the Astrophysics Science Division at NASA/GSFC. This work made use of data
supplied by the UK Swift Science Data Centre at the University of Leicester
(Evans et al. 2007, 2009). Part of this work is based on archival data,
software or online services provided by the Space Science Data Center - ASI.
This work was supported in part by NASA through the NICER mission and the
Astrophysics Explorers Program. NICER data used in this work were gathered
under a Guest Observer (GO) approved programme. This research has made use of
the NASA/IPAC Extragalactic Database (NED), which is funded by the National
Aeronautics and Space Administration and operated by the California Institute
of Technology. The Skynet Robotic Telescope Network is supported by the
National Science Foundation, the Department of Defense, the North Carolina
Space Grant Consortium, and the Mount Cuba Astronomical Foundation. This
publication makes use of data products from the Wide-field Infrared Survey
Explorer, which is a joint project of the University of California, Los
Angeles, and the Jet Propulsion Laboratory/California Institute of Technology,
funded by the National Aeronautics and Space Administration. This publication
also makes use of data products from NEOWISE, which is a project of the Jet
Propulsion Laboratory/California Institute of Technology, funded by the
Planetary Science Division of the National Aeronautics and Space
Administration. This research has made use of ISIS functions (ISISscripts)
provided by ECAP/Remeis observatory and MIT (http://www.sternwarte.uni-
erlangen.de/isis/).

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##  Appendix A Optical/UV continuum photometric data tables

Table 10: Optical/UV continuum magnitudes from Swift-UVOT and XMM-Newton-OM

Obs. | Date | Filter | Flux Dens. | Flux Dens. | Flux Dens.
---|---|---|---|---|---
| (MJD) |  | (Vega mag) | (mJy) | (erg cm-2 s-1 Å-1)
Sw1 | 59130.95 | B | $16.39\pm 0.03$ | $1.16\pm 0.03$ | $(1.82\pm 0.05)\times 10^{-15}$
Sw2 | 59137.49 | B | $16.34\pm 0.03$ | $1.22\pm 0.03$ | $(1.91\pm 0.05)\times 10^{-15}$
XM1 | 59209.82 | M2 | $14.721\pm 0.003$ | $1.108\pm 0.003$ | $(6.229\pm 0.014)\times 10^{-15}$
XM2 | 59242.45 | M2 | $14.644\pm 0.003$ | $1.111\pm 0.004$ | $(6.254\pm 0.016)\times 10^{-15}$
Sw3 | 59391.00 | V | $16.14\pm 0.03$ | $1.31\pm 0.04$ | $(1.34\pm 0.04)\times 10^{-15}$
|  | B | $16.24\pm 0.04$ | $1.34\pm 0.04$ | $(2.14\pm 0.07)\times 10^{-15}$
|  | U | $15.11\pm 0.04$ | $1.36\pm 0.05$ | $(3.32\pm 0.11)\times 10^{-15}$
|  | W1 | $14.85\pm 0.04$ | $1.08\pm 0.04$ | $(4.74\pm 0.19)\times 10^{-15}$
|  | M2 | $14.67\pm 0.04$ | $1.12\pm 0.04$ | $(6.77\pm 0.26)\times 10^{-15}$
|  | W2 | $14.71\pm 0.04$ | $1.03\pm 0.03$ | $(7.15\pm 0.25)\times 10^{-15}$
XM3 | 59465.41 | M2 | $14.731\pm 0.004$ | $1.097\pm 0.004$ | $(6.169\pm 0.020)\times 10^{-15}$
Sw4 | 59662.59 | V | $16.16\pm 0.04$ | $1.28\pm 0.05$ | $(1.32\pm 0.05)\times 10^{-15}$
|  | B | $16.20\pm 0.04$ | $1.39\pm 0.05$ | $(2.22\pm 0.07)\times 10^{-15}$
|  | U | $15.05\pm 0.04$ | $1.43\pm 0.05$ | $(3.50\pm 0.12)\times 10^{-15}$
|  | W1 | $14.86\pm 0.04$ | $1.07\pm 0.04$ | $(4.69\pm 0.16)\times 10^{-15}$
|  | M2 | $14.71\pm 0.04$ | $1.08\pm 0.04$ | $(6.58\pm 0.26)\times 10^{-15}$
|  | W2 | $14.76\pm 0.04$ | $0.99\pm 0.03$ | $(7.18\pm 0.24)\times 10^{-15}$
XM4 | 59694.64 | B | $16.38\pm 0.36$ | $1.24\pm 0.27$ | $(1.84\pm 0.60)\times 10^{-15}$
| 59695.04 | M2 | $14.793\pm 0.004$ | $1.036\pm 0.004$ | $(5.827\pm 0.020)\times 10^{-15}$
Sw5 | 59733.06 | V | $16.18\pm 0.06$ | $1.26\pm 0.07$ | $(1.29\pm 0.07)\times 10^{-15}$
|  | B | $16.25\pm 0.04$ | $1.32\pm 0.05$ | $(2.12\pm 0.09)\times 10^{-15}$
|  | U | $15.17\pm 0.04$ | $1.28\pm 0.05$ | $(3.14\pm 0.13)\times 10^{-15}$
|  | W1 | $14.99\pm 0.04$ | $0.95\pm 0.04$ | $(4.16\pm 0.16)\times 10^{-15}$
|  | M2 | $14.92\pm 0.04$ | $0.89\pm 0.04$ | $(5.38\pm 0.21)\times 10^{-15}$
|  | W2 | $15.01\pm 0.04$ | $0.79\pm 0.03$ | $(5.70\pm 0.22)\times 10^{-15}$
Sw6 | 59783.49 | V | $16.27\pm 0.05$ | $1.16\pm 0.05$ | $(1.19\pm 0.06)\times 10^{-15}$
|  | B | $16.34\pm 0.04$ | $1.22\pm 0.04$ | $(1.95\pm 0.07)\times 10^{-15}$
|  | U | $15.27\pm 0.04$ | $1.17\pm 0.04$ | $(2.86\pm 0.10)\times 10^{-15}$
|  | W1 | $15.11\pm 0.04$ | $0.85\pm 0.03$ | $(3.73\pm 0.15)\times 10^{-15}$
|  | M2 | $14.94\pm 0.04$ | $0.88\pm 0.03$ | $(5.28\pm 0.21)\times 10^{-15}$
|  | W2 | $15.09\pm 0.04$ | $0.73\pm 0.03$ | $(5.30\pm 0.21)\times 10^{-15}$
Sw7 | 59873.53 | V | $16.24\pm 0.04$ | $1.19\pm 0.05$ | $(1.22\pm 0.05)\times 10^{-15}$
|  | B | $16.35\pm 0.04$ | $1.21\pm 0.04$ | $(1.93\pm 0.06)\times 10^{-15}$
|  | U | $15.24\pm 0.04$ | $1.20\pm 0.04$ | $(2.94\pm 0.10)\times 10^{-15}$
|  | W1 | $15.07\pm 0.04$ | $0.88\pm 0.03$ | $(3.87\pm 0.15)\times 10^{-15}$
|  | M2 | $14.95\pm 0.04$ | $0.87\pm 0.03$ | $(5.23\pm 0.20)\times 10^{-15}$
|  | W2 | $15.07\pm 0.04$ | $0.74\pm 0.03$ | $(5.39\pm 0.18)\times 10^{-15}$
Sw8 | 59992.20 | V | $16.04\pm 0.04$ | $1.42\pm 0.05$ | $(1.47\pm 0.06)\times 10^{-15}$
|  | B | $16.14\pm 0.04$ | $1.46\pm 0.05$ | $(2.34\pm 0.08)\times 10^{-15}$
|  | U | $14.99\pm 0.04$ | $1.51\pm 0.05$ | $(3.70\pm 0.12)\times 10^{-15}$
|  | W1 | $14.76\pm 0.04$ | $1.17\pm 0.04$ | $(5.15\pm 0.17)\times 10^{-15}$
|  | M2 | $14.60\pm 0.04$ | $1.20\pm 0.05$ | $(7.23\pm 0.28)\times 10^{-15}$
|  | W2 | $14.63\pm 0.04$ | $1.11\pm 0.04$ | $(8.09\pm 0.27)\times 10^{-15}$
262626All magnitudes are Vega magnitudes, and are observed magnitudes, i.e.,
not corrected for Galactic absorption. In contrast, all flux densities are
corrected for Galactic absorption. Uncertainties are statistical only (see
Appendix B).

Table 11: B- and V-band photometric observations at PROMPT6

Date | Filter | Expo. | Obsd.
---|---|---|---
(MJD) |  | (s) | Mag.
|  |  | (Vega)
59164.06 | B | 1030 | $16.42\pm 0.02$
59174.08 | B | 1224 | $16.48\pm 0.02$
59180.16 | B | 1350 | $16.42\pm 0.02$
59182.01 | B | 1350 | $16.40\pm 0.03$
59202.04 | B | 1251 | $16.37\pm 0.02$
59205.35 | B | 1251 | $16.46\pm 0.03$
59207.14 | B | 1251 | $16.39\pm 0.02$
59211.19 | B | 1251 | $16.39\pm 0.02$
59215.04 | B | 1350 | $16.35\pm 0.02$
59217.10 | B | 1350 | $16.35\pm 0.02$
59219.11 | B | 1350 | $16.43\pm 0.02$
59221.04 | B | 1350 | $16.41\pm 0.02$
59223.10 | B | 1350 | $16.35\pm 0.02$
59230.09 | B | 1371 | $16.39\pm 0.01$
59231.14 | B | 1200 | $16.43\pm 0.02$
59233.18 | B | 1200 | $16.43\pm 0.02$
59234.15 | B | 1200 | $16.19\pm 0.01$
59236.19 | B | 1200 | $16.29\pm 0.02$
59237.18 | B | 1200 | $16.54\pm 0.02$
59241.09 | B | 600 | $16.54\pm 0.03$
59242.09 | B | 900 | $16.46\pm 0.02$
59243.21 | B | 1200 | $16.48\pm 0.03$
59245.08 | B | 1200 | $16.50\pm 0.02$
59249.08 | B | 1200 | $16.25\pm 0.01$
59256.16 | B | 1200 | $16.77\pm 0.03$
59263.16 | B | 1200 | $16.68\pm 0.03$
59272.07 | B | 1800 | $16.52\pm 0.04$
59274.08 | B | 720 | $16.54\pm 0.05$
59279.02 | B | 720 | $16.33\pm 0.10$
59283.99 | B | 692 | $16.44\pm 0.06$
59291.03 | B | 720 | $16.52\pm 0.05$
59294.00 | B | 900 | $16.51\pm 0.05$
59308.97 | B | 764 | $16.72\pm 0.10$
59389.38 | V | 1440 | $16.25\pm 0.03$
59393.34 | V | 1440 | $16.29\pm 0.02$
59464.23 | V | 1440 | $16.23\pm 0.01$
59467.28 | V | 1603 | $16.19\pm 0.05$
59489.25 | V | 1620 | $16.24\pm 0.03$
59537.06 | V | 1746 | $16.13\pm 0.02$
59541.19 | V | 1620 | $16.10\pm 0.07$
272727All magnitudes in Col. 4 are Vega magnitudes, and are observed
magnitudes, i.e., not corrected for Galactic absorption, and not corrected for
contamination by the host galaxy.

