that, if $C=\mathbb{P}^{1}$ and there are only two anti-diagonal matrices among $M_{1},\ldots ,M_{n}$, say $M_{n-1}$ and $M_{n}$, then again the (algebraic) locus of $D_{\infty }$ consists in a finite number of isomonodromic leaves. Indeed, the monodromy representation is determined by the matrices $M_{i}$; for $i=1,\ldots ,n-2$, we have

$$\begin{eqnarray}\displaystyle M_{i}=\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D706}_{i} & 0\\ 0 & \unicode[STIX]{x1D706}_{i}^{-1}\end{array}\right)\quad \text{with}~\unicode[STIX]{x1D706}_{i}=e^{\pm \sqrt{-1}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D703}_{i}}; & & \displaystyle \nonumber\end{eqnarray}$$

we can rescale $M_{n}=\left(\!\begin{smallmatrix}0 & 1\\ -1 & 0\end{smallmatrix}\!\right)$ by diagonal conjugacy, and $M_{n-1}$ is determined by the relation. The connection is also determined by the irregular curve in this case (principal parts).

2.2 Garnier algebraic solutions and apparent singular points

So far, we have neglected to consider apparent singular points in isomonodromic deformations since we were dealing with normalized equations. However, Garnier systems are derived from isomonodromic deformations of scalar differential equations $d^{2}u/dx^{2}+f(x)(du/dx)+g(x)u=0$ with $N$ poles (counted with multiplicity) and $N-3$ apparent singular points (see [Reference KimuraKim89, Reference KawamukoKaw09]). This can be reinterpreted as the data of a connection $\unicode[STIX]{x1D6FB}$ on the bundle $E={\mathcal{O}}_{\mathbb{P}^{1}}\oplus {\mathcal{O}}_{\mathbb{P}^{1}}(-1)$ with $N$ poles, and we recover the scalar equation by taking ${\mathcal{O}}_{\mathbb{P}^{1}}\subset E$ as a cyclic vector. Equivalently, and closer to our point of view, one can consider a $\text{SL}_{2}$-connection $(E,\unicode[STIX]{x1D6FB})$ with $E={\mathcal{O}}_{\mathbb{P}^{1}}\oplus {\mathcal{O}}_{\mathbb{P}^{1}}$, and the cyclic vector (with the right number of apparent singular points) is chosen to be the constant line bundle $L\subset E$ fitting with one of the eigendirections over one pole (typically at $\infty$ when normalizing the position of poles on $\mathbb{P}^{1}$). Here, the poles are of type $\mathbf{Log}$, $\mathbf{Log}^{\mathbf{res}}$, $\mathbf{Irr}^{\mathbf{un}}$ or $\mathbf{Irr}^{\mathbf{ram}}$, but in the case $\mathbf{Log}$, we also allow $\unicode[STIX]{x1D703}\in \mathbb{Z}$ as a degenerate case of $\mathbf{Log}^{\mathbf{res}}$ when the singular point becomes apparent. These considerations lead us to the following facts.

Proposition 6. Let $(E={\mathcal{O}}_{\mathbb{P}^{1}}\oplus {\mathcal{O}}_{\mathbb{P}^{1}},\unicode[STIX]{x1D6FB})$ be an irregular $\text{SL}_{2}$-connection and $L\subset E$ be a cyclic vector like above. The (local) isomonodromic deformation of $(E,\unicode[STIX]{x1D6FB})$ provides a (local) solution of the Garnier system if the line bundle $L$ is not $\unicode[STIX]{x1D6FB}$-invariant. If $\text{Gal}(E,\unicode[STIX]{x1D6FB})=C_{\infty }$ and there is no apparent singular point, then $L$ is $\unicode[STIX]{x1D6FB}$-invariant and the deformation fails to provide a solution of the corresponding Garnier system.

Proof. The condition that $L$ is not $\unicode[STIX]{x1D6FB}$-invariant is equivalent to the fact that it defines a cyclic vector, which allows us to define the scalar system with $N$ poles and $N-3$ apparent singular points. This condition is preserved under isomonodromic deformations and the deformation of the scalar system leads to a Garnier solution. In the case where $(E,\unicode[STIX]{x1D6FB})$ is normalized (i.e. without apparent singular points) and the Galois group is $C_{\infty }$, then $E=L_{0}\oplus L_{0}^{-1}$ with $L_{0},L_{0}^{-1}$ two $\unicode[STIX]{x1D6FB}$-invariant line bundles. Automatically, we have $L_{0}\simeq {\mathcal{O}}_{\mathbb{P}^{1}}$ (a constant line bundle) and $L_{0}$ (respectively $L_{0}^{-1}$) coincides with an eigendirection over each pole. It follows that $L_{0}$ or $L_{0}^{-1}$ coincides with $L$ everywhere and $L$ is therefore $\unicode[STIX]{x1D6FB}$-invariant.◻

In the presence of apparent singular points, the condition for a deformation to be isomonodromic and giving rise to a Garnier solution is more subtle. Let $(E_{0},\unicode[STIX]{x1D6FB}_{0})$ be a meromorphic $\text{SL}_{2}$-connection, which is normalized except at an apparent singular point $p_{0}\in C$: it is of type $\mathbf{Log}$ with $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D708}\in \mathbb{Z}_{{>}0}$. Then, after a birational bundle transformation $\unicode[STIX]{x1D719}:E_{0}{\dashrightarrow}E_{0}^{\prime }$, we can erase the singular point; moreover, we can assume that $\unicode[STIX]{x1D719}$ is supported by $p_{0}$, i.e. inducing a biholomorphic bundle transformation outside of $p_{0}$. Generic local sections of $E_{0}$ at $p_{0}$ are transformed into sections of $E_{0}^{\prime }$, all tangent at the order $\unicode[STIX]{x1D708}-1$ to a given $\unicode[STIX]{x1D6FB}$-invariant analytic subbundle $L_{0}\subset E_{0}^{\prime }$. Then we have the following result.

Proposition 7. Under the notation above, given a deformation of $(E_{0},\unicode[STIX]{x1D6FB}_{0})$ induced by a flat connection $(E,\unicode[STIX]{x1D6FB})$, the following are equivalent:

• ∙ $(E,\unicode[STIX]{x1D6FB})$ is logarithmic near $p$;

• ∙ the deformation $p$ of the singular point $p_{0}$ remains apparent and the corresponding deformation of the line bundle $L_{0}\subset E_{0}^{\prime }$ is induced by a local analytic $\unicode[STIX]{x1D6FB}$-invariant line bundle $L\subset E^{\prime }$.

In this case we say that the deformation is isomonodromic.

Consequently, we will encode the data of such an equation $(E_{0},\unicode[STIX]{x1D6FB}_{0})$ by the data of the normalized equation $(E_{0}^{\prime },\unicode[STIX]{x1D6FB}_{0}^{\prime })$ together with the data of $p_{0}$ and the local $\unicode[STIX]{x1D6FB}_{0}^{\prime }$-invariant analytic line bundle $L_{0}$, or better by its fiber $l_{0}:=L_{0}|_{p_{0}}\subset E_{0}^{\prime }|_{p_{0}}$ (initial condition) that is usually called parabolic data. This method to deal with apparent singular points is used in [Reference Inaba, Iwasaki and SaitoIIS06].

Remark 8. The case $\unicode[STIX]{x1D708}=0$ also occurs in solutions of Garnier systems as a degenerate case of singular points of type $\mathbf{Log}^{\mathbf{res}}$. Garnier solutions can also be interpreted as before as deformation of a normalized equation $(E_{0},\unicode[STIX]{x1D6FB}_{0})$ together with the parabolic data $(p_{0}\in C,l_{0}\in E_{0}|_{p_{0}})$ and there are as many Garnier solutions as choices of initial condition $l_{0}$.

If we have several apparent singular points, the condition above must be imposed for each of them to get an isomonodromic solution, and therefore giving rise to Garnier solutions. By the way, we can have Garnier solutions corresponding to an equation with Galois group $C_{\infty }$ in the presence of apparent singular points. This is the case for the Laguerre polynomial solution of the Painlevé V equation (see [Reference Ohyama and OkumuraOO13]).

Proposition 9. The isomonodromic deformation of an irregular normalized connection $(E_{0},\unicode[STIX]{x1D6FB}_{0})$ with an apparent singular point $(p_{0},l_{0})$ (notation above) is algebraic if and only if:

• ∙ the deformation of $(E_{0},\unicode[STIX]{x1D6FB}_{0})$ is algebraic with Galois group $C_{\infty }$ or $D_{\infty }$;

• ∙ the local $\unicode[STIX]{x1D6FB}_{0}$-invariant analytic line bundle $L_{0}$ defined by $l_{0}$ has algebraic closure, i.e. corresponds to one of the two $\unicode[STIX]{x1D6FB}_{0}$ invariant line bundles in the diagonal case $C_{\infty }$, or of the two multivalued line bundles in the dihedral case $D_{\infty }$.