##  Appendix B Cross-calibration of UVOT and OM M2 magnitudes

In this work, we combine photometry data from both XMM-Newton OM and Swift
UVOT. For purposes of cross calibration between the two instruments, and for
determining if the variability observed in the B and M2 light curves is real,
we additionally examined the optical/UV data of two stars in the field of
view, each located 1$\aas@@fstack{\prime}$3–1$\aas@@fstack{\prime}$9 from
$\textrm{J}0458\mbox{--}5202$. These stars are TYC 8083-1113-1
($\alpha=04^{\rm h}58^{\rm m}08{\aas@@fstack{s}}3$,
$\delta=-52^{\circ}02\arcmin 12{\aas@@fstack{\prime\prime}}1$) and TYC
8083-1608-1 ($\alpha=04^{\rm h}58^{\rm m}14{\aas@@fstack{s}}7$,
$\delta=-52^{\circ}03\arcmin 20{\aas@@fstack{\prime\prime}}9$).

For the UVOT B-band, we converted UVOT B-band magnitudes to Johnson B-band
magnitudes following the ”color correction” document available at the Swift
UVOT Calibration Documents
webpage282828https://heasarc.gsfc.nasa.gov/docs/heasarc/caldb/swift/docs/uvot/
and using observed $B-V$ and $U-B$ colors for each object. This action
resulted in magnitude corrections of +0.017, +0.012, and +0.015 for
$\textrm{J}0458\mbox{--}5202$, TYC 8083-1113-1, and TYC 8083-1608-1,
respectively.

We used star TYC 8083-1113-1 to estimate systematic uncertainties for the
Swift UVOT data, under the assumption that the star is not variable. We
estimated systematic uncertainties to be 0.015, 0.012, 0.013, 0.015, 0.029,
and 0.023 mag for the V, B, U, W1, M2, and W2 bands, respectively. Star TYC
8083-1608-1 seems to display some energy-dependent variability during Sw7, so
we did not use this star to estimate systematics. The four OM observations
each used the M2 band; from TYC 8083-1113-1, we estimated the systematic
uncertainty to be $\sim$0.015 mag.

A direct comparison between the UVOT and OM M2 magnitudes, however, was still
hindered by the fact that the energy peaks of the two instruments’ effective
areas differ by $\sim$10 percent: OM M2 peaks near 2300 $\AA$ (5.4 eV), while
UVOT M2 peaks near 2140 $\AA$ (5.8 eV). To extrapolate from the OM M2 peak
energy to the UVOT M2 peak energy, we took advantage of having observed the
field with all six UVOT filters in Sw3–8. For the stars, we used the time-
averaged V, B, U, W1, M2, and W1 magnitudes to construct SEDs, and fit simple
power-law models via linear regression. For $\textrm{J}0458\mbox{--}5202$, we
used the Sw5 SED to estimate adjustments for XM4, as Sw5 occurs closest in
time to XM4; we used Sw3 to estimates adjustments to XM1–3. We obtained
spectral indices of $+0.97\pm 0.05$ ($\textrm{J}0458\mbox{--}5202$, Sw3),
$+0.42\pm 0.10$ ($\textrm{J}0458\mbox{--}5202$, Sw5), $-3.75\pm 0.07$ (TYC
8083-1113-1, average of Sw3–8), and $-2.61\pm 0.07$ (TYC 8083-1608-1, average
of Sw3–8). These values correspond to magnitude corrections applied to the OM
M2 light curves of $-0.070$ ($\textrm{J}0458\mbox{--}5202$, XM1–3), $-0.031$
($\textrm{J}0458\mbox{--}5202$, XM4), +0.276 (TYC 8083-1113-1), and +0.192
(TYC 8083-1608-1).

The final combined M2 and B light curves are shown in Fig. 17. Magnitudes are
not corrected for Galactic extinction, just for the aforementioned magnitude
shifts. In the M2 band, there is agreement between the UVOT and OM magnitudes
to within 0.05–0.1 mag. In addition, with the exception of a 0.1 mag drop
between Sw6 and Sw7 in object C, both stars’ light curves display minimal
variability – $\sim$0.1 mag or less within any one instrument or between
instruments. We thus conclude that the M2 flux drop by $>$0.2 mag in
$\textrm{J}0458\mbox{--}5202$ from Sw4/XM4 to Sw5–7 is real and not an
artifact.

Finally, for completeness, in Fig. 18, we display the two stars’ V, B, U, W1,
M2, and W2-band light curves, from observations Sw3–8.

Figure 17: B-band and M2-band Vega magnitude light curves for
$\textrm{J}0458\mbox{--}5202$ and for nearby stars TYC 8083-1113-1 and TYC
8083-1608-1. XMM-Newton OM points are denoted by filled circles; Swift UVOT
points are denoted by open circles. Uncertainties for
$\textrm{J}0458\mbox{--}5202$ are statistical only. Some XMM-Newton OM M2-band
data points have error bars smaller than their data point symbol. Figure 18:
Same as Fig. 2, but for two stars nearby in the field of view, TYC
8083-1113-1, and TYC 8083-1608-1. For plotting purposes, the magnitudes for
the V, B, W1, M2, and W2 bands have been offset by the magnitude values listed
in the top left corner, such that their mean magnitudes match that for U-band.

##  Appendix C Optical spectral fit result tables

Table 12: Optical spectral fit parameters, H$\beta$ region: continuum

Spec. | Power-law continuum
---|---
# | ${\nu}F_{5100\AA}$ | Slope
1 | $6.64\pm 0.02$ | $-1.71\pm 0.02$
2 | $6.96\pm 0.16$ | $-1.77\pm 0.05$
3 | $5.96\pm 0.02$ | $-1.51\pm 0.02$
4 | $7.61\pm 0.14$ | $-1.58\pm 0.04$
5 | $7.24\pm 0.31$ | $-1.94\pm 0.09$
6 | $6.73\pm 0.23$ | $-1.55\pm 0.07$
7 | $7.26\pm 0.28$ | $-1.76\pm 0.08$
8 | $7.86\pm 0.03$ | $-1.81\pm 0.02$
9 | $8.41\pm 0.02$ | $-1.60\pm 0.02$
10 | $7.51\pm 0.02$ | $-1.60\pm 0.05$
292929${\nu}F_{5100\AA}$ is in units of $10^{-16}$ erg cm-2 s-1 Å-1.

Table 13: Optical spectral fit parameters, H$\beta$ region: [O iii] narrow
line emission

Spec. | Narrow [O iii]$\lambda$5007 | Narrow [O iii]$\lambda$4959
---|---|---
| Centroid | Width $\sigma$ | Flux ($10^{-14}$ | Centroid | Width $\sigma$ | Flux ($10^{-14}$
| (Å) | (km s-1) | erg cm-2 s-1) | (Å) | (km s-1) | erg cm-2 s-1)
1 | $5006.1\pm 0.1$ | $214\pm 7$ | $3.51\pm 0.20$ | $4958.8\pm 0.3$ | (tied) | $1.04\pm 0.12$
2 | $5006.1\pm 0.1$ | $214\pm 7$ | $3.53\pm 0.20$ | $4958.3\pm 0.4$ | – | $0.81\pm 0.12$
3 | $5006.0\pm 0.1$ | $220\pm 11$ | $3.08\pm 0.26$ | $4958.9\pm 0.4$ | – | $0.97\pm 0.15$
4 | $5005.7\pm 0.1$ | $275\pm 13$ | $5.33\pm 0.55$ | $4958.8\pm 0.5$ | – | $1.52\pm 0.28$
5 | $5006.3\pm 0.1$ | $195\pm 10$ | $3.31\pm 0.27$ | $4959.4\pm 0.4$ | – | $1.10\pm 0.09$
6 | $5006.0\pm 0.1$ | $215\pm 9$ | $3.66\pm 0.27$ | $4958.6\pm 0.4$ | – | $1.17\pm 0.16$
7 | $5006.1\pm 0.1$ | $202\pm 10$ | $3.37\pm 0.27$ | $4958.3\pm 0.5$ | – | $0.87\pm 0.16$
8 | $5004.7\pm 0.2$ | $238\pm 16$ | $3.97\pm 0.50$ | $4959.5\pm 0.9$ | – | $0.76\pm 0.24$
9 | $5004.5\pm 0.2$ | $378\pm 12$ | $10.53\pm 0.51$ | $4955.0\pm 0.5$ | – | $3.51\pm 0.17$
10 | $5006.2\pm 0.1$ | $202\pm 7$ | $3.27\pm 0.18$ | $4959.1\pm 0.3$ | – | $1.09\pm 0.06$
303030Line parameters are reported for the rest frame. The velocity width
$\sigma$ for the 4959 $\AA$ line was tied to that for the 5007 $\AA$ line.

Table 14: Optical spectral fit parameters, H$\beta$ region: [O iii] broad,
blueshifted line emission

Spec. | Broad [O iii]$\lambda$5007 | Broad [O iii]$\lambda$4959
---|---|---
| Centroid | Width $\sigma$ | Flux ($10^{-14}$ | Centroid | Width $\sigma$ | Flux ($10^{-14}$
| (Å) | (km s-1) | erg cm-2 s-1) | (Å) | (km s-1) | erg cm-2 s-1)
1 | $4996.2\pm 0.9$ | $653\pm 24$ | $12.68\pm 0.90$ | $4951.0\pm 0.8$ | (tied) | $9.27\pm 0.59$
2 | $4995.9\pm 0.9$ | $643\pm 23$ | $12.60\pm 0.86$ | $4953.4\pm 0.7$ | – | $10.32\pm 0.58$
3 | $4996.0\pm 1.6$ | $688\pm 40$ | $10.82\pm 1.26$ | $4950.0\pm 1.4$ | – | $8.14\pm 0.79$
4 | $4994.7\pm 1.9$ | $671\pm 40$ | $12.01\pm 1.60$ | $4950.9\pm 1.3$ | – | $9.55\pm 0.98$
5 | $4995.1\pm 1.4$ | $662\pm 40$ | $14.74\pm 1.45$ | $4950.3\pm 1.1$ | – | $9.70\pm 1.57$
6 | $4994.2\pm 1.3$ | $665\pm 35$ | $13.62\pm 1.21$ | $4949.9\pm 1.2$ | – | $9.74\pm 0.57$
7 | $4996.5\pm 1.2$ | $670\pm 32$ | $15.54\pm 1.43$ | $4951.1\pm 1.1$ | – | $11.64\pm 0.94$
8 | $4995.6\pm 1.9$ | $620\pm 39$ | $10.67\pm 1.64$ | $4951.6\pm 1.3$ | – | $9.14\pm 0.94$
9 | $5008.2\pm 3.9$ | $1602\pm 226$ | $41.60\pm 33.57$ | $4956.5\pm 23.7$ | – | $24.23\pm 5.11$
10 | $4996.9\pm 0.8$ |  $665\pm 0$aa$a$Unobscured 1–10 keV observed-frame flux of the hard power-law component.Unobscured 1–10 keV observed-frame flux of the hard power-law component. | $15.46\pm 0.80$ | $4951.4\pm 0.7$ | – | $9.72\pm 0.49$

313131Line parameters are reported for the rest frame. The velocity widths
($\sigma$) for both broad components were kept tied in fits.