Proof. The global deformation space of the irregular curve including the apparent singular point may be viewed as a fiber bundle over the deformation space of the strict irregular curve, with fiber $C$ corresponding to the position of the apparent singular point for a fixed normalized equation. Clearly, the deformation of $(E_{0},\unicode[STIX]{x1D6FB}_{0})$ is algebraic if and only if it is algebraic in each of these two directions. But to be algebraic along $C$-fibers implies, due to Proposition 7, that the local analytic line bundle $L_{0}$ has algebraic closure. And, this implies that the Galois group must be $C_{\infty }$ or $D_{\infty }$ (the third possibility $\text{SL}_{2}$ in the irregular case is excluded here).◻

Proposition 9 can immediately be generalized to the case of several apparent singular points.

Remark 10. In the dihedral case $D_{\infty }$, an algebraic solution of a Garnier system with apparent singular point, like in Proposition 9, always arises in a family as a limit of algebraic solutions with linear monodromy $D_{\infty }$ and no apparent singular points like in § 2.1. Indeed, from the monodromy side, this can be written as

$$\begin{eqnarray}\displaystyle & \displaystyle M_{1}\cdots M_{n}=I\quad \text{with}~M_{n-2}=\left(\begin{array}{@{}cc@{}}0 & -\unicode[STIX]{x1D706}\\ \unicode[STIX]{x1D706}^{-1} & 0\end{array}\right),\quad M_{n-1}=\left(\begin{array}{@{}cc@{}}0 & 1\\ -1 & 0\end{array}\right), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \underbrace{M_{n}=\left(\begin{array}{@{}cc@{}}1 & 0\\ 0 & 1\end{array}\right)}_{\text{apparent}}{\rightsquigarrow}\tilde{M}_{n-2}=\left(\begin{array}{@{}cc@{}}0 & -\unicode[STIX]{x1D706}_{1}\\ \unicode[STIX]{x1D706}_{1}^{-1} & 0\end{array}\right),\quad \tilde{M}_{n-1}=\left(\begin{array}{@{}cc@{}}0 & 1\\ -1 & 0\end{array}\right),\quad \underbrace{\tilde{M}_{n}=\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D706}_{2} & 0\\ 0 & \unicode[STIX]{x1D706}_{2}^{-1}\end{array}\right)}_{\text{nonapparent}}, & \displaystyle \nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}=\unicode[STIX]{x1D706}$. For the differential equation, the apparent point is replaced by a nonapparent logarithmic singular point.

3 Structure theorem

The key result for our classification is the following theorem.

The following result is a direct consequence.

This result has been recently proved in the Painlevé case in [Reference Ohyama and OkumuraOO13] by checking a posteriori the known list of algebraic solutions. As we shall see, the result above also holds for isomonodromy equations for curves of genus $g>0$; however, we do not know the corresponding form of polynomial differential equation, i.e. higher genus Garnier systems. Some analytic form is described in [Reference KricheverKri02]; we knew from [Reference Inaba and SaitoIS13] that we should have an algebraic differential equation though.

To prove Corollary 12, we just notice that any solution (algebraic of not) comes from an isomonodromic deformation, or equivalently a flat connection over the total space $\boldsymbol{C}$ of the deformation curve. If the solution is algebraic, then $\boldsymbol{C}$ is algebraic, as well as the flat connection. We can therefore apply Theorem 11 and deduce Corollary 12.

4 Ramified covers and differential equations

Given a ramified cover $\unicode[STIX]{x1D719}:C\rightarrow C_{0}$ of degree $d$, and given a point $c_{0}\in C_{0}$, we can associate the pull-back divisor $\unicode[STIX]{x1D719}^{\ast }[c_{0}]=m_{1}[t_{1}]+\cdots +m_{s}[t_{s}]$ with $t_{i}\in C$ pair-wise distinct for $i=1,\ldots ,s$. This means that $\{t_{1},\ldots ,t_{s}\}$ is the fiber of $\unicode[STIX]{x1D719}$ and, through convenient local coordinates $x_{i}$ near $t_{i}$, $\unicode[STIX]{x1D719}(x_{i})=c_{0}+(x_{i})^{m_{i}}$; we have $m_{1}+\cdots +m_{s}=d$ and, for generic $c_{0}$, we have all $m_{i}=1$. We can therefore associate to $\unicode[STIX]{x1D719}$ its passport

(8)$$\begin{eqnarray}\left(\begin{array}{@{}ccc@{}}\left[\begin{array}{@{}c@{}}m_{1,1}\\ \vdots \\ m_{1,s_{1}}\end{array}\right] & \cdots \, & \left[\begin{array}{@{}c@{}}m_{\unicode[STIX]{x1D708},1}\\ \vdots \\ m_{\unicode[STIX]{x1D708},s_{\unicode[STIX]{x1D708}}}\end{array}\right]\end{array}\right),\quad \mathop{\sum }_{l=1}^{s_{k}}m_{k,l}=d\quad \text{for}~k=1,\ldots ,\unicode[STIX]{x1D708}.\end{eqnarray}$$

It is the data, for each critical value $c_{1},\ldots ,c_{\unicode[STIX]{x1D708}}\in C_{0}$ of $\unicode[STIX]{x1D719}$, of the corresponding partition $m_{k,1}+\cdots +m_{k,s_{k}}=d$: there are $s_{k}$ points in the fiber $\unicode[STIX]{x1D719}^{-1}(c_{k})$ with multiplicities $m_{k,l}$, $l=1,\ldots ,s_{k}$. The Riemann–Hurwitz formula writes

(9)$$\begin{eqnarray}\underbrace{2-2g(C)}_{\unicode[STIX]{x1D712}(C)}=d(\underbrace{2-2g(C_{0})}_{\unicode[STIX]{x1D712}(C_{0})})-R,\quad \text{where}~R=\text{total ramification}:=\mathop{\sum }_{k=1}^{n}(d-s_{k}).\end{eqnarray}$$

Given a differential equation $(E_{0},\unicode[STIX]{x1D6FB}_{0})$ with polar divisor $D$ on $C_{0}$, we can consider the pull-back $(E,\unicode[STIX]{x1D6FB}):=\unicode[STIX]{x1D719}^{\ast }(E_{0},\unicode[STIX]{x1D6FB}_{0})$. It is easy to deduce the local formal data of $(E,\unicode[STIX]{x1D6FB})$ from that one of $(E_{0},\unicode[STIX]{x1D6FB}_{0})$ and the passport. Precisely, let $p\in C$ be a point of multiplicity $m$ for $\unicode[STIX]{x1D719}$ (ramification $r=m-1$); then the local formal data of $(E_{0},\unicode[STIX]{x1D6FB}_{0})$ at $\unicode[STIX]{x1D719}(p)$ and $(E,\unicode[STIX]{x1D6FB})$ at $p$ are related by

$$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D705}\\ \unicode[STIX]{x1D703}\end{array}\right)=\left(\begin{array}{@{}c@{}}m\unicode[STIX]{x1D705}_{0}\\ m\unicode[STIX]{x1D703}_{0}\end{array}\right) & & \displaystyle \nonumber\end{eqnarray}$$

except when $\unicode[STIX]{x1D705}=0$, $\unicode[STIX]{x1D703}_{0}\in \mathbb{Q}\setminus \mathbb{Z}$ and $m\unicode[STIX]{x1D703}_{0}\in \mathbb{Z}$, where the singular point becomes apparent, i.e. can be deleted by bundle transformation; in that latter case, we indeed delete the pole. We deduce, for the respective order of poles, that

$$\begin{eqnarray}\displaystyle \operatorname{ord}_{p}(\unicode[STIX]{x1D6FB})\leqslant m\cdot \operatorname{ord}_{\unicode[STIX]{x1D719}(p)}(\unicode[STIX]{x1D6FB}_{0})+1-m & & \displaystyle \nonumber\end{eqnarray}$$

with strict inequality in the special case above, and the ramified case $\unicode[STIX]{x1D705}_{0}\not \in \mathbb{Z}$ and $m>1$. On the other hand, when $\unicode[STIX]{x1D719}(p)$ is not a pole for $\unicode[STIX]{x1D6FB}_{0}$, then $p$ is also nonsingular for $\unicode[STIX]{x1D6FB}$.

In the following, we still fix the differential equation $(C_{0},E_{0},\unicode[STIX]{x1D6FB}_{0})$ and we want to deform the ramified cover and pull-back equation

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{t}:C_{t}\rightarrow C_{0},\quad (E_{t},\unicode[STIX]{x1D6FB}_{t}):=\unicode[STIX]{x1D719}_{t}^{\ast }(E_{0},\unicode[STIX]{x1D6FB}_{0}). & & \displaystyle \nonumber\end{eqnarray}$$

We consider an irreducible algebraic family, parametrized by say $P\ni t$ (irreducible and projective), and there is a Zariski open subset $U\subset P$ where the passport of $\unicode[STIX]{x1D719}_{t}$ is locally constant, as well as the number $B$ of its critical values outside of the (fixed) polar locus of $\unicode[STIX]{x1D6FB}_{0}$. Note that $B$ bounds the dimension of $P$, and maybe switching to a larger family, i.e. with a larger parameter space $P$, we can assume that $B=\dim (P)$. The deformation $t\mapsto (C_{t},E_{t},\unicode[STIX]{x1D6FB}_{t})$, being automatically isomonodromic, locally factors through the universal isomonodromic deformation (see [Reference HeuHeu10]). In general, it defines a projective subvariety contained in the transcendental isomonodromy leaf ${\mathcal{L}}$ of the corresponding isomonodromy foliation ${\mathcal{F}}$. But, if $P\rightarrow {\mathcal{L}}$ is locally dominant, then it is globally dominant and the entire leaf ${\mathcal{L}}$ itself is algebraic, giving rise to an algebraic solution of the corresponding isomonodromy equation. A necessary condition for this is that the dimension $T$ of ${\mathcal{L}}$ is bounded by the dimension $B$ of $P$. We will call admissible the data of a differential equation $(C_{0},E_{0},\unicode[STIX]{x1D6FB}_{0})$ and a passport (8) such that $B\geqslant T$. In the following, it is convenient to add in the passport all trivial fibers

$$\begin{eqnarray}\displaystyle \left.\left[\begin{array}{@{}c@{}}1\\ \vdots \\ 1\end{array}\right]\right\}~d~\text{times} & & \displaystyle \nonumber\end{eqnarray}$$

appearing over poles, so that we may assume that $\unicode[STIX]{x1D708}=n+B$ in (8) with entries $k=1,\ldots ,n$ corresponding to fibers over the poles of $\unicode[STIX]{x1D6FB}_{0}$, and entries $k=n+1,\ldots ,\unicode[STIX]{x1D708}$ corresponding to free critical points that are deformed along the family. Our aim now is to classify those admissible data such that the pull-back differential equation has irregular Teichmüller dimension $T\leqslant B$. We will see in the next section that this inequality gives very strong constraints.

5 Scattering ramifications

Suppose that we are given a normalized differential equation $(C_{0},E_{0},\unicode[STIX]{x1D6FB}_{0})$ with poles $p_{1},\ldots ,p_{n}\in C_{0}$ and local formal data $(\unicode[STIX]{x1D705}_{i},\unicode[STIX]{x1D703}_{i})_{i=1,\ldots ,n}$ like (3). Suppose that we are given a ramified covering $\unicode[STIX]{x1D719}:C\rightarrow C_{0}$ with passport $(d=m_{k,1}+\cdots +m_{k,s_{k}})_{k=1,\ldots ,\unicode[STIX]{x1D708}}$ over $c_{1},\ldots ,c_{\unicode[STIX]{x1D708}}\in C_{0}$ like (8), where:

• ∙ $(m_{k,l})_{l}$ for $k=1,\ldots ,n$ correspond to the multiplicities of $\unicode[STIX]{x1D719}$ along fibers over the poles of $\unicode[STIX]{x1D6FB}_{0}$ ($c_{k}=p_{k}$), some of them being possibly unbranched, i.e. $m_{k,l}=1$ for all $l$;

• ∙ $(m_{k,l})_{l}$ for $k=n+1,\ldots ,n+b$ correspond to fibers of $\unicode[STIX]{x1D719}$ over nonsingular points of $\unicode[STIX]{x1D6FB}_{0}$, all of which are branching, i.e. $m_{k,l}>1$ for at least one $l$.

Denote by $R_{k}:=d-s_{k}$ the total ramification number of the fiber over $c_{k}$. We denote by $(C,E,\unicode[STIX]{x1D6FB})$ the normalized equation that can be deduced from the pull-back $\unicode[STIX]{x1D719}^{\ast }(E_{0},\unicode[STIX]{x1D6FB}_{0})$ by bundle transformation. Let $N_{k}$ denote the number of poles counted with multiplicity in the fiber over $c_{k}$. The irregular Teichmüller dimension of $(C,E,\unicode[STIX]{x1D6FB})$ is given by

$$\begin{eqnarray}\displaystyle T=3g-3+N=3g-3+N_{1}+\cdots +N_{n}, & & \displaystyle \nonumber\end{eqnarray}$$

where $g$ is the genus of $C$, given by the Riemann–Hurwitz formula (9) with $R:=\sum _{k=1}^{\unicode[STIX]{x1D708}}R_{k}$. We now explain how to compute $N_{k}$ by means of $(\unicode[STIX]{x1D705}_{k},\unicode[STIX]{x1D703}_{k})$ and $(m_{k,l})_{l}$.

• ∙ If $\unicode[STIX]{x1D705}_{k}=0$ and $\unicode[STIX]{x1D703}_{k}\not \in \mathbb{Q}\setminus \mathbb{Z}$, then

  $$\begin{eqnarray}\displaystyle N_{k}=s_{k}=d-R_{k}. & & \displaystyle \nonumber\end{eqnarray}$$

• ∙ If $\unicode[STIX]{x1D705}_{k}=0$ and $\unicode[STIX]{x1D703}_{k}$ has order $m>1$ modulo $\mathbb{Z}$, then

  $$\begin{eqnarray}\displaystyle N_{k}=d-R_{k}-\#\{l=1,\ldots ,s_{k}~|~m~\text{divides}~m_{k,l}\}. & & \displaystyle \nonumber\end{eqnarray}$$

• ∙ If $\unicode[STIX]{x1D705}_{k}\in \mathbb{Z}_{{>}0}$, then

  $$\begin{eqnarray}\displaystyle N_{k}=\mathop{\sum }_{l=1}^{s_{k}}(m_{k,l}+1)=d(\unicode[STIX]{x1D705}_{k}+1)-R_{k}. & & \displaystyle \nonumber\end{eqnarray}$$

• ∙ If $\unicode[STIX]{x1D705}_{k}\in \mathbb{Z}_{{>}0}-\frac{1}{2}$, then

  $$\begin{eqnarray}\displaystyle N_{k}=d(\unicode[STIX]{x1D705}_{k}+1)-R_{k}+\#\{l=1,\ldots ,s_{k}~|~2~\text{does not divide}~m_{k,l}\}. & & \displaystyle \nonumber\end{eqnarray}$$

We assume that $\unicode[STIX]{x1D719}$ admissible, i.e. that $T\leqslant B$.

In this section, we show that we can replace $\unicode[STIX]{x1D719}$ by another ramified cover $\unicode[STIX]{x1D719}^{\prime }$ with more critical points but less multiplicity in fibers in such a way that $T-B$ can only decrease. The total ramification will be unchanged, but will be scattered outside of the polar locus. This will allow us to replace the deformation of $\unicode[STIX]{x1D719}$ by the wider deformation of $\unicode[STIX]{x1D719}^{\prime }$, so that we will be able to recover $\unicode[STIX]{x1D719}$ (and its deformation) by confluence of critical values. By the way, the passport of $\unicode[STIX]{x1D719}^{\prime }$ will be as simple as possible and it will be easy to classify such covers.

The first step consists in scattering the branching points over critical values outside of the polar locus. We call simple branching a fiber of the form

$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}m_{k,1}\\ m_{k,2}\\ \vdots \\ m_{k,s_{k}}\end{array}\right]=\left[\begin{array}{@{}c@{}}2\\ 1\\ \vdots \\ 1\end{array}\right], & & \displaystyle \nonumber\end{eqnarray}$$

i.e. with $R_{k}=1$.