Table 15: Optical spectral fit parameters, H$\beta$ region: broad and narrow
H$\beta$ $\lambda$4861 emission

Spec. | Broad H$\beta$ | Narrow H$\beta$
---|---|---
# | Centroid | Width $\sigma$ | Flux ($10^{-14}$ | Centroid | Width $\sigma$ | Flux ($10^{-14}$
| (Å) | (km s-1) | erg cm-2 s-1) | (Å) | (km s-1) | erg cm-2 s-1)
1 | $4858.5\pm 0.2$ | $1996\pm 13$ | $400.86\pm 2.58$ | (4861.3) | (tied) | $0.53\pm 0.07$
2 | $4857.9\pm 0.2$ | $2019\pm 15$ | $416.12\pm 2.83$ | – | – | $0.58\pm 0.07$
3 | $4857.2\pm 0.3$ | $2124\pm 21$ | $380.62\pm 3.73$ | – | – | $0.50\pm 0.09$
4 | $4856.5\pm 0.2$ | $2081\pm 17$ | $462.87\pm 3.54$ | – | – | $0.87\pm 0.13$
5 | $4857.1\pm 0.4$ | $2111\pm 25$ | $482.24\pm 5.50$ | – | – | $0.52\pm 0.10$
6 | $4857.1\pm 0.3$ | $2151\pm 22$ | $451.67\pm 4.48$ | – | – | $0.64\pm 0.09$
7 | $4856.3\pm 0.3$ | $2167\pm 23$ | $508.53\pm 5.35$ | – | – | $0.61\pm 0.10$
8 | $4856.5\pm 0.3$ | $1988\pm 20$ | $412.25\pm 3.95$ | – | – | $0.11\pm 0.12$
9 | $4859.1\pm 0.5$ | $2087\pm 29$ | $488.98\pm 6.91$ | – | – | $0.69\pm 0.26$
10 | $4860.0\pm 0.3$ | $2057\pm 19$ | $405.33\pm 3.62$ | – | – | $0.33\pm 0.08$
323232Line parameters are reported for the rest frame. The energy centroid for
narrow H$\beta$ was kept fixed at 4861.3 $\AA$. The velocity width for the
narrow H$\beta$ component was tied to that for the narrow [O iii]$\lambda$5007
line.

Table 16: Optical spectral fit parameters, H$\alpha$ region: broad, ultra-
broad, and narrow H$\alpha$ $\lambda$6563 emission

Spec. | Broad H$\alpha$ | Ultra-Broad H$\alpha$ | Narrow H$\alpha$
---|---|---|---
# | Centroid | Width $\sigma$ | Flux ($10^{-14}$ | Centroid | Width $\sigma$ | Flux ($10^{-14}$ | Flux ($10^{-14}$
| (Å) | (km s-1) | erg cm-2 s-1) | (Å) | (km s-1) | erg cm-2 s-1) | erg cm-2 s-1)
1 | $6565.9\pm 0.6$ | $1495\pm 49$ | $833.85\pm 148.83$ | $6529.0\pm 20.4$ | $2111\pm 268$ | $396.63\pm 225.80$ | $3.84\pm 0.14$
2 | $6564.5\pm 0.4$ | $1450\pm 24$ | $742.41\pm 44.69$ | $6538.2\pm 5.1$ | $2415\pm 100$ | $521.70\pm 84.30$ | $2.80\pm 0.13$
3 | $6561.5\pm 0.2$ | $1536\pm 14$ | $737.51\pm 11.90$ | $6562.2\pm 2.4$ | $4548\pm 190$ | $1416.61\pm 61.80$ | $3.13\pm 0.15$
4 | $6561.7\pm 0.2$ | $1524\pm 15$ | $1007.48\pm 16.81$ | $6564.1\pm 3.5$ | $4684\pm 126$ | $2057.88\pm 87.36$ | $3.63\pm 0.21$
5 | $6563.2\pm 0.2$ | $1548\pm 20$ | $1084.18\pm 29.80$ | $6567.7\pm 2.8$ | $4250\pm 296$ | $1807.65\pm 100.85$ | $5.08\pm 0.28$
6 | $6560.7\pm 0.3$ | $1371\pm 38$ | $751.19\pm 60.11$ | $6559.8\pm 1.2$ | $2811\pm 138$ | $1603.46\pm 116.41$ | $5.19\pm 0.34$
7 | $6562.3\pm 0.2$ | $1501\pm 23$ | $1044.80\pm 39.05$ | $6565.0\pm 2.0$ | $3658\pm 224$ | $1606.93\pm 0.08$ | $5.23\pm 0.27$
8 | $6560.9\pm 0.3$ | $1477\pm 15$ | $918.87\pm 14.03$ | $6566.3\pm 2.7$ |  $4571\pm 0$aa$a$Width was kept frozen to enable reasonable constraints on other parameters. | $2145.13\pm 101.36$ | $2.72\pm 0.27$
9 | $6562.1\pm 0.2$ | $1538\pm 11$ | $1126.17\pm 13.26$ | $6563.8\pm 2.5$ |  $4571\pm 0$aa$a$Width was kept frozen to enable reasonable constraints on other parameters. | $1888.62\pm 97.62$ | $2.88\pm 0.21$
10 | $6563.7\pm 0.2$ | $1528\pm 12$ | $1229.23\pm 13.34$ | $6534.7\pm 2.1$ |  $4571\pm 0$aa$a$Width was kept frozen to enable reasonable constraints on other parameters. | $2782.34\pm 76.25$ | $5.16\pm 0.28$

333333Line parameters are reported for the rest frame. The velocity width for
the narrow H$\alpha$ component was tied to that for the narrow [O
iii]$\lambda$5007 line; the energy centroid was kept frozen at 6562.8 $\AA$.


#  $A$-analyticity of separatricies of foliations

Stéphane Druel Stéphane Druel: Univ Lyon, CNRS, Université Claude Bernard Lyon
1, UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne, France.
<EMAIL_ADDRESS>
###### Abstract.

Let $X$ be a smooth quasi-projective surface over a number field $K$, and let
$L$ be a foliation on $X$. We prove that if $L$ is closed under $p$-th powers
for almost all primes $p$, then any $L$-invariant smooth formal curve is
$A$-analytic. Building on prior work of Bost we obtain an algebraicity
criterion for those curves.

###### 2010 Mathematics Subject Classification:

14G40, 37F75

###### Contents

  1. 1 Introduction
  2. 2 Notation, conventions and used facts
  3. 3 Power series solutions of certain $p$-adic differential equations
  4. 4 Proof of Theorem 1.1

##  1\. Introduction

Let $X$ be a smooth quasi-projective surface over a number field $K$, and let
$L$ be a foliation on $X$ (defined over $K$).

If $L$ is closed under $p$-th powers for almost all primes $p$, then the
generalization to foliations by Ekedahl, Shepherd-Barron and Taylor (see
[ESBT99, Conjecture F]) of the classical Grothendieck-Katz conjecture predicts
that $L$ has algebraic leaves. The conjecture has been shown in some special
cases, e.g. if $X$ is a abelian surface and $L$ is induced by a non-zero
vector field ([Bos01, Theorem 2.3]), if $X$ is the total space of a line
bundle over an algebraic curve and $L$ is induced by a linear connection
([And04, Corollaire 4.3.6]), and if $X$ is a $\mathbb{P}^{1}$-bundle over an
elliptic curve $E$ and $L$ is an Ehresmann connection on $X\to E$ ([Dru21,
Proposition 9.3]).

Actually, [Bos01, Theorem 2.3] follows from an algebraicity criterion for
smooth formal schemes in algebraic varieties over number fields ([Bos01,
Theorem 3.4]), which we recall now. Let $P\in X(K)$, and let $\widehat{V}$ be
a smooth formal subscheme of the completion $\widehat{X}_{P}$ of $X$ at $P$.
If $\widehat{V}$ is A-analytic (we refer to Section 2 for this notion) and
$\widehat{V}_{\mathbb{C}}$ satisfies the Liouville condition for some
embedding $K\subset\mathbb{C}$, then the formal scheme $\widehat{V}$ is
algebraic.

If $L$ is regular at $P\in X(K)$, then the formal leaf $\widehat{V}$ of $L$
through $P$ is smooth but $\widehat{V}$ is not A-analytic in general.
Nevertheless, if $L$ is closed under $p$-th powers for almost all primes $p$,
then $\widehat{V}$ is $A$-analytic by [Bos01, Proposition 3.9].

Suppose now that $P\in X(K)$ be a singular point of $L$, and let
$K\subset\mathbb{C}$ be an embedding. In [CS82], Camacho and Sad proved that
there is at least one separatrix of $L_{\mathbb{C}}$ through
$P_{\mathbb{C}}\in X_{\mathbb{C}}$, i.e. a (local) irreducible complex curve
in $X_{\mathbb{C}}$ passing through $P_{\mathbb{C}}$ and tangent to
$L_{\mathbb{C}}$. Suppose in addition that $P_{\mathbb{C}}$ is a non-
degenerate reduced singularity of $L_{\mathbb{C}}$. Then, by [MM80, Appendice
II], $L_{\mathbb{C}}$ has exactly two separatrices through $P_{\mathbb{C}}$.
They are smooth and intersect transversely at $P_{\mathbb{C}}$. In this paper,
we extend [Bos01, Proposition 3.9] to this setting.

######  Theorem 1.1.

Let $X$ be a smooth quasi-projective surface over a number field $K$, and let
$L$ be a foliation on $X$. Suppose that $L$ is closed under $p$-th powers for
almost all primes $p$. Let $P\in X(K)$ be a singular point of $L$. Suppose
furthermore that $P_{\mathbb{C}}$ is a non-degenerate reduced singularity of
$L_{\mathbb{C}}$ for some embedding $K\subset\mathbb{C}$. Then the following
holds.

  1. (1)

There exist smooth formal subschemes $\widehat{V}$ and $\widehat{W}$ over $K$
of the completion $\widehat{X}_{P}$ of $X$ at $P$ such that
$\widehat{V}_{\mathbb{C}}$ and $\widehat{W}_{\mathbb{C}}$ are the formal
completion at $P$ of the separatrices of $L_{\mathbb{C}}$ through
$P_{\mathbb{C}}$ for any embedding $K\subset\mathbb{C}$.

  2. (2)

The formal curves $\widehat{V}$ and $\widehat{W}$ are $A$-analytic.

In fact, a slightly more general statement is true (see Theorem 4.1 and
Corollary 4.2).

The following is an immediate consequence of [Bos01, Theorem 3.4] and Theorem
1.1 together.

######  Corollary 1.2.

Let $X$ be a smooth quasi-projective surface over a number field $K$, and let
$L$ be a foliation on $X$. Suppose that $L$ is closed under $p$-th powers for
almost all primes $p$. Let $P\in X(K)$ be a singular point of $L$. Suppose
that $P_{\mathbb{C}}$ is a non-degenerate reduced singularity of
$L_{\mathbb{C}}$ for some embedding $K\subset\mathbb{C}$. Let
$\widehat{X}_{P}$ be the completion of $X$ at $P$, and let
$\widehat{V}\subset\widehat{X}_{P}$ be a formal separatrix through $P$.
Suppose in addition that there exists an embedding $K\subset\mathbb{C}$ such
that $\widehat{V}_{\mathbb{C}}$ satisfies the Liouville property. Then
$\widehat{V}$ is algebraic.

###  1.1. Structure of the paper

Section 2 gathers notation, known results and global conventions that will be
used throughout the paper. In section 3 we provide technical tools for the
proof of the main results. More precisely, we study formal power series
solutions of certain $p$-adic nonlinear differential equation using a Newton
iteration procedure. Section 4 is devoted to the proof of Theorem 1.1.

##  2\. Notation, conventions and used facts

###  2.1. Global convention

Throughout the paper a variety is a reduced and irreducible scheme separated
and of finite type over a field.