Proof. Fix a disc $\unicode[STIX]{x1D6E5}\subset C_{0}$ in which $c_{k}$ is the unique critical value of $\unicode[STIX]{x1D719}$. The monodromy of the ramified covering is given by the permutation that decomposes into the product of $s_{k}$ cyclic permutations of orders $m_{k,1},\ldots ,m_{k,s_{k}}$ with disjoint support. In order to construct $\unicode[STIX]{x1D719}^{\prime }$, it is enough to define its monodromy in $\unicode[STIX]{x1D6E5}$, namely the data of $R_{k}$ transpositions whose product is the monodromy of $\unicode[STIX]{x1D719}$. But it suffices to decompose each cyclic permutation of length $m_{k,l}$ above as the product of $m_{k,l}-1$ transpositions, which indeed gives

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{l=1}^{s_{k}}(m_{k,l}-1)=d-s_{k}=R_{k} & & \displaystyle \nonumber\end{eqnarray}$$

transpositions making the job. ◻

After applying Lemma (13) to each critical fiber of $\unicode[STIX]{x1D719}$ outside the polar locus, we can now assume that all fibers $\unicode[STIX]{x1D719}^{-1}(c_{k})$ are simple branching for $k>n$. In this way, we have maximized $B$ without touching fibers over poles of $\unicode[STIX]{x1D6FB}_{0}$ so far, so $N$ has not changed. We now discuss how to simplify fibers over poles of $\unicode[STIX]{x1D6FB}$ by putting some of their branch points out, in additional simple branching fibers, without increasing $N-B$; this will however increase $B$. We do this in successive lemmas discussing the type of poles.

The proof is the same as before.

Lemma 16. Let $0\leqslant k\leqslant n$, i.e. $c_{k}=p_{k}$ is a pole of $\unicode[STIX]{x1D6FB}$, and assume that $\unicode[STIX]{x1D705}_{k}=0$ and $\unicode[STIX]{x1D703}_{k}$ has order $m>1$ modulo $\mathbb{Z}$. We can deform $\unicode[STIX]{x1D719}{\rightsquigarrow}\unicode[STIX]{x1D719}^{\prime }$ over a neighborhood of $p_{k}$ so that the single fiber of $\unicode[STIX]{x1D719}$ with total ramification $R_{k}$ is replaced by the fiber with passport

$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}m_{k,1}^{\prime }\\ \vdots \\ m_{k,s_{k}^{\prime }}^{\prime }\end{array}\right]=\left[\begin{array}{@{}c@{}}m\\ \vdots \\ m\\ 1\\ \vdots \\ 1\end{array}\right] & & \displaystyle \nonumber\end{eqnarray}$$

and only simple branching fibers nearby. We have $N_{k}^{\prime }\geqslant N_{k}$, $B^{\prime }\geqslant B$ and $N_{k}^{\prime }-B^{\prime }\leqslant N_{k}-B$.

Proof. Here, we cannot just proceed as before. Indeed, if $m$ divides $m_{k,l}$, then there is no pole on the preimage (or an apparent one that disappears after normalization); however, replacing by $m_{k,l}$ nonbranching points would increase $N_{k}$ by $m_{k,l}$, but increase $B$ only by $m_{k,l}-1$, so that $N-B$ increases by 1. We thus have to take care of those points with $m$ dividing $m_{k,l}$.

Consider the euclidean division $m_{k,l}=s^{0}\cdot m+s^{1}$. Then we can replace the point with multiplicity $m_{k,l}$ by:

• ∙ $s^{0}$ points of multiplicity $m$ in the fiber $\unicode[STIX]{x1D719}^{-1}(p_{k})$ (contributing to no pole);

• ∙ $s^{1}$ nonbranching points in the fiber $\unicode[STIX]{x1D719}^{-1}(p_{k})$ (contributing to $s^{1}$ poles); and

• ∙ $s^{0}+s^{1}-1$ additional simple branching fibers around it.

To realize the deformation of $\unicode[STIX]{x1D719}$, we have to realize the corresponding monodromy representation, which is easy in this case. Indeed, the concatenation of a cyclic permutation (or a cycle inside a permutation) runs as follows:

$$\begin{eqnarray}\displaystyle \underbrace{(1\ldots \unicode[STIX]{x1D707})}_{\text{over}~p_{k}}=\underbrace{(1\ldots \unicode[STIX]{x1D707}^{\prime })(\unicode[STIX]{x1D707}^{\prime }+1\ldots \unicode[STIX]{x1D707})}_{\text{over}~p_{k}~\text{after concatenation}}\cdot \underbrace{(1,\unicode[STIX]{x1D707}^{\prime }+1)}_{\text{simple branching fiber}}, & & \displaystyle \nonumber\end{eqnarray}$$

where $1<\unicode[STIX]{x1D707}^{\prime }<\unicode[STIX]{x1D707}$. We just have to repeat this procedure $s_{0}+s_{1}-1$ times. By the way, we get $N_{k}^{\prime }=N_{k}+s^{1}-1$ (or $N_{k}^{\prime }=N_{k}=0$ if $m$ divides $m_{k,l}$) and $B^{\prime }=B+s^{0}+s^{1}-1$. We proceed similarly with all $m_{k,l}\not =m,1$ in the fiber.◻

Lemma 17. Let $0\leqslant k\leqslant n$, i.e. $c_{k}=p_{k}$ is a pole of $\unicode[STIX]{x1D6FB}$, and assume that $\unicode[STIX]{x1D705}_{k}\in \mathbb{Z}_{{>}0}-\frac{1}{2}$. We can deform $\unicode[STIX]{x1D719}{\rightsquigarrow}\unicode[STIX]{x1D719}^{\prime }$ over a neighborhood of $p_{k}$ so that the single fiber of $\unicode[STIX]{x1D719}$ with total ramification $R_{k}$ is replaced by the fiber with passport

$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}m_{k,1}^{\prime }\\ \vdots \\ m_{k,s_{k}^{\prime }}^{\prime }\end{array}\right]=\left[\begin{array}{@{}c@{}}2\\ \vdots \\ 2\\ 1\\ \vdots \\ 1\end{array}\right] & & \displaystyle \nonumber\end{eqnarray}$$

and only simple branching fibers nearby. We have $N_{k}^{\prime }-B^{\prime }=N_{k}-B$.

The proof is similar to that of Lemma 16. We call a scattered admissible covering (with respect to $(C_{0},E_{0},\unicode[STIX]{x1D6FB}_{0})$) an admissible covering $\unicode[STIX]{x1D719}$ satisfying the conclusion for $\unicode[STIX]{x1D719}^{\prime }$ in Lemmas 13–17. We assume from now on that $\unicode[STIX]{x1D719}$ is scattered. We now deform the differential equation $(E_{0},\unicode[STIX]{x1D6FB}_{0})$ on $C_{0}$ into a logarithmic one $(E_{0}^{\prime },\unicode[STIX]{x1D6FB}_{0}^{\prime })$ without changing the ramified cover, in such a way that $N-B$ does not increase. This will allow us to conclude with the classification established by the first author [Reference DiarraDia13] in the logarithmic case.

Proof. It is similar to the previous proofs: instead of dealing with the monodromy of the covering, we use the monodromy of the differential equation. We can write the monodromy $M$ of $\unicode[STIX]{x1D6FB}_{0}$ around $p_{k}$ as the product of $\unicode[STIX]{x1D705}_{k}+1$ nontrivial linear transformations:

$$\begin{eqnarray}\displaystyle M=M_{0}\cdot M_{1}\cdots M_{\unicode[STIX]{x1D705}_{k}},\quad M_{i}\in \text{SL}_{2}(\mathbb{C}),\quad M_{i}\not =\pm I. & & \displaystyle \nonumber\end{eqnarray}$$

By standard arguments à la Riemann–Hilbert, we can first realize these matrices as local monodromy of a differential equation over a disc with $(\unicode[STIX]{x1D705}_{k}+1)$ simple poles in normal form and total monodromy $M$. Next, by surgery over the disc, we replace the single pole $p_{k}$ in $(E_{0},\unicode[STIX]{x1D6FB}_{0})$ by this new differential equation and get $(E_{0}^{\prime },\unicode[STIX]{x1D6FB}_{0}^{\prime })$ with the desired properties.◻

6 Irregular Euler characteristic

Following Poincaré, we can associate, to a fuchsian differential equation, an orbifold structure on the base curve $C_{0}$ (see also [Reference DiarraDia13]) and get the notion of orbifold Euler characteristic. Here, we define an irregular version of it. Given $(C_{0},E_{0},\unicode[STIX]{x1D6FB}_{0})$ in normal form, we define the orbifold order $\unicode[STIX]{x1D708}_{k}$ at a logarithmic singular point as the order of the local monodromy:

• ∙ $\unicode[STIX]{x1D708}_{k}\in \mathbb{Z}_{{>}1}$ is the order of $[\unicode[STIX]{x1D703}_{k}~\hspace{0.6em}{\rm mod}\hspace{0.2em}~\mathbb{Z}]$ if $\unicode[STIX]{x1D703}_{k}\in \mathbb{Q}\setminus \mathbb{Z}$;

• ∙ $\unicode[STIX]{x1D708}_{k}=\infty$ if not.