###  2.2. Foliations

Let $K$ be a number field, and let $R$ be its ring of integers. Let $X$ be a
smooth quasi-projective variety over $K$. A foliation on $X$ is a line bundle
$L\subseteq T_{X}$ such that the quotient $T_{X}/L$ is torsion free.

Let $U\subseteq X$ be the open subset where $L|_{U}$ is a subbundle of
$T_{U}$. We say that $L$ is singular at $P\in X$ if $P\in X\setminus U$.

Let $Y\subseteq X$ be a closed subvariety, and let $D$ be a derivation on $X$.
Say that $Y$ is invariant under $D$ if
$D(\mathscr{I}_{Y})\subseteq\mathscr{I}_{Y}$.

Say that $Y$ is invariant under $L$ if for any local section $D$ of $L$ over
some open subset $U$ of $X$, $D(\mathscr{I}_{Y\cap
U})\subseteq\mathscr{I}_{Y\cap U}$. To prove that $Y$ is invariant under $L$
it is enough to show that $Y\cap U$ of $Y$ is invariant under $L|_{U}$ for
some open set $U\subseteq X$ such that $Y\cap U$ is dense in $Y$. If $X$ and
$Y$ are smooth and $L\subseteq T_{X}$ is a subbundle, then $Y$ is invariant
under $L$ if and only if $L|_{Y}\subseteq T_{Y}\subseteq T_{X}|_{Y}$.

Let $N\geqslant 1$ be a sufficiently divisible integer, and let $\mathscr{X}$
(resp. $\mathscr{L}$) be a smooth model of $X$ over $R[1/N]$ (resp. a line
bundle on $\mathscr{X}$ contained in $T_{\mathscr{X}/S}$ such that
$\mathscr{L}\otimes_{R}K$ coincides with $L$ and the quotient
$T_{\mathscr{X}/S}/\mathscr{L}$ is torsion free), where
$S:=\textup{Spec}\,R[1/N]$. Let $\mathfrak{p}$ be a maximal ideal of $R$ with
$\mathfrak{p}\,\nmid\,N$, and let $k_{\mathfrak{p}}:=R/\mathfrak{p}$ denote
the residue field at $\mathfrak{p}$. Let $p$ denote the characteristic of
$k_{\mathfrak{p}}$. The sheaf of derivations
$\textup{Der}_{k_{\mathfrak{p}}}(\mathscr{O}_{\mathscr{X}_{\mathfrak{p}}})\cong
T_{\mathscr{X}_{\mathfrak{p}}}$ is endowed with the $p$-th power operation,
which maps any local $k_{\mathfrak{p}}$-derivation of
$\mathscr{O}_{\mathscr{X}_{\mathfrak{p}}}$ to its $p$-th iterate.

We say that $L$ is closed under $p$-th powers for almost all primes $p$ if
there exists $N\mid N^{\prime}$ such that, for any
$\mathfrak{p}\,\nmid\,N^{\prime}$,
$\mathscr{L}|_{\mathscr{X}_{\mathfrak{p}}}\subseteq
T_{\mathscr{X}_{\mathfrak{p}}}$ is closed under $p$-th powers. This condition
is independent of the choices of $\mathscr{X}$ and $\mathscr{L}$.

###  2.3. $A$-analyticity of formal smooth schemes

We briefly recall a number of definitions and facts concerning $A$-analytic
formal curves from [Bos01] (see also [BCL09]). We refer to loc. cit. for
further explanations concerning these notions.

Let $K$ be field equipped with some complete ultrametric absolute value
$|\cdot|$ and assume that its valuation ring $R$ is a discrete valuation ring.

Let $N$ be a positive integer, and let $r$ be a positive real number $r$. For
any formal power series
$f=\sum_{I\in\mathbb{N}^{N}}a_{I}X^{I}\in\mathbb{C}_{p}[[X_{1},\ldots,X_{N}]]$,
we define $||f||_{r}$ by the formula

$||f||_{r}=\sup_{I}\,|a_{I}|r^{|I|}\in\mathbb{R}_{\geqslant 0}\cup{+\infty}.$

The power series $f$ such that $||f||_{r}<+\infty$ are those that are
convergent and bounded on the open $N$-ball of radius $r$ in $K^{\,N}$.

######  Fact 2.1.

Let $r>0$ be a real number, and let $f\in\mathbb{C}_{p}[[X_{1},\ldots,X_{N}]]$
and $g\in\mathbb{C}_{p}[[X_{1},\ldots,X_{N}]]$ be formal power series with
$||f||_{r}<+\infty$ and $||g||_{r}<+\infty$. Then $||fg||_{r}<+\infty$, and
$||fg||_{r}\leqslant||f||_{r}||g||_{r}$.

For any positive real number $r$, we denote by
$G_{\textup{an},r}<\textup{Aut}\big{(}{\widehat{\mathbb{A}^{N}_{K}}}_{0}\big{)}$
the subgroup consisting of all $N$-tuples $f=(f_{1},\ldots,f_{N})$ such that
$f(0)=0$, $Df(0)\in\textup{GL}_{2}(R)$, and $||f_{i}||_{r}\leqslant r$ for
each $i$. This subgroup may be identified with the group of all analytic
automorphisms, preserving the origin, of the open $N$-dimensional ball of
radius $r$. Set also $G_{\textup{an}}:=\cup_{r>0}G_{\textup{an},r}$.

Notice that a smooth formal subscheme $\widehat{V}$ of dimension $d$ of
${\widehat{\mathbb{A}^{N}_{K}}}_{0}$ is $K$-analytic if and only if there
exists $f\in G_{\textup{an}}$ such that $f^{-1}\widehat{V}$ is the formal
subscheme ${\widehat{\mathbb{A}^{d}_{K}}}_{0}\times\\{0\\}$ of
${\widehat{\mathbb{A}^{N}_{K}}}_{0}$.

Let $\mathscr{X}$ be a quasi-projective $R$-scheme, and
$X=:\mathscr{X}\otimes_{R}K$ its generic fiber. Let
$\mathscr{P}\in\mathscr{X}(R)$, and let $P:=\mathscr{P}\otimes_{R}K$. Let
$\widehat{V}$ be a smooth formal subscheme of the completion $\widehat{X}_{P}$
of $X$ at $P$. There is a unique way to attach a number
$S_{\mathscr{X}}(\widehat{V})\in[0,1]$ such that the following holds (see
[Bos01]).

  1. (1)

We have $S_{\mathscr{X}}(\widehat{V})>0$ if and only if $\widehat{V}$ is
$K$-analytic.

  2. (2)

If $(\mathscr{X},\mathscr{P})=(\mathbb{A}^{N}_{R},0)$ and $\widehat{V}$ is
$K$-analytic, then $S_{\mathscr{X}}(\widehat{V})$ is supremum of the set of
real numbers $0<r\leqslant 1$ for which there exists $f\in G_{\textup{an}}$
such that $f^{-1}\widehat{V}$ is the formal subscheme
${\widehat{\mathbb{A}^{d}_{K}}}_{0}\times\\{0\\}$ of
${\widehat{\mathbb{A}^{N}_{K}}}_{0}$.

  3. (3)

If $\mathscr{X}\to\mathscr{X}^{\prime}$ is an immersion, then
$S_{\mathscr{X}}(\widehat{V})=S_{\mathscr{X}^{\prime}}(\widehat{V})$.

  4. (4)

For any two triples $(\mathscr{X},\mathscr{P},\widehat{V})$ and
$(\mathscr{X}^{\prime},\mathscr{P}^{\prime},\widehat{V}^{\prime})$ as above,
if there exists an $R$-morphism $f\colon\mathscr{X}\to\mathscr{X}^{\prime}$
mapping $\mathscr{P}$ to $\mathscr{P}^{\prime}$, étale along $\mathscr{P}$,
and inducing an isomorphism $\widehat{V}\cong\widehat{V}^{\prime}$, then
$S_{\mathscr{X}}(\widehat{V})=S_{\mathscr{X}^{\prime}}(\widehat{V}^{\prime})$.

We will refer to $S_{\mathscr{X}}(\widehat{V})$ as the size of $\widehat{V}$
with respect to the model $\mathscr{X}$ of $X$.

We will need the following easy observations.

######  Lemma 2.2 ([BCL09]).

Let $\varphi=\sum_{m\geqslant 1}c_{m}X^{m}\in K[[X]]$ be formal power series
such that $c_{1}\in R$, and let $\widehat{V}$ be its graph in
${\widehat{\mathbb{A}^{2}_{K}}}_{0}$. Set $\lambda:=\inf_{m\geqslant
1}-\frac{\log|c_{m+1}|}{m}\in\mathbb{R}\cup\\{-\infty\\}$, and let $\rho$ be
the radius of convergence of $\varphi$. Then the following holds.

  1. (1)

The size $S_{\mathbb{A}^{2}_{R}}(\widehat{V})$ of $\widehat{V}$ with respect
to $(\mathbb{A}^{2}_{R},0)$ satisfies
$S_{\mathbb{A}^{2}_{R}}(\widehat{V})\leqslant\rho$.

  2. (2)

If $\rho>0$, then
$S_{\mathbb{A}^{2}_{R}}(\widehat{V})\geqslant\min(1,\exp\lambda)$. In
particular, $\widehat{V}$ is $K$-analytic. Moreover, if $\varphi\in R[[X]]$,
then $S_{\mathbb{A}^{2}_{R}}(\widehat{V})=1$.

  3. (3)

If $\rho>0$ and $\varphi^{\prime}(0)$ is a unit in $R$, then
$S_{\mathbb{A}^{2}_{R}}(\widehat{V})=\min(1,\exp\lambda)$.

###### Proof.

Items (1) and (3) are shown in [BCL09, Proposition 3.5]. To prove Item (2),
let

$f(X_{1},X_{2}):=(X_{1}+X_{2},\varphi(X_{1}))\in\textup{Aut}\big{(}{\widehat{\mathbb{A}^{2}_{K}}}_{0}\big{)}.$

Then $f^{-1}\widehat{V}$ is the formal subscheme
${\widehat{\mathbb{A}^{1}_{K}}}_{0}\times\\{0\\}$ of
${\widehat{\mathbb{A}^{2}_{K}}}_{0}$. If $\rho>0$, then
$\lambda\in\mathbb{R}$. Set $r:=\min(1,\exp\lambda)\in[0,1]$. Then
$||f||_{r}\leqslant r$, and hence $f\in G_{\textup{an}}$. This finishes the
proof the lemma. ∎

######  Lemma 2.3.

Let $\varphi=\sum_{m\geqslant 2}c_{m}X^{m}\in K[[X]]$ be formal power series
such that $c_{2}\in R$, and set $\varphi:=\varphi(X)/X=\sum_{m\geqslant
1}c_{m+1}X^{m}$. Let $\widehat{V}$ $($resp. $\widehat{W})$ be the graph of
$\varphi$ $($resp. $\psi)$ in ${\widehat{\mathbb{A}^{2}_{K}}}_{0}$. Then
$S_{\mathbb{A}^{2}_{R}}(\widehat{V})\geqslant
S_{\mathbb{A}^{2}_{R}}(\widehat{W})$.