We define the irregular Euler characteristic of the differential equation as

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}^{\text{irr}}(C_{0},E_{0},\unicode[STIX]{x1D6FB}_{0}):=2-2g_{0}-\mathop{\sum }_{k=1}^{n}(1+\unicode[STIX]{x1D705}_{k})+\mathop{\sum }_{\unicode[STIX]{x1D705}_{k}=0}\frac{1}{\unicode[STIX]{x1D708}_{k}}. & & \displaystyle \nonumber\end{eqnarray}$$

In the logarithmic case, this notion coincides with the orbifold Euler characteristic $\unicode[STIX]{x1D712}^{\text{irr}}=\unicode[STIX]{x1D712}^{\text{orb}}$. We note that the two operations of Lemmas 18 and 19, replacing $(E_{0},\unicode[STIX]{x1D6FB}_{0})$ by a logarithmic equation $(E_{0}^{\prime },\unicode[STIX]{x1D6FB}_{0}^{\prime })$, do not change the irregular Euler characteristic. Moreover, like as in [Reference DiarraDia13, Proposition 2.5], we have the following characterization.

Proof. In the logarithmic case, the curve $C_{0}$ with its orbifold structure is a finite quotient of the sphere or the torus. The differential equation lifts as a differential equation with trivial or abelian monodromy group, respectively. Since the Galois group is the Zariski closure of the monodromy group in the logarithmic case, we get that $\text{Gal}(E_{0},\unicode[STIX]{x1D6FB}_{0})$ is virtually abelian. In the irregular case, $\unicode[STIX]{x1D712}^{\text{irr}}\geqslant 0$ gives us

$$\begin{eqnarray}\displaystyle -\unicode[STIX]{x1D712}^{\text{irr}}=2g_{0}-2+\underbrace{\mathop{\sum }_{\unicode[STIX]{x1D705}_{k}=0}\biggl(1-\frac{1}{\unicode[STIX]{x1D708}_{k}}\biggr)}_{{\geqslant}0}+\underbrace{\mathop{\sum }_{\unicode[STIX]{x1D705}_{k}>0}(1+\unicode[STIX]{x1D705}_{k})}_{{\geqslant}1+1/2}\leqslant 0. & & \displaystyle \nonumber\end{eqnarray}$$

We promptly see that $g_{0}=0$, like as in the logarithmic case, and we have the following possible formal local data up to bundle transformation:

$$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D705}_{1}\cdots \unicode[STIX]{x1D705}_{n}\\ \unicode[STIX]{x1D703}_{1}\cdots \unicode[STIX]{x1D703}_{n}\end{array}\right)=\left(\begin{array}{@{}c@{}}1\\ 0\end{array}\right),\quad \left(\begin{array}{@{}c@{}}\frac{1}{2}\\ 0\end{array}\right)\quad \text{or}\quad \left(\begin{array}{@{}cc@{}}0 & \frac{1}{2}\\ \frac{1}{2} & 0\end{array}\right). & & \displaystyle \nonumber\end{eqnarray}$$

In the first case, we assume that we have an unramified irregular singular point, and the inequality for $\unicode[STIX]{x1D712}^{\text{irr}}$ gives no place for any other singular point; moreover, $\unicode[STIX]{x1D705}=1$. The monodromy around the unique singular point decomposes as

$$\begin{eqnarray}\displaystyle M=\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D706} & 0\\ 0 & \unicode[STIX]{x1D706}^{-1}\end{array}\right)\left(\begin{array}{@{}cc@{}}1 & s\\ 0 & 1\end{array}\right)\left(\begin{array}{@{}cc@{}}1 & 0\\ t & 1\end{array}\right)=\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D706}(1+st) & \unicode[STIX]{x1D706}s\\ \unicode[STIX]{x1D706}^{-1}t & \unicode[STIX]{x1D706}^{-1}\end{array}\right), & & \displaystyle \nonumber\end{eqnarray}$$

which must be trivial, implying that $\unicode[STIX]{x1D706}=e^{i\unicode[STIX]{x1D70B}\unicode[STIX]{x1D703}}=1$ and $s=t=0$. This means that $\unicode[STIX]{x1D703}\in \mathbb{Z}$, or equivalently $\unicode[STIX]{x1D703}=0$ after bundle transformation, and we have trivial Stokes matrices. The local (and therefore global) Galois group is diagonal, like the differential equation.

In the second and third cases, the irregular point is ramified and there is a place for a single logarithmic pole with orbifold order $\unicode[STIX]{x1D708}=2$. The local monodromy decomposes as

$$\begin{eqnarray}\displaystyle M=\left(\begin{array}{@{}cc@{}}0 & 1\\ -1 & 0\end{array}\right)\left(\begin{array}{@{}cc@{}}1 & s\\ 0 & 1\end{array}\right)=\left(\begin{array}{@{}cc@{}}0 & 1\\ -1 & -s\end{array}\right), & & \displaystyle \nonumber\end{eqnarray}$$

which is never the identity. The second case is therefore impossible. In the third case, $M$ is also the local monodromy at the logarithmic pole: the trace must be zero, $s=0$, implying trivial Stokes again.◻

Proposition 21. Let $(C_{0},E_{0},\unicode[STIX]{x1D6FB}_{0})$ be a normalized differential equation with local formal data $(\unicode[STIX]{x1D705}_{k},\unicode[STIX]{x1D703}_{k})_{k}$. Let $\unicode[STIX]{x1D719}:C\rightarrow C_{0}$ be a degree-$d$ ramified cover and $(E,\unicode[STIX]{x1D6FB})$ be the pull-back equation. Let $T$ be the dimension of the irregular Teichmüller deformation space for $(C,E,\unicode[STIX]{x1D6FB})$. Let $B$ be the number of critical values of $\unicode[STIX]{x1D719}$ outside the poles of $\unicode[STIX]{x1D6FB}_{0}$. Then we have

$$\begin{eqnarray}\displaystyle T-B\geqslant g-1-d\cdot \unicode[STIX]{x1D712}^{\text{irr}}(C_{0},E_{0},\unicode[STIX]{x1D6FB}_{0}), & & \displaystyle \nonumber\end{eqnarray}$$

where $g$ is the genus of $C$.

Proof. Let us decompose

$$\begin{eqnarray}\displaystyle N & = & \displaystyle N_{1}+\cdots +N_{n},\nonumber\\ \displaystyle R & = & \displaystyle R_{1}+\cdots +R_{n}+B,\nonumber\end{eqnarray}$$

where $N_{k}$ is the number of poles of $\unicode[STIX]{x1D6FB}$ (counted with multiplicity) along the fiber $\unicode[STIX]{x1D719}^{-1}(p_{k})$ and $R_{k}$ is the total ramification along $\unicode[STIX]{x1D719}^{-1}(p_{k})$. Now we have

$$\begin{eqnarray}\displaystyle T=3g-3+N=g-1+\underbrace{2g-2}_{=d(2g_{0}-2)+R}+N & & \displaystyle \nonumber\end{eqnarray}$$

(by Riemann–Hurwitz), which gives

$$\begin{eqnarray}\displaystyle T-B=g-1+d(2g_{0}-2)+\mathop{\sum }_{k=1}^{n}(N_{k}+R_{k}). & & \displaystyle \nonumber\end{eqnarray}$$

Let us lower bound $N_{k}+R_{k}$ in a function of the type of pole $p_{k}$ for $\unicode[STIX]{x1D6FB}_{0}$. We note that, along scattering in Lemmas 13–17, the value of $N_{k}+R_{k}$ can only decrease, so that it is enough to estimate a lower bound for a scattered covering $\unicode[STIX]{x1D719}$.

If $\unicode[STIX]{x1D705}_{k}=0$ and $\unicode[STIX]{x1D708}_{k}\in \mathbb{Z}_{{>}0}\cup \{\infty \}$ is the orbifold order, then we have

$$\begin{eqnarray}\displaystyle d=N_{k}+m\cdot \unicode[STIX]{x1D708}_{k}\quad \text{and}\quad R_{k}=m\cdot (\unicode[STIX]{x1D708}_{k}-1), & & \displaystyle \nonumber\end{eqnarray}$$

so that

$$\begin{eqnarray}\displaystyle N_{k}+R_{k}=N_{k}+\frac{d-N_{k}}{\unicode[STIX]{x1D708}_{k}}(\unicode[STIX]{x1D708}_{k}-1)=d\biggl(1-\frac{1}{\unicode[STIX]{x1D708}_{k}}\biggr)+\frac{N_{k}}{\unicode[STIX]{x1D708}_{k}}\geqslant d\biggl(1-\frac{1}{\unicode[STIX]{x1D708}_{k}}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$

If $\unicode[STIX]{x1D705}_{k}\in \mathbb{Z}_{{>}0}$, then we find (after scattering) that

$$\begin{eqnarray}\displaystyle N_{k}=d(1+\unicode[STIX]{x1D705}_{k})\quad \text{and}\quad R_{k}=0. & & \displaystyle \nonumber\end{eqnarray}$$