###### Proof.

Let be a positive real number such that
$r<S_{\mathbb{A}^{2}_{R}}(\widehat{W})$. By assumption, there exist formal
power series $f_{1}$ and $f_{2}$ in $K[[X_{1},X_{2}]]$ such that
$f=(f_{1},f_{2})\in G_{\textup{an},r}$ and
$f^{-1}\widehat{W}=\widehat{A^{1}_{K}}\times\\{0\\}$. This last condition is
actually equivalent to the identity

$f_{2}(T,0)=\psi(f_{1}(T,0))$

in $K[[T]]$. Let us write $f_{1}(T,0)=\sum_{m\geqslant 1}a_{m}T^{m}$, and
$f_{2}(T,0)=\sum_{m\geqslant 1}b_{m}T^{m}$. We have $b_{1}=c_{2}a_{1}$. Since
$f\in G_{\textup{an},r}$, we must have $Df(0)\in\textup{GL}_{2}(R)$. This
immediately implies that $a_{1}$ is a unit in $R$.

Let us set

$g_{1}(X_{1},X_{2})=f_{1}(X_{1},X_{2}),\quad
g_{2}(X_{1},X_{2})=Y+f_{1}(X_{1},X_{2})f_{2}(X_{1},X_{2}),\quad\textup{and}\quad
g=(g_{1},g_{2}).$

Then $||g_{1}||_{r}=||f_{1}||_{r}\leqslant r$ and $||g_{2}||_{r}\leqslant r$
since $||f_{1}f_{2}||_{r}\leqslant||f_{1}||_{r}||f_{2}||_{r}\leqslant
r^{2}\leqslant r$. Moreover,

$Dg(0)=\begin{pmatrix}a_{1}&\partial_{X_{2}}f(0)\\\
0&1\end{pmatrix}\in\textup{GL}_{2}(R),$

and hence $g\in G_{\textup{an},r}$. Finally, notice that
$g^{-1}\widehat{V}=\widehat{A^{1}_{K}}\times\\{0\\}$ since
$f_{2}(T,0)f_{1}(T,0)=\varphi(f_{1}(T,0))$ in $K[[T]]$. This shows that
$S_{\mathbb{A}^{2}_{R}}(\widehat{V})\geqslant
S_{\mathbb{A}^{2}_{R}}(\widehat{W})$, finishing the proof of the lemma. ∎

######  Remark 2.4.

In the setup of Lemma 2.3, note that the (formal) curve $T\mapsto(T,\psi(T))$
is the proper transform of the (formal) curve $T\mapsto(T,\varphi(T))$ in the
blow-up of $\mathbb{A}^{1}_{K}$ at $0$.

Let $K$ be a number field, and let $R$ be its ring of integers. Let
$\mathfrak{p}$ be a maximal ideal of $R$, let $|\cdot|_{\mathfrak{p}}$ be the
$\mathfrak{p}$-adic absolute value, normalized by the condition
$|\varpi_{\mathfrak{p}}|_{\mathfrak{p}}=\frac{1}{\sharp(R/\mathfrak{p})}$ for
any uniformizing element $\varpi_{\mathfrak{p}}$ at $\mathfrak{p}$. Let
$K_{\mathfrak{p}}$ and $R_{\mathfrak{p}}$ be the $\mathfrak{p}$-adic
completions of $K$ and $R$, and
$k_{\mathfrak{p}}:=R_{\mathfrak{p}}/(\varpi_{\mathfrak{p}})$ the residue field
at $\mathfrak{p}$. Let $p$ denote the characteristic of $k_{\mathfrak{p}}$.
Let $X$ be a quasi-projective algebraic variety over $K$, and let $P\in X(K)$.
Let $\widehat{V}$ be a smooth formal subscheme (defined over $K$) of the
formal completion $\widehat{X}_{P}$ of $X$ at $P$. Let $N\geqslant 1$ be a
sufficiently divisible integer, and let $\mathscr{X}$ be a quasi-projective
model of $X$ over $R[1/N]$, such that $P$ extends to a point
$\mathscr{P}\in\mathscr{X}(R[1/N])$. The smooth formal scheme $\widehat{V}$ is
said to be $A$-analytic (see [BCL09, Definition 3.7]) if

  1. (1)

for any place $v$ of $K$, the formal scheme $\widehat{V}_{K_{v}}$ is
$K_{v}$-analytic, where $K_{v}$ denotes the completion of $K$ with respect to
$v$, and

  2. (2)

we have

$\sum_{\mathfrak{p}\,\nmid\,N}\log\frac{1}{S_{\mathscr{X}_{R_{\mathfrak{p}}}}(\widehat{V}_{K_{\mathfrak{p}}})}<+\infty.$

This condition is independent of the choices of $\mathscr{X}$, $\mathscr{P}$,
and $N$.

##  3\. Power series solutions of certain $p$-adic differential equations

Let $p$ be a prime number. Let $\mathbb{Q}_{p}$ (resp. $\mathbb{C}_{p}$) be
the field of $p$-adic rational numbers (resp. complex numbers). We denote by
$|\cdot|$ the ultrametric absolute value on $\mathbb{C}_{p}$ normalized by
$|p|=p^{-1}$. We denote by $v$ the $p$-adic valuation on $\mathbb{C}_{p}$
normalized by $v(p)=1$.

In this section, we study formal power series solutions of certain $p$-adic
nonlinear differential equations at a regular singular point. The following is
the main result of this section.

######  Proposition 3.1.

Let $p$ be an odd prime integer, let $1\leqslant s\leqslant p-1$ and
$1\leqslant t\leqslant p-1$ be relatively prime integers, and set
$\alpha:=\frac{s}{t}$. Let $a$, $b$, and $c_{m}$ for any integer $m\geqslant
2$ be power series in $\mathbb{C}_{p}[[X]]$ such that
$||a||_{r}\leqslant\frac{r}{p}$, $||b||_{r}\leqslant\frac{r}{p}$, and
$||c_{m}||_{r}\leqslant\frac{1}{p}$ for some real number $r\in]0,1]$. Suppose
in addition that $a(0)=a^{\prime}(0)=0$ and that $b(0)=0$. Then the following
holds.

  1. (1)

There exists a unique formal power series $y$ in $\mathbb{C}_{p}[[X]]$ with
$y(0)=y^{\prime}(0)=0$ solution of the differential equation

(3.1) $xy^{\prime}+\alpha y=a+by+\sum_{m\geqslant 2}c_{m}y^{m}.$

  2. (2)

There exists a constant $C>0$ such that, letting
$R:=r\exp\Big{(}-Ct\frac{(\log p)^{2}}{p^{2}}\Big{)}$, we have
$||y||_{R}\leqslant R$.

We will need the following well-known facts.

######  Fact 3.2.

Let $f=\sum_{m\geqslant 0}a_{m}X^{m}\in\mathbb{C}_{p}[[X]]$ and
$g=\sum_{m\geqslant 0}b_{m}X^{m}\in\mathbb{C}_{p}[[X]]$. Suppose that there
exists a positive real number $r$ such that both $|a_{m}|r^{m}$ and
$|b_{m}|r^{m}$ goes to $0$ as $m$ goes to $+\infty$. Then
$||fg||_{r}=||f||_{r}||g||_{r}$ (see [Rob00, Proposition 2 of Section 6.1.4]).

######  Fact 3.3.

Let $f\in\mathbb{C}_{p}[[X]]$. If $f(0)=0$ and $||f||_{r}\leqslant Cr$ for
some positive real numbers $r$ and $C$, then $||f||_{r_{1}}\leqslant Cr_{1}$
for $0<r_{1}\leqslant r$.

######  Fact 3.4.

Let $f=\sum_{m\geqslant 0}a_{m}X^{m}\in\mathbb{C}_{p}[[X]]$. Suppose that
there exists a positive real number $r$ such that $|a_{m}|r^{m}$ goes to $0$
as $m$ goes to $+\infty$. Then $||f||_{r}=||f_{r}||_{1}=\sup_{|x|\leqslant
1}|f_{r}(x)|=\sup_{|x|\leqslant r}|f(x)|$, where
$f_{r}(X):=f(rX)=\sum_{m\geqslant 0}a_{m}r^{m}X^{m}$ (see [Rob00, Proposition
1 of Section 6.1.4]).

Before we give the proof of Proposition 3.1, we need the following auxiliary
statements.

######  Lemma 3.5.

Let $k$ be a positive integer, and let $b=\sum_{m\geqslant
2^{k}}b_{m}X^{m}\in\mathbb{C}_{p}[[X]]$ be a power series such that
$||b||_{r}\leqslant\frac{r}{p}$ for some real number $r\in]0,1]$. Set
$B=\sum_{m\geqslant 2^{k}}\frac{b_{m}}{m}X^{m}$.

  1. (1)

Let $r_{1}:=r\exp\big{(}-\frac{2}{p(p-1)}\log p\big{)}$. Then
$||B||_{r_{1}}\leqslant p^{-\frac{2}{p-1}}$.

  2. (2)

Let $r_{1}:=r\exp\big{(}-\frac{k+1}{2^{k}}\log 2\big{)}$. Then
$||B||_{r_{2}}\leqslant p^{-\frac{2}{p-1}}$.

###### Proof.

Let $R$ be a positive real number. Then $||B||_{R}\leqslant
p^{-\frac{2}{p-1}}$ if and only if

$\log R\leqslant\inf_{m\geqslant
2^{k}}\left\\{\frac{1}{m}\Big{(}-\log|b_{m}|+\log|m|-\frac{2}{p-1}\log
p\Big{)}\right\\}.$

By assumption, we have

$\log|b_{m}|+(m-1)\log r\leqslant-\log p$

for any integer $m\geqslant 2^{k}$, and hence

$\displaystyle\inf_{m\geqslant
2^{k}}\left\\{\frac{1}{m}\Big{(}-\log|b_{m}|+\log|m|-\frac{2}{p-1}\log
p\Big{)}\right\\}$ $\displaystyle\geqslant\inf_{m\geqslant
2^{k}}\left\\{\frac{1}{m}\Big{(}(m-1)\log r+\log|m|+\frac{p-3}{p-1}\log
p\Big{)}\right\\}$ $\displaystyle\geqslant\log r+\inf_{m\geqslant
2^{k}}\left\\{\frac{1}{m}\Big{(}\log|m|+\frac{p-3}{p-1}\log p\Big{)}\right\\}$

using the fact that $r\leqslant 1$.

Notice that $\log|m|+\frac{p-3}{p-1}\log p\leqslant-\log p+\frac{p-3}{p-1}\log
p=-\frac{2}{p-1}\log p<0$ if $|m|<1$, and hence

$\inf_{m\geqslant 2^{k}}\left\\{\frac{1}{m}\Big{(}\log|m|+\frac{p-3}{p-1}\log
p\Big{)}\right\\}<0.$

Then

$\displaystyle\inf_{m\geqslant
2^{k}}\left\\{\frac{1}{m}\Big{(}\log|m|+\frac{p-3}{p-1}\log p\Big{)}\right\\}$
$\displaystyle\geqslant\inf_{m\geqslant
1}\left\\{\frac{1}{m}\Big{(}\log|m|+\frac{p-3}{p-1}\log p\Big{)}\right\\}$
$\displaystyle=\inf_{m\geqslant 1\text{ such that
}|m|<1}\left\\{\frac{1}{m}\Big{(}\log|m|+\frac{p-3}{p-1}\log
p\Big{)}\right\\}$ $\displaystyle=\inf_{t\geqslant 1\text{ and }u\geqslant
1\text{ such that
}|u|=1}\left\\{\frac{1}{p^{t}u}\Big{(}-t+\frac{p-3}{p-1}\Big{)}\log
p\right\\}$ $\displaystyle\geqslant\inf_{t\geqslant
1}\left\\{\frac{1}{p^{t}}\Big{(}-t+\frac{p-3}{p-1}\Big{)}\log p\right\\}.$

Now observe that $x\mapsto\frac{1}{p^{x}}\Big{(}-x+\frac{p-3}{p-1}\Big{)}$ is
an increasing function on $x>\frac{p-3}{p-1}+\frac{1}{\log p}$. On the other
hand, we have $\frac{p-3}{p-1}+\frac{1}{\log p}<2$. Thus

$\inf_{t\geqslant
1}\left\\{\frac{1}{p^{t}}\Big{(}-t+\frac{p-3}{p-1}\Big{)}\log
p\right\\}=\min_{t\in\\{1,2\\}}\left\\{\frac{1}{p^{t}}\Big{(}-t+\frac{p-3}{p-1}\Big{)}\log
p\right\\}=\frac{-2}{p(p-1)}\log p,$

and hence

$\inf_{m\geqslant
2^{k}}\left\\{\frac{1}{m}\Big{(}-\log|b_{m}|+\log|m|-\frac{2}{p-1}\log
p\Big{)}\right\\}\geqslant\log r-\frac{2}{p(p-1)}\log p.$

This proves (1).

We proceed to show (2). We have

$\displaystyle\inf_{m\geqslant
2^{k}}\left\\{\frac{1}{m}\Big{(}\log|m|+\frac{p-3}{p-1}\log p\Big{)}\right\\}$
$\displaystyle\geqslant\inf_{m\geqslant 2^{k}}\left\\{\frac{1}{m}\Big{(}-\log
m+\frac{p-3}{p-1}\log p\Big{)}\right\\}$
$\displaystyle\geqslant\inf_{m\geqslant 2^{k}}\left\\{\frac{1}{m}\Big{(}-\log
m+\frac{p-3}{p-1}\log p\Big{)}\right\\}$
$\displaystyle\geqslant\inf_{m\geqslant 2^{k}}\left\\{-\frac{\log
m}{m}\right\\}-\frac{1}{2^{k}}\log 2.$

Notice that the function $x\mapsto-\frac{\log x}{x}$ is increasing on $\log
x>1$. Therefore, if $k\geqslant 2$, then

$\inf_{m\geqslant 2^{k}}\left\\{\frac{1}{m}\Big{(}-\log
m\Big{)}\right\\}=-\frac{k}{2^{k}}\log 2.$

If $k=1$, then

$\inf_{m\geqslant 2^{k}}\left\\{\frac{1}{m}\Big{(}-\log
m\Big{)}\right\\}=\min\left\\{-\frac{1}{2}\log 2,-\frac{2}{2^{2}}\log
2\right\\}=-\frac{1}{2}\log 2\geqslant-\log 2.$

This proves (2), completing the proof of the lemma. ∎

######  Remark 3.6.

In the setup of Lemma 3.5, notice that $\exp(B(x))$ is the unique formal power
series solution of the differential equation $xy^{\prime}=by$ such that
$y(0)=1$. We will denote $B(x)$ by $\int_{0}^{x}\frac{b(u)}{u}du$.

######  Lemma 3.7.

Let $k$ be a positive integer, and let $a=\sum_{m\geqslant
2^{k}}a_{m}X^{m}\in\mathbb{C}_{p}[[X]]$ be a power series such that
$||a||_{r}\leqslant\frac{r}{p}$ for some positive real number $r$. Let
$1\leqslant s\leqslant p-1$ and $1\leqslant t\leqslant p-1$ be relatively
prime integers, and set $\alpha:=\frac{s}{t}$ and $A=\sum_{m\geqslant
2^{k}}\frac{a_{m}}{m+\alpha}X^{m}$.

  1. (1)

Let $r_{1}:=r\exp\big{(}-\frac{t}{(p-1)^{2}}\log p\big{)}$. Then
$||A||_{r_{1}}\leqslant r_{1}$.

  2. (2)

Suppose that $2^{k}\geqslant\alpha+2$, and let
$r_{2}:=r\exp\big{(}-\frac{kt}{2^{k-1}}\log 2\big{)}$. Then
$||A||_{r_{2}}\leqslant r_{2}$.

###### Proof.

Let $R$ be a positive real number. Then $||A||_{R}\leqslant R$ if and only if

$\log R\leqslant\inf_{m\geqslant
2^{k}}\left\\{-\frac{1}{m-1}\log\frac{|a_{m}|}{|m+\alpha|}\right\\}.$

By assumption, we have $\log|a_{m}|+(m-1)\log r\leqslant-\log p$ for any
$m\geqslant 2^{k}$. Notice also that $|s|=|t|=1$. Thus

$\displaystyle\inf_{m\geqslant
2^{k}}\left\\{-\frac{1}{m-1}\log\frac{|a_{m}|}{|m+\alpha|}\right\\}$
$\displaystyle\geqslant\log r+\inf_{m\geqslant
2^{k}}\left\\{\frac{1}{m-1}\big{(}\log p+\log|tm+s|\big{)}\right\\}$
$\displaystyle\geqslant\log r+\inf_{m\geqslant
2^{k}t+s}\left\\{\frac{t}{m-s-t}\big{(}\log p+\log|m|\big{)}\right\\}$
$\displaystyle\geqslant\log r+\inf_{m\geqslant
2t+s}\left\\{\frac{t}{m-s-t}\big{(}\log p+\log|m|\big{)}\right\\}.$

Note that $\log p+\log|m|<0$ if $|m|<|p|$, and that $m-s-t\geqslant t\geqslant
1$. Hence

$\inf_{m\geqslant 2t+s}\left\\{\frac{t}{m-s-t}\big{(}\log
p+\log|m|\big{)}\right\\}<0,$

and thus

$\displaystyle\inf_{m\geqslant 2t+s}\left\\{\frac{t}{m-s-t}\big{(}\log
p+\log|m|\big{)}\right\\}$ $\displaystyle=\inf_{m\geqslant 2t+s\text{ and
}|m|<|p|}\left\\{\frac{t}{m-s-t}\big{(}\log p+\log|m|\big{)}\right\\}$
$\displaystyle=\inf_{k\geqslant 2\text{ and }u\geqslant 1\text{ such that
}|u|=1}\left\\{\frac{t}{p^{k}u-s-t}\big{(}1-k\big{)}\log p\right\\}$
$\displaystyle\geqslant\inf_{k\geqslant
2}\left\\{\frac{t}{p^{k}-2(p-1)}\big{(}1-k\big{)}\log p\right\\}$
$\displaystyle\geqslant\inf_{k\geqslant
2}\left\\{\frac{t}{p^{k}\Big{(}1-\frac{2(p-1)}{p^{2}}\Big{)}}\big{(}1-k\big{)}\log
p\right\\}$ $\displaystyle=\frac{t\log
p}{\Big{(}1-\frac{2(p-1)}{p^{2}}\Big{)}}\inf_{k\geqslant 2}\frac{1-k}{p^{k}}.$

On the other hand, the function $x\mapsto\frac{1-x}{p^{x}}$ is increasing on
$x>1+\frac{1}{\log p}$. This immediately implies

$\inf_{k\geqslant 2}\frac{1-k}{p^{k}}=-\frac{1}{p^{2}},$

and hence

$\displaystyle\frac{t\log
p}{\Big{(}1-\frac{2(p-1)}{p^{2}}\Big{)}}\inf_{k\geqslant 2}\frac{1-k}{p^{k}}$
$\displaystyle=-\frac{t}{p^{2}\Big{(}1-\frac{2(p-1)}{p^{2}}\Big{)}}\log p$
$\displaystyle=-\frac{t}{p^{2}-2p+2}\log p$
$\displaystyle\geqslant-\frac{t}{(p-1)^{2}}\log p.$

This proves (1).

We proceed to show (2). Suppose that $2^{k}\geqslant\alpha+2$. If moreover
$m\geqslant 2^{k}t+s$, then

$\frac{m}{2}-s-t\geqslant\frac{1}{2}(2^{k}t+s)-s-t\geqslant\frac{1}{2}(s+2t+s)-s-t=0,$

and hence $m-s-t\geqslant\frac{m}{2}$. Notice in addition that we must have
$k\geqslant 2$ since $\alpha>0$ by assumption. Then, we have

$\displaystyle\inf_{m\geqslant 2^{k}t+s}\left\\{\frac{t}{m-s-t}\big{(}\log
p+\log|m|\big{)}\right\\}$ $\displaystyle\geqslant\inf_{m\geqslant
2^{k}t+s}\left\\{\frac{t}{m-s-t}\big{(}\log p-\log m\big{)}\right\\}$
$\displaystyle\geqslant-\inf_{m\geqslant 2^{k}t+s}\left\\{\frac{t}{m-s-t}\log
m\right\\}$ $\displaystyle\geqslant-2t\inf_{m\geqslant 2^{k}t+s}\frac{\log
m}{m}$ $\displaystyle\geqslant-2t\inf_{m\geqslant 2^{k}}\frac{\log m}{m}$
$\displaystyle=-\frac{kt}{2^{k-1}}\log 2.$

This completes the proof of the lemma. ∎

######  Remark 3.8.

In the setup of Lemma 3.7, note that $A$ is the unique formal power series
solution of the differential equation

$xy^{\prime}+\alpha y=a$

with $y(0)=y^{\prime}(0)=0$.

We are now in position to prove Proposition 3.1.

###### Proof of Proposition 3.1.

Notice that $m+\alpha\neq 0$ for any integer $m\geqslant 2$. Item (1) then
follows easily.

The proof of Item (2) is subdivided into a number of steps.

Step 1. Let $m\geqslant 2$ be an integer. Observe that
$v(m+\alpha)=v(tm+s)\leqslant\frac{\log(tm+s)}{\log p}$. Therefore, [SS81,
Theorem 2] applies to show that the formal solution $y$ is convergent.

Set

$c(X_{1},X_{2}):=\sum_{m\geqslant
2}c_{m}(X_{1})X_{2}^{m}\in\mathbb{C}_{p}[[X_{1},X_{2}]],$

and consider the formal power series

$B(x):=\int_{0}^{x}\frac{b(u)}{u}du\quad\text{and}\quad
z(x):=y(x)\exp(-B(x)).$

Note that $z(0)=z^{\prime}(0)=0$. Then $y$ is solution of Equation (3.1) if
and only if $z$ is solution of

$xz^{\prime}(x)+\alpha
z(x)=\exp(-B(x))\big{(}a(x)+c(x,z(x)\exp(B(x)))\big{)}.$

By Lemma 3.5 (1), $B(x)$ converges if $|x|\leqslant
r_{0}:=r\exp\big{(}-\frac{2}{p(p-1)}\log p\big{)}$, and $|B(x)|\leqslant
p^{-\frac{2}{p-1}}<p^{-\frac{1}{p-1}}$. In particular, $\exp(\pm B(x))$ is
well-defined. Set

$a_{0}(x):=a(x)\exp(-B(x)))\quad\text{and}\quad
c_{0,m}(x):=c_{m}(x)\exp((m-1)B(x))$

for all $m\geqslant 2$, so that

$\exp(-B(x))c(x,z(x)\exp(B(x)))=\sum_{m\geqslant 2}c_{0,m}(x)z(x)^{m}.$

Let $r_{1}:=r\exp\big{(}-\frac{3}{p(p-1)}\log p\big{)}<r_{0}$. Then [Rob00,
Theorem of Section 6.1.5] applies to show that $\exp(\pm B)(x)$ converges and
that $\exp(\pm B)(x)=\exp(\pm(B(x))$ if $|x|\leqslant r_{1}$. In addition, we
have $||\exp(\pm B)||_{r_{1}}=\sup_{|x|\leqslant r_{1}}|\exp(\pm
B)(x)|=\sup_{|x|\leqslant r_{1}}|\exp(\pm(B(x))|$. On the other hand,
$\sup_{|x|\leqslant r_{1}}|\exp(\pm(B(x))|\leqslant 1$ since $|m!|\geqslant
p^{-\frac{m}{p-1}}$ for every integer $m\geqslant 1$. It follows that

$||\exp(\pm B)||_{r_{1}}=1$

since $B(0)=0$. As a consequence, we have

$||a_{0}||_{r_{1}}=||a||_{r_{1}}||\exp(-B)||_{r_{1}}=||a||_{r_{1}}\leqslant\frac{r_{1}}{p},$
$||c_{0,m}||_{r_{1}}=||c_{m}||_{r_{1}}\leqslant||c_{m}||_{r}\leqslant\frac{1}{p},$

and

$||y||_{R}=||z||_{R}$

for any real number $0<R\leqslant r_{1}$.