If $\unicode[STIX]{x1D705}_{k}\in \mathbb{Z}_{{>}0}-\frac{1}{2}$, then we have

$$\begin{eqnarray}\displaystyle d=m_{0}+2m_{1},\quad N_{k}=m_{0}(1+\unicode[STIX]{x1D705}_{k}+{\textstyle \frac{1}{2}})+m_{1}(1+2\unicode[STIX]{x1D705}_{k})\quad \text{and}\quad R_{k}=m_{1}, & & \displaystyle \nonumber\end{eqnarray}$$

so that $m_{1}=(d-m_{0})/2$ and, after substitution, we find that

$$\begin{eqnarray}\displaystyle N_{k}+R_{k}=d(1+\unicode[STIX]{x1D705}_{k})+\frac{m_{0}}{2}\geqslant d(1+\unicode[STIX]{x1D705}_{k}). & & \displaystyle \nonumber\end{eqnarray}$$

After summing for $k=1,\ldots ,n$, we find the expected lower bound.◻

Corollary 22. Under the assumptions of Proposition 21, if $\unicode[STIX]{x1D6FB}_{0}$ is irregular with nontrivial Stokes matrices and $T-B\leqslant 0$, then we have $\unicode[STIX]{x1D712}^{\text{irr}}<0$, $g_{0}=g=0$ and

$$\begin{eqnarray}\displaystyle d|\unicode[STIX]{x1D712}^{\text{irr}}|\leqslant 1. & & \displaystyle \nonumber\end{eqnarray}$$

Proof. The inequality $\unicode[STIX]{x1D712}^{\text{irr}}<0$ directly follows from Proposition 20 and the fact that we are assuming nontrivial Stokes matrices. Then Proposition 21 gives

$$\begin{eqnarray}\displaystyle 0\geqslant T-B\geqslant g-1+\underbrace{d|\unicode[STIX]{x1D712}^{\text{irr}}|}_{{>}0}, & & \displaystyle \nonumber\end{eqnarray}$$

which implies that $g=0$ (and therefore $g_{0}=0$). Then we deduce the expected inequality.◻

7 Classification of covers

In this section, we classify pull-back algebraic solutions of irregular Garnier systems (see § 2). In other words, we list all differential equations $(C_{0},E_{0},\unicode[STIX]{x1D6FB}_{0})$ and ramified coverings $\unicode[STIX]{x1D719}:C\rightarrow C_{0}$ such that, by deforming $\unicode[STIX]{x1D719}{\rightsquigarrow}\unicode[STIX]{x1D719}_{t}$, we get a complete isomonodromic deformation $\unicode[STIX]{x1D719}_{t}^{\ast }(E_{0},\unicode[STIX]{x1D6FB}_{0})$. In fact, we omit classical solutions that will be discussed in § 2.1 and therefore assume that $\unicode[STIX]{x1D712}^{\text{irr}}(C_{0},E_{0},\unicode[STIX]{x1D6FB}_{0})<0$ (see Proposition 20). Moreover, the equations $(E_{0},\unicode[STIX]{x1D6FB}_{0})$ are listed up to bundle transformation; in particular, we can assume without loss of generality that $(E_{0},\unicode[STIX]{x1D6FB}_{0})$ is a normalized equation. In the following, we use the notation of previous sections. In particular, $T$ is the dimension of the irregular Teichmüller space of the irregular curve given by $\unicode[STIX]{x1D719}^{\ast }(C_{0},E_{0},\unicode[STIX]{x1D6FB}_{0})$, and $B$ is the dimension of deformation of $\unicode[STIX]{x1D719}$, obtained by moving the critical values of $\unicode[STIX]{x1D719}$ outside the poles of $\unicode[STIX]{x1D6FB}_{0}$. To get a complete deformation, we need $T\leqslant B$ and we will assume that $T>0$ (otherwise there is no deformation).

Table 1. Logarithmic classification.

By confluence of the poles of the differential equation, we deduce the following result.

Table 2. Irregular classification with scattered cover.

Finally, by confluence of ramification fibers of $\unicode[STIX]{x1D719}$, we complete the list for pull-back solutions.

Table 3. Irregular classification with confluent cover.

8 Classification of classical solutions in the rank-2 case

In § 7, we have given a complete classification of algebraic solutions of irregular Garnier systems whose linear Galois group is $\text{SL}_{2}$. Indeed, we have classified those solutions of type (1) in Corollary 12. It does not make sense to do the same for solutions of type (2) or (3) in Corollary 12, since there are infinitely many, for arbitrary large rank $N$. However, for a given rank, it makes sense to classify, and we do this in this section for the case $N=2$, which is the first open case after the Painlevé equations. Recall that we have two solutions of type (1) in the case $N=2$, one for each formal data:

$$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}ccc@{}}0 & 1 & 1\\ \frac{1}{3} & 0 & 1\end{array}\right)\quad \text{and}\quad \left(\begin{array}{@{}cc@{}}1 & 2\\ 0 & 1\end{array}\right). & & \displaystyle \nonumber\end{eqnarray}$$

8.1 Without apparent singular point

They correspond to the type (2) of Corollary 12 and have Galois group $D_{\infty }$.

(10)$$\begin{eqnarray}\!\!\!\left\{\begin{array}{@{}lr@{}}\left(\begin{array}{@{}cccc@{}}0 & 0 & 0 & \frac{1}{2}\\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & 0\end{array}\right) & \text{infinite discrete family},\\ \left(\begin{array}{@{}cccc@{}}0 & 0 & 0 & 1\\ \frac{1}{2} & \frac{1}{2} & \unicode[STIX]{x1D703}_{1} & \unicode[STIX]{x1D703}_{2}\end{array}\right),\quad \left(\begin{array}{@{}cccc@{}}0 & \frac{1}{2} & 0 & 0\\ \frac{1}{2} & 0 & \unicode[STIX]{x1D703}_{1} & \unicode[STIX]{x1D703}_{2}\end{array}\right) & \text{two-parameter families},\\ \left(\begin{array}{@{}ccc@{}}0 & 0 & 2\\ \frac{1}{2} & \frac{1}{2} & \unicode[STIX]{x1D703}\end{array}\right),\quad \left(\begin{array}{@{}ccc@{}}0 & \frac{1}{2} & 1\\ \frac{1}{2} & 0 & \unicode[STIX]{x1D703}\end{array}\right),\quad \left(\begin{array}{@{}ccc@{}}\frac{1}{2} & \frac{1}{2} & 0\\ 0 & 0 & \unicode[STIX]{x1D703}\end{array}\right),\quad \left(\begin{array}{@{}ccc@{}}0 & \frac{3}{2} & 0\\ \frac{1}{2} & 0 & \unicode[STIX]{x1D703}\end{array}\right) & \text{one-parameter families},\\ \left(\begin{array}{@{}cc@{}}\frac{1}{2} & \frac{3}{2}\\ 0 & 0\end{array}\right)\quad \text{and}\quad \left(\begin{array}{@{}cc@{}}0 & \frac{5}{2}\\ \frac{1}{2} & 0\end{array}\right) & \text{sporadic solutions}.\end{array}\right.\quad\end{eqnarray}$$

These formal data correspond to those for which the dihedral group $D_{\infty }$ occurs as a Galois group of the linear equation, up to bundle transformation. In order to find this list, we have to take into account the following constraints:

• ∙ there are two or four poles where the local monodromy (or Galois group) is anti-diagonal, and the local formal type must be

  $$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}0\\ \frac{1}{2}\end{array}\right)\quad \text{or}\quad \left(\begin{array}{@{}c@{}}{\displaystyle \frac{k}{2}}\\ 0\end{array}\right)\quad \text{with}~k\in \mathbb{Z}_{{>}0}~\text{odd}; & & \displaystyle \nonumber\end{eqnarray}$$

• ∙ other poles are of local formal type

  $$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}k\\ \unicode[STIX]{x1D703}\end{array}\right)\quad \text{with}~k\in \mathbb{Z}_{{\geqslant}0},\,\unicode[STIX]{x1D703}\in \mathbb{C}. & & \displaystyle \nonumber\end{eqnarray}$$

We would like to insist that it is not necessary to consider particular values for $\unicode[STIX]{x1D703}$ as normalized equations with differential Galois group $D_{\infty }$ having poles with diagonal local monodromy occurring in a family where each exponent $\unicode[STIX]{x1D703}$ can be deformed arbitrarily. This comes from the fact that the monodromy representation itself can be deformed as well.