Replacing $r$ by $r_{1}$, if necessary, we may therefore assume without loss
of generality that $b=0$, so that Equation (3.1) reads

(3.2) $xy^{\prime}+\alpha y=a+\sum_{m\geqslant 2}c_{m}y^{m}.$

Step 2. The proof of [SS81, Theorem 2] then goes as follows. Let
$z_{0}\in\mathbb{C}_{p}[[X]]$ be the unique formal power series solution of

$xz_{0}^{\prime}(x)+\alpha z_{0}(x)=a(x)$

with $z_{0}(x)=O(x^{2})$ as $x$ goes to $0$. Set $b_{0}=0$. Observe that
$a_{1}(x):=c(x,z_{0}(x))=O\big{(}x^{2^{2}}\big{)}$ as $x$ goes to $0$. Set
$b_{1}(x):=\partial_{x_{2}}c(x,z_{0}(x))$, and let
$z_{1}\in\mathbb{C}_{p}[[X]]$ be the unique formal power series solution of

$xz_{1}^{\prime}(x)+\alpha z_{1}(x)=a_{1}(x)+b_{1}(x)z_{1}(x)$

with $z_{1}(x)=O\big{(}x^{2^{2}}\big{)}$ as $x$ goes to $0$. Next, one defines
inductively $z_{k}\in\mathbb{C}_{p}[[X]]$ for all integer $k\geqslant 2$ as
follows. Set

$y_{k}:=\sum_{i=0}^{k}z_{i}$

for $k\geqslant 0$. If $k\geqslant 2$, set

$a_{k}(x):=c(x,y_{k-1}(x))-c(x,y_{k-2}(x))-z_{k-1}(x)\partial_{x_{2}}c(x,y_{k-2}(x))$

and

$b_{k}(x):=\partial_{x_{2}}c(x,y_{k-1}(x)).$

For $k\geqslant 2$, one then proves that there exists a unique formal power
series $z_{k}$ solution of

(3.3) $xz_{k}^{\prime}(x)+\alpha z_{k}(x)=a_{k}(x)+b_{k}(x)z_{k}(x)$

with $z_{k}(x)=O\big{(}x^{2^{k+1}}\big{)}$. Finally, one proves that $z_{k}$
converges for all $k\geqslant 0$ as well as $y:=\sum_{k\geqslant 0}z_{k}$ and
that $y$ is the unique solution formal power series solution of Equation (3.2)
with $y(x)=O(x^{2})$ as $x$ goes to $0$.

Step 3. Set $B_{k}(x):=\int_{0}^{x}\frac{b_{k}(u)}{u}du$. Notice that, for any
integer $k\geqslant 1$, the formal power series $z_{k}$ is solution of
Equation $\eqref{eq:edo_k}$ if and only if

$w_{k}(x):=z_{k}(x)\exp(-B_{k}(x))$

is solution of

$xw_{k}^{\prime}(x)+\alpha w_{k}(x)=\exp(-B_{k}(x))a_{k}(x).$

Let $k_{1}$ be the smallest positive integer such that
$\frac{k_{1}+1}{2^{k_{1}}}\leqslant\frac{1}{p^{2}}$. Then $2^{k_{1}}\geqslant
p^{2}+1\geqslant p+1\geqslant\alpha+2$. Notice that $k_{1}\leqslant 5\log p$.
Indeed, let $k$ be any integer such that $k\geqslant\frac{2}{\log 2}\log p+1$.
Then $\frac{k+1}{2^{k}}\leqslant\frac{1}{2^{k-1}}\leqslant\frac{1}{p^{2}}$. It
follows that $k_{1}\leqslant\frac{2}{\log 2}\log p+2\leqslant 3\log
p+2\leqslant 5\log p$.

Then, we define inductively a decreasing sequence $(r_{k})_{k\geqslant 0}$ of
positive real numbers such that the following holds. For any integer
$k\geqslant 0$, if $|x|\leqslant r_{k}$, then $\exp(\pm B_{k})(x)$ converges,
$\exp(\pm B_{k})(x)=\exp(\pm(B_{k}(x))$, and $z_{k}(x)$ converges as well. In
addition,

$||\exp(\pm B_{k})||_{r_{k}}=1,$

and

$||z_{k}||_{r_{k}}\leqslant r_{k}.$

Finally, we will show that the limit $r_{\infty}$ of the sequence
$(r_{k})_{k\geqslant 0}$ satisfies

$\log r-\log r_{\infty}\leqslant Ct\frac{(\log p)^{2}}{p^{2}}$

for some constant $C>0$. In particular, $r_{\infty}>0$.

Set $r_{0}:=r\exp\big{(}-\frac{t}{(p-1)^{2}}\log p\big{)}$. By Lemma 3.7 (1),
$||z_{0}||_{r_{0}}\leqslant r_{0}$. Moreover, $B_{0}=0$ since $b_{0}=0$.

Let now $k$ be a positive integer. Suppose $r_{k-1}<r_{k-2}<\cdots<r_{0}$ have
already been defined. For any integer $0\leqslant j\leqslant k-1$, we have

$||y_{j}||_{r_{k-1}}\leqslant\max_{0\leqslant i\leqslant
j}||z_{i}||_{r_{k-1}}\leqslant r_{k-1},$

and $y_{j}(x)$ converges if $|x|\leqslant r_{k-1}$. Moreover,

$\displaystyle||b_{k}||_{r_{k-1}}$
$\displaystyle=||\partial_{x_{2}}c\big{(}x,y_{k-1}(x)\big{)}||_{r_{k-1}}$
$\displaystyle=||\sum_{m\geqslant 2}mc_{m}y_{k-1}^{m-1}||_{r_{k-1}}$
$\displaystyle\leqslant\max_{m\geqslant
2}\left\\{|m|||c_{m}||_{r_{k-1}}||y_{k-1}||^{m-1}_{r_{k-1}}\right\\}$
$\displaystyle\leqslant\max_{m\geqslant
2}\left\\{||c_{m}||_{r}r_{k-1}^{m-1}\right\\}$
$\displaystyle\leqslant\frac{r_{k-1}}{p}.$

Similarly, we have

$||a_{k}||_{r_{k-1}}=||c(x,y_{k-1}(x))-c(x,y_{k-2}(x))-z_{k-1}(x)\partial_{x_{2}}c(x,y_{k-2}(x))||_{r_{k-1}}\leqslant\frac{r_{k-1}}{p}.$

Suppose first $k<k_{1}$. Set

$r_{k}:=r_{k-1}\exp\Big{(}-\frac{2t}{(p-1)^{2}}\log p\Big{)}\leqslant
r_{k-1}\exp\Big{(}-\frac{2}{p(p-1)}\log p\Big{)}<r_{k-1}.$

By Lemma 3.5 (1) applied to $b_{k}$ and Lemma 3.7 (1) applied to
$\exp(-B_{k}(x))a_{k}(x)$ and arguing as in Step 1, we see that $r_{k}$
satisfies all the conditions listed above.

Suppose now that $k\geqslant k_{1}$, and set

$r_{k}:=r_{k-1}\exp\Big{(}-\frac{(k+1)t}{2^{k-1}}\log 2\Big{)}\leqslant
r_{k-1}\exp\Big{(}-\frac{k+1}{2^{k}}\log 2\Big{)}<r_{k-1}.$

Applying Lemma 3.5 (2) to $b_{k}$ and Lemma 3.7 (2) to
$\exp(-B_{k}(x))a_{k}(x)$ and arguing again as in Step 1, we see that $r_{k}$
also satisfies all the conditions listed above.

Finally, we have

$\sum_{1\leqslant k\leqslant k_{1}-1}(\log r_{k-1}-\log
r_{k})=\sum_{1\leqslant k\leqslant k_{1}-1}\frac{2t}{(p-1)^{2}}\log
p=\frac{2(k_{1}-1)t}{(p-1)^{2}}\log p\leqslant 10t\frac{(\log
p)^{2}}{(p-1)^{2}},$

and

$\displaystyle\sum_{k\geqslant k_{1}}(\log r_{k-1}-\log r_{k})$
$\displaystyle=\sum_{i\geqslant k_{1}}\frac{(k+1)t}{2^{k-1}}\log 2$
$\displaystyle\leqslant t\log 2\int_{k_{1}}^{+\infty}\frac{x+1}{2^{x-1}}dx$
$\displaystyle=\frac{t}{2^{k_{1}-1}}\left(\frac{1}{\log 2}+k_{1}+1\right)$
$\displaystyle\leqslant 4t\frac{k_{1}+1}{2^{k_{1}}}$
$\displaystyle\leqslant\frac{4t}{p^{2}}.$

Therefore, we have

$\log r-\log r_{\infty}=\sum_{k\geqslant 1}(\log r_{k-1}-\log r_{k})\leqslant
10t\frac{(\log p)^{2}}{(p-1)^{2}}+\frac{4t}{p^{2}}\leqslant 14t\frac{(\log
p)^{2}}{(p-1)^{2}}.$

Set $C:=14$, and $R:=r\exp\big{(}-tC\frac{(\log p)^{2}}{p^{2}}\big{)}$. Then
$R\leqslant r_{k}$ for any $k\geqslant 0$, and $||y||_{R}\leqslant R$ since
$||y_{k}||_{R}\leqslant R$ for each $k\geqslant 0$. This completes the proof
of the proposition. ∎

##  4\. Proof of Theorem 1.1

In this section we prove our main result. Note that Theorem 1.1 is an
immediate consequence of Theorem 4.1 below.

Let $X$ a smooth complex quasi-projective surface, and let $L$ be a foliation
on $X$. Let $P$ be a singular point of $L$, and let $D$ be a vector field on
some open subset $U\ni P$ such that $L|_{U}=\mathscr{O}_{U}D$. Let
$\alpha_{1}$ and $\alpha_{2}$ be the eigenvalues of the linear part of $D$ at
$P$. Recall that $P$ is a reduced singularity of $L$ if at least one of the
$\alpha_{i}$’s is non-zero, say $\alpha_{2}$, and
$\frac{\alpha_{1}}{\alpha_{2}}$ is not a positive rational number. A reduced
singularity $P$ is called non-degenerate if both $\alpha_{1}$ and $\alpha_{2}$
are non-zero. Then $\big{\\{}\alpha,\frac{1}{\alpha}\big{\\}}$ does not depend
on the choice of $D$, where $\alpha:=\frac{\alpha_{1}}{\alpha_{2}}$. Finally,
recall from [MM80, Appendice II], that if $P$ is a non-degenerate reduced
singularity, then there are exactly two analytic curves in $X$ passing through
$P$ and invariant under $L$. They are smooth and intersect transversely at
$P$.