The first entry corresponds to the unique case with four poles having local anti-diagonal monodromy. It is an irregular version of the Picard–Painlevé equation (see [Reference MazzoccoMaz01b, Reference Loray, van der Put and UlmerLPU08]): there are infinitely many algebraic solutions, in bijection with the orbits of $\mathbb{Q}\times \mathbb{Q}$ under the standard action of $\text{SL}_{2}(\mathbb{Z})$. In fact, if $\tilde{C}\rightarrow C\simeq \mathbb{P}^{1}$ denotes the elliptic curve given by the 2-fold cover ramifying over the four poles of $(E,\unicode[STIX]{x1D6FB})$, then Picard solutions are related with torsion points on $\tilde{C}$ and how they vary when deforming the poles, and the curve $\tilde{C}$. Here, the story is the same. Indeed, the locus of the $D_{\infty }$ Galois group in the moduli space is closed algebraic, and a differential equation in this closed set has trivial Stokes matrices and is characterized by its monodromy representation.

For all other cases, recall that the differential equation with dihedral Galois group can be determined by means of the exponents $\unicode[STIX]{x1D703}_{i}$, as for its monodromy, once we know the irregular curve. Therefore, for each exponent $\unicode[STIX]{x1D703}_{i}$, we get exactly one algebraic solution.

8.2 With apparent singular point

They correspond to the type (3) of Corollary 12 and have Galois group $C_{\infty }$ or $D_{\infty }$. However, as noticed in Remark 10, algebraic solutions with Galois group $D_{\infty }$ and apparent singular points always arise as particular cases of more general algebraic solutions with Galois group $D_{\infty }$ and arbitrary singular points as listed in § 8.1. It just remains to complete the list with those algebraic solutions with Galois group $C_{\infty }$ and (at least one) apparent singular point:

(11)$$\begin{eqnarray}\left\{\begin{array}{@{}lr@{}}\left(\begin{array}{@{}cccc@{}}0 & 0 & 0 & 1\\ 0 & \unicode[STIX]{x1D703}_{1} & \unicode[STIX]{x1D703}_{2} & \unicode[STIX]{x1D703}_{3}\end{array}\right),\quad \unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{2}+\unicode[STIX]{x1D703}_{3}=0 & \text{two-parameter family},\\ \left(\begin{array}{@{}ccc@{}}0 & 0 & 2\\ 0 & \unicode[STIX]{x1D703} & -\unicode[STIX]{x1D703}\end{array}\right),\quad \left(\begin{array}{@{}ccc@{}}0 & 1 & 1\\ 0 & \unicode[STIX]{x1D703} & -\unicode[STIX]{x1D703}\end{array}\right) & \text{one-parameter families},\\ \left(\begin{array}{@{}cc@{}}0 & 3\\ 0 & 0\end{array}\right) & \text{sporadic solution}.\end{array}\right.\end{eqnarray}$$

For each value of the exponents $\unicode[STIX]{x1D703}_{i}$, there is exactly one normalized equation once the irregular curve is fixed, and therefore exactly one algebraic solution. Solutions with two or more apparent singular points arise as particular cases of these ones by specifying the exponents $\unicode[STIX]{x1D703}_{i}$.

9 Explicit Hamiltonians for some irregular Garnier systems

Here, we provide the linear differential equation and the Hamiltonians for some particular formal types (the complete list comes from [Reference KimuraKim89, Reference KawamukoKaw09]). We translate our notation with Kimura and Kawamuko’s.

9.1 $\text{Kim}_{4}(1,2,2)$

The general linear differential equation with (nonapparent) poles $x=0,1,\infty$ and corresponding formal type $\left(\!\begin{smallmatrix}2 & 2 & 1\\ \unicode[STIX]{x1D703}_{0} & \unicode[STIX]{x1D703}_{1} & \unicode[STIX]{x1D703}_{\infty }\end{smallmatrix}\!\right)$ can be normalized into the formFootnote ²

(12)$$\begin{eqnarray}L(1,2,2):\left\{\begin{array}{@{}l@{}}u^{\prime \prime }+f(x)u^{\prime }+g(x)u=0\quad \text{with}\\ f(x)={\displaystyle \frac{t_{2}}{x^{2}}}+{\displaystyle \frac{2-\unicode[STIX]{x1D703}_{0}}{x}}+{\displaystyle \frac{t_{1}}{(x-1)^{2}}}+{\displaystyle \frac{2-\unicode[STIX]{x1D703}_{1}}{x-1}}-\displaystyle \mathop{\sum }_{k=1,2}{\displaystyle \frac{1}{x-q_{k}}},\\ \displaystyle g(x)=\frac{(\unicode[STIX]{x1D703}_{0}+\unicode[STIX]{x1D703}_{1}-1)^{2}-\unicode[STIX]{x1D703}_{\infty }^{2}}{4x(x-1)}-\frac{t_{1}H_{1}}{x(x-1)^{2}}+\frac{t_{2}H_{2}}{x^{2}(x-1)}+\mathop{\sum }_{k=1,2}\frac{q_{k}(q_{k}-1)p_{k}}{x(x-1)(x-q_{k})}.\end{array}\right.\end{eqnarray}$$

Singular points $x=q_{1},q_{2}$ are apparent if and only if the coefficients $H_{1},H_{2}$ are given by

(13)$$\begin{eqnarray}H(1,2,2):\left\{\begin{array}{@{}rcl@{}}H_{1}\, & =\, & \displaystyle -\frac{q_{1}^{2}(q_{1}-1)^{2}(q_{2}-1)}{t_{1}(q_{1}-q_{2})}\biggl(p_{1}^{2}-\biggl(\frac{\unicode[STIX]{x1D703}_{0}}{q_{1}}-\frac{t_{2}}{q_{1}^{2}}+\frac{\unicode[STIX]{x1D703}_{1}-1}{q_{1}-1}-\frac{t_{1}}{(q_{1}-1)^{2}}\biggr)p_{1}+\frac{(\unicode[STIX]{x1D703}_{0}+\unicode[STIX]{x1D703}_{1}-1)^{2}-\unicode[STIX]{x1D703}_{\infty }^{2}}{4q_{1}(q_{1}-1)}\biggr)\\ \, & \, & \displaystyle +\,\frac{(q_{1}-1)q_{2}^{2}(q_{2}-1)^{2}}{t_{1}(q_{1}-q_{2})}\biggl(p_{2}^{2}-\biggl(\frac{\unicode[STIX]{x1D703}_{0}}{q_{2}}-\frac{t_{2}}{q_{2}^{2}}+\frac{\unicode[STIX]{x1D703}_{1}-1}{q_{2}-1}-\frac{t_{1}}{(q_{2}-1)^{2}}\biggr)p_{1}+\frac{(\unicode[STIX]{x1D703}_{0}+\unicode[STIX]{x1D703}_{1}-1)^{2}-\unicode[STIX]{x1D703}_{\infty }^{2}}{4q_{2}(q_{2}-1)}\biggr),\\ H_{2}\, & =\, & \displaystyle -\frac{q_{1}^{2}(q_{1}-1)^{2}q_{2}}{t_{2}(q_{1}-q_{2})}\biggl(p_{1}^{2}-\biggl(\frac{\unicode[STIX]{x1D703}_{0}-1}{q_{1}}-\frac{t_{2}}{q_{1}^{2}}+\frac{\unicode[STIX]{x1D703}_{1}}{q_{1}-1}-\frac{t_{1}}{(q_{1}-1)^{2}}\biggr)p_{1}+\frac{(\unicode[STIX]{x1D703}_{0}+\unicode[STIX]{x1D703}_{1}-1)^{2}-\unicode[STIX]{x1D703}_{\infty }^{2}}{4q_{1}(q_{1}-1)}\biggr)\\ \, & \, & \displaystyle +\,\frac{q_{1}q_{2}^{2}(q_{2}-1)^{2}}{t_{2}(q_{1}-q_{2})}\biggl(p_{2}^{2}-\biggl(\frac{\unicode[STIX]{x1D703}_{0}-1}{q_{2}}-\frac{t_{2}}{q_{2}^{2}}+\frac{\unicode[STIX]{x1D703}_{1}}{q_{2}-1}-\frac{t_{1}}{(q_{2}-1)^{2}}\biggr)p_{1}+\frac{(\unicode[STIX]{x1D703}_{0}+\unicode[STIX]{x1D703}_{1}-1)^{2}-\unicode[STIX]{x1D703}_{\infty }^{2}}{4q_{2}(q_{2}-1)}\biggr).\end{array}\right.\qquad\end{eqnarray}$$

A deformation of (12) is isomonodromic if and only if the parameters satisfy the Hamiltonian system (4),

(14)$$\begin{eqnarray}\frac{dq_{j}}{dt_{i}}=\frac{\unicode[STIX]{x2202}H_{i}}{\unicode[STIX]{x2202}p_{j}}\quad \text{and}\quad \frac{dp_{j}}{dt_{i}}=-\frac{\unicode[STIX]{x2202}H_{i}}{\unicode[STIX]{x2202}q_{j}}\quad \forall i,j=1,2.\end{eqnarray}$$

To construct the pull-back solution (second line of Table 2), we start with the differential equation

$$\begin{eqnarray}\displaystyle \frac{d^{2}u}{dz^{2}}+\frac{2}{3z}\frac{du}{dz}-\frac{1}{z}u=0 & & \displaystyle \nonumber\end{eqnarray}$$

and consider its pull-back by the branch cover

$$\begin{eqnarray}\displaystyle z=\unicode[STIX]{x1D719}(x)=\frac{t_{2}^{2}(2x-4q_{1}x+q_{1}^{2}+q_{1})^{3}}{16q_{1}^{3}(q_{1}+1)^{3}x^{2}(x-1)^{2}}. & & \displaystyle \nonumber\end{eqnarray}$$

Comparing with (12), we get the first solution of Theorem 2.

9.2 $\text{Kim}_{6}(2,3)$

The general linear differential equation with (nonapparent) poles $x=0,\infty$ and corresponding formal type $\left(\!\begin{smallmatrix}2 & 3\\ \unicode[STIX]{x1D703}_{0} & \unicode[STIX]{x1D703}_{\infty }\end{smallmatrix}\!\right)$ can be normalized into the formFootnote ³

(15)$$\begin{eqnarray}L(2,3):\left\{\begin{array}{@{}l@{}}u^{\prime \prime }+f(x)u^{\prime }+g(x)u=0\quad \text{with}\\ \displaystyle f(x)=\frac{t_{2}}{x^{2}}+\frac{2-\unicode[STIX]{x1D703}_{0}}{x}-t_{1}-\frac{x}{2}-\mathop{\sum }_{k=1,2}\frac{1}{x-q_{k}},\\ \displaystyle g(x)=\frac{\unicode[STIX]{x1D703}_{0}+\unicode[STIX]{x1D703}_{\infty }-1}{8}-\frac{H_{1}}{2x}+\frac{t_{2}H_{2}}{x^{2}}+\mathop{\sum }_{k=1,2}\frac{q_{k}(q_{k}-1)p_{k}}{x(x-1)(x-q_{k})}.\end{array}\right.\end{eqnarray}$$

Singular points $x=q_{1},q_{2}$ are apparent if and only if the coefficients $H_{1},H_{2}$ are given by

(16)$$\begin{eqnarray}H(2,3):\left\{\begin{array}{@{}rcl@{}}H_{1}\ & =\ & \displaystyle \frac{2q_{1}^{2}}{q_{1}-q_{2}}\biggl(p_{1}^{2}-\biggl(\frac{\unicode[STIX]{x1D703}_{0}}{q_{1}}-\frac{t_{2}}{q_{1}^{2}}+\frac{q_{1}}{2}+t_{1}\biggr)p_{1}+\frac{\unicode[STIX]{x1D703}_{0}+\unicode[STIX]{x1D703}_{\infty }-1}{8}\biggr)\\ \ & \ & \displaystyle -\,\frac{2q_{2}^{2}}{q_{1}-q_{2}}\biggl(p_{2}^{2}-\biggl(\frac{\unicode[STIX]{x1D703}_{0}}{q_{2}}-\frac{t_{2}}{q_{2}^{2}}+\frac{q_{2}}{2}+t_{1}\biggr)p_{2}+\frac{\unicode[STIX]{x1D703}_{0}+\unicode[STIX]{x1D703}_{\infty }-1}{8}\biggr),\\ H_{2}\ & =\ & \displaystyle -\frac{q_{1}^{2}q_{2}}{t_{2}(q_{1}-q_{2})}\biggl(p_{1}^{2}-\biggl(\frac{\unicode[STIX]{x1D703}_{0}-1}{q_{1}}-\frac{t_{2}}{q_{1}^{2}}+\frac{q_{1}}{2}+t_{1}\biggr)p_{1}+\frac{\unicode[STIX]{x1D703}_{0}+\unicode[STIX]{x1D703}_{\infty }-1}{8}\biggr)\\ \ & \ & \displaystyle +\,\frac{q_{1}q_{2}^{2}}{t_{2}(q_{1}-q_{2})}\biggl(p_{2}^{2}-\biggl(\frac{\unicode[STIX]{x1D703}_{0}-1}{q_{2}}-\frac{t_{2}}{q_{2}^{2}}+\frac{q_{2}}{2}+t_{1}\biggr)p_{2}+\frac{\unicode[STIX]{x1D703}_{0}+\unicode[STIX]{x1D703}_{\infty }-1}{8}\biggr)\end{array}\right.\end{eqnarray}$$

and isomonodromic deformations are defined by (14). To construct the pull-back solution (first line of Table 3), we start with the differential equation

$$\begin{eqnarray}\displaystyle \frac{d^{2}u}{dz^{2}}+\frac{2}{3z}\frac{du}{dz}-\frac{1}{z}u=0 & & \displaystyle \nonumber\end{eqnarray}$$

and consider its pull-back by the branch cover

$$\begin{eqnarray}\displaystyle z=\unicode[STIX]{x1D719}(x)=\frac{(3x^{2}+8t_{1}x+4q_{1}t_{1}+6q_{1}^{2})^{3}}{6912x^{2}}. & & \displaystyle \nonumber\end{eqnarray}$$

Comparing with (15), we get the second solution of Theorem 2.

9.3 $\text{Kaw}_{4}(5/2,3/2)$

The general linear differential equation with (nonapparent) poles $x=0,\infty$ and corresponding formal type $\left(\!\begin{smallmatrix}\frac{3}{2} & \frac{1}{2}\\ 0 & 0\end{smallmatrix}\!\right)$ can be normalized into the formFootnote ⁴ $u^{\prime \prime }=g(x)u$ with

(17)$$\begin{eqnarray}g(x)=\frac{t_{2}^{2}}{4x^{3}}+\frac{H_{1}}{x^{2}}+\frac{H_{2}}{x}+\frac{t_{1}}{2}+\frac{x}{4}+\mathop{\sum }_{k=1,2}\biggl(\frac{3}{4(x-q_{k})^{2}}-\frac{p_{k}}{x-q_{k}}\biggr).\end{eqnarray}$$

Then set

$$\begin{eqnarray}\displaystyle & \displaystyle u_{1}=q_{1}+q_{2},\quad u_{2}=q_{1}q_{2},\quad v_{1}=\frac{q_{1}+q_{2}}{2(q_{1}-q_{2})^{2}}+\frac{p_{1}q_{1}-p_{2}q_{2}}{q_{1}-q_{2}} & \displaystyle \nonumber\\ \displaystyle & \displaystyle v_{2}=-\frac{1}{(q_{1}-q_{2})^{2}}-\frac{p_{1}-p_{2}}{q_{1}-q_{2}} & \displaystyle \nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle K_{1}=2u_{1}v_{1}^{2}+4u_{2}v_{1}v_{2}-4v_{1}-\frac{u_{1}^{2}}{2}-t_{1}u_{1}+\frac{u_{2}}{2}+\frac{t_{2}^{2}}{2u_{2}},\\ \displaystyle t_{2}K_{2}=-2u_{2}v_{1}^{2}+2u_{2}^{2}v_{2}^{2}-2u_{2}v_{2}+\frac{u_{1}u_{2}}{2}+t_{1}u_{2}-\frac{t_{2}^{2}u_{1}}{2u_{2}}.\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$

Deformation of (17) is isomonodromic if and only if

$$\begin{eqnarray}\displaystyle \frac{du_{j}}{dt_{i}}=\frac{\unicode[STIX]{x2202}K_{i}}{\unicode[STIX]{x2202}v_{j}}\quad \text{and}\quad \frac{dv_{j}}{dt_{i}}=-\frac{\unicode[STIX]{x2202}K_{i}}{\unicode[STIX]{x2202}u_{j}}\quad \forall i,j=1,2. & & \displaystyle \nonumber\end{eqnarray}$$

The first classical sporadic solution of Theorem 3, with dihedral linear Galois group, can be constructed by pulling back the differential equation $d^{2}u/dz^{2}=(1/z-3/16x^{2})u$ by the ramified covering

$$\begin{eqnarray}\displaystyle z=\unicode[STIX]{x1D719}(x)=\frac{(x^{2}+3t_{1}x-3t_{2})^{2}}{36x}; & & \displaystyle \nonumber\end{eqnarray}$$

after normalizing, and comparing with (17), we get the rational solution

$$\begin{eqnarray}\displaystyle (t_{1},t_{2})\mapsto (u_{1},u_{2},v_{1},v_{2}):=\biggl(-t_{1},t_{2},0,\frac{3}{4t_{2}}\biggr). & & \displaystyle \nonumber\end{eqnarray}$$