######  Theorem 4.1.

Let $X$ be a smooth quasi-projective surface over a number field $K$, and let
$L$ be a foliation on $X$. Suppose that $L$ is closed under $p$-th powers for
almost all primes $p$. Let $P\in X(K)$ be a singular point of $L$. Suppose
that $P_{\mathbb{C}}$ is a reduced singularity of $L_{\mathbb{C}}$ for some
embedding $K\subset\mathbb{C}$. Then the following holds.

  1. (1)

The foliation $L$ has a non-degenerate singularity at $P$, and
$-\alpha_{P_{\mathbb{C}}}(L_{\mathbb{C}})\in\mathbb{Q}_{>0}$. In particular,
$P_{\mathbb{C}}$ is a reduced singularity of $L_{\mathbb{C}}$ for any
embedding $K\subset\mathbb{C}$.

  2. (2)

There exist two $\widehat{L}$-invariant smooth formal subschemes $\widehat{V}$
and $\widehat{W}$ of the completion $\widehat{X}_{P}$ of $X$ at $P$ defined
over $K$, where $\widehat{L}$ denotes the $($formal$)$ foliation induced by
$L$ on $\widehat{X}_{P}$. In particular, $\widehat{V}_{\mathbb{C}}$ and
$\widehat{W}_{\mathbb{C}}$ are the formal completion at $P$ of the two
separatrices of $L_{\mathbb{C}}$ through $P_{\mathbb{C}}$ for any embedding
$K\subset\mathbb{C}$.

  3. (3)

The formal curves $\widehat{V}$ and $\widehat{W}$ are $A$-analytic.

###### Proof.

Replacing $X$ by a Zariski open neighborhood of $P$ in $X$, if necessary, we
may assume without loss of generality that $X$ is affine, and that
$L=\mathscr{O}_{X}D$ for some vector field $D\in
T_{X}\cong\textup{Der}_{k}(\mathscr{O}_{X})$. Let $\overline{K}$ be an
algebraic closure of $K$, and let $\alpha_{1}\in\overline{K}$ and
$\alpha_{2}\in\overline{K}$ be the eigenvalues of the linear part of $D$ at
$P$. Suppose $\alpha_{2}\neq 0$. By [McQ08, Proposition II.1.3], $D$ is
formally linearisable at $P$ and $\frac{\alpha_{1}}{\alpha_{2}}\in\mathbb{Q}$.
This immediately implies that $\alpha_{1}\neq 0$ since $D$ has isolated zeroes
by assumption. In addition, we must have
$-\alpha_{P_{\mathbb{C}}}(L_{\mathbb{C}})=-\frac{\alpha_{1}}{\alpha_{2}}\in\mathbb{Q}_{>0}$
for any embedding $K\subset\mathbb{C}$, proving (1).

Let us write $\alpha=-\frac{s}{t}$, where $s$ and $t$ are relatively prime
positive integers, and let $K\subset\mathbb{C}$ be an embedding. Let $x_{1}$
and $x_{2}$ be regular functions on $X$ such that the induced map
$X\to\mathbb{A}_{K}^{2}$ is étale at $P$, and maps $P$ to $0$. Shrinking $X$
further, we may assume that $X\to\mathbb{A}_{K}^{2}$ is étale. Let
$\widehat{D}$ denote the $K$-derivation of $\widehat{\mathscr{O}}_{X,P}\cong
K[[X_{1},X_{2}]]$ induced by $D$. We may assume without loss of generality
that

$\widehat{D}=(X_{1}+f_{1}(X_{1},X_{2}))\partial_{X_{1}}+(\lambda
X_{2}+f_{2}(X_{1},X_{2}))\partial_{X_{2}},$

where $f_{1}$ and $f_{2}$ are formal power series with coefficients in $K$
vanishing to order at least $2$ at $P$. Let $\varphi_{1}$ and $\varphi_{2}$ be
formal power series with coefficients in $K$ vanishing to order at least $2$
at $0$. Then the formal (smooth) curves

$T\mapsto(\varphi_{1}(T),T)\quad\text{and}\quad T\mapsto(T,\varphi_{2}(T))$

are invariant under $\widehat{D}$ if and only if

$\widehat{D}(X_{1}-\varphi_{1}(X_{2}))\in(X_{1}-\varphi_{1}(X_{2}))\quad\text{and}\quad\widehat{D}(X_{2}-\varphi_{2}(X_{1}))\in(X_{2}-\varphi_{2}(X_{1}))$

if and only if

(4.1)
$\varphi_{1}(T)+f_{1}(\varphi_{1}(T),T)-\varphi^{\prime}_{1}(T)\big{(}\lambda
T+f_{2}(\varphi_{1}(T),T)\big{)}=0$

and

(4.2)
$\lambda\varphi_{2}(T)+f_{2}(T,\varphi_{2}(T))-\varphi^{\prime}_{2}(T)\big{(}T+f_{1}(T,\varphi_{2}(T))\big{)}=0.$

One readily checks that there is a unique formal power series $\varphi_{1}\in
K[[T]]$ (resp. $\varphi_{2}\in K[[T]]$) vanishing to order at least $2$ at $0$
solution of Equation (4.1) (resp. Equation (4.2)) since
$-\alpha\in\mathbb{Q}_{>0}$. This proves (2).

We will denote by $\widehat{V}$ (resp. $\widehat{W}$) the formal curve
$T\mapsto(\varphi_{1}(T),T)$ (resp. $T\mapsto(T,\varphi_{2}(T))$).

The proof of Item (3) is subdivided into a number of steps.

Step 1. Let $v$ be a place of $K$, and let $K_{v}$ be the completion of $K$ at
$v$. Notice first that $\widehat{V}_{K_{v}}$ and $\widehat{W}_{K_{v}}$ are
$K_{v}$-analytic by [SS81, Theorem 2] if $v$ is a finite place and [MM80,
Appendice II] if $v$ is archimedean.

Step 2. Suppose that the (algebraic) curve $\\{x_{1}\\}=0$ is
$\mathscr{L}$-invariant. Shrinking $X$ further, if necessary, we may therefore
assume that

$D=x_{1}\partial_{x_{1}}+f(x_{1},x_{2})\partial_{x_{2}},$

where $f$ is a regular function on $X$.

Let $R$ be the ring of integers of $K$. If $N$ denotes a sufficiently
divisible positive integer, there exists a model $\mathscr{X}$ of $X$, smooth
and quasi-projective over $R[1/N]$, such that $P$ extends to a point
$\mathscr{P}\in\mathscr{X}(R[1/N])$. We may also assume that
$X\to\mathbb{A}^{2}_{K}$ extends to an étale morphism
$\mathscr{X}\to\mathbb{A}^{2}_{R[1/N]}$, and that $D$ extends to a vector
field $\mathscr{D}\in
H^{0}(\mathscr{X},T_{\mathscr{X}/\textup{Spec}\,R[1/N]})$ with isolated
zeroes. Let $\mathfrak{p}$ be a maximal ideal of $R$, and let
$|\cdot|_{\mathfrak{p}}$ be the $\mathfrak{p}$-adic absolute value, normalized
by the condition
$|\varpi_{\mathfrak{p}}|_{\mathfrak{p}}=\frac{1}{\sharp(R/\mathfrak{p})}$ for
any uniformizing element $\varpi_{\mathfrak{p}}$ at $\mathfrak{p}$. Let
$K_{\mathfrak{p}}$ and $R_{\mathfrak{p}}$ be the $\mathfrak{p}$-adic
completions of $K$ and $R$, and
$k_{\mathfrak{p}}:=R_{\mathfrak{p}}/(\varpi_{\mathfrak{p}})$ the residue field
at $\mathfrak{p}$. Let finally $p$ denote the characteristic of
$k_{\mathfrak{p}}$.

Suppose that $\mathfrak{p}\nmid N$, and that $K_{\mathfrak{p}}$ is absolutely
unramified. Let $\widehat{\mathscr{D}}_{\mathfrak{p}}$ denote the
$K_{\mathfrak{p}}$-derivation of
$\widehat{\mathscr{O}}_{\mathscr{X}_{K_{\mathfrak{p}}},\mathscr{P}_{K_{\mathfrak{p}}}}\cong
K_{\mathfrak{p}}[[X_{1},X_{2}]]$ induced by
$\mathscr{D}_{K_{\mathfrak{p}}}=D_{K_{\mathfrak{p}}}$. Then

$\widehat{\mathscr{D}}_{\mathfrak{p}}=X_{1}\partial_{X_{1}}+f_{\mathfrak{p}}(X_{1},X_{2})\partial_{X_{2}}$

where $f_{\mathfrak{p}}\in R_{\mathfrak{p}}[[X_{1},X_{2}]]$. Let also
$\overline{\mathscr{D}}_{\mathfrak{p}}$ be the $k_{\mathfrak{p}}$-derivation
of $k_{\mathfrak{p}}[[X_{1},X_{2}]]$ induced by
$\widehat{\mathscr{D}}_{\mathfrak{p}}$. By assumption,
$\overline{\mathscr{D}}_{\mathfrak{p}}$ is p-closed. This immediately implies

$\overline{\mathscr{D}}_{\mathfrak{p}}^{\,p}=\overline{\mathscr{D}}_{\mathfrak{p}}$

since
$\overline{\mathscr{D}}_{\mathfrak{p}}^{\,p}(X_{1})=\overline{\mathscr{D}}_{\mathfrak{p}}(X_{1})=X_{1}$.
By [AA86, Lemma 6.4] applied to $\overline{\mathscr{D}}_{\mathfrak{p}}$, there
exists a formal power series $\overline{Y}_{2}$ in
$k_{\mathfrak{p}}[[X_{1},X_{2}]]$ such that
$\overline{\mathscr{D}}_{\mathfrak{p}}(\overline{Y}_{2})=\alpha\overline{Y}_{2}$.
Notice that $\overline{Y}_{2}$ may a priori depend on our choice of
$\mathfrak{p}$. Let $Y_{2}\in R_{\mathfrak{p}}[[X_{1},X_{2}]]$ be any formal
power series whose reduction modulo $\varpi_{\mathfrak{p}}$ is
$\overline{Y}_{2}$. By construction, we have
$\widehat{\mathscr{D}}_{\mathfrak{p}}(Y_{2})=\lambda Y_{2}$ modulo
$(\varpi_{\mathfrak{p}})$. Set $Y_{1}:=X_{1}$. Then, we have

$\widehat{\mathscr{D}}_{\mathfrak{p}}=Y_{1}\partial_{Y_{1}}+(\lambda
Y_{2}+g_{\mathfrak{p}}(Y_{1},Y_{2}))\partial_{Y_{2}},$

where
$g_{\mathfrak{p}}\in\varpi_{\mathfrak{p}}R_{\mathfrak{p}}[[Y_{1},Y_{2}]]$
vanishes to order at least $2$ at $0$. On the other hand, recall that
$\widehat{V}_{K_{\mathfrak{p}}}$ is defined by $Y_{1}=\varphi(Y_{2})$ in
$\textup{Spf}\,\widehat{\mathscr{O}}_{\mathscr{X}_{K_{\mathfrak{p}}},\mathscr{P}_{K_{\mathfrak{p}}}}\cong\textup{Spf}\,K_{\mathfrak{p}}[[Y_{1},Y_{2}]]$,
where $\varphi$ is the unique formal power series in $K_{\mathfrak{p}}[[T]]$
solution of the differential equation

$T\varphi^{\prime}(T)-\alpha\varphi(T)=g_{\mathfrak{p}}(X,\varphi(X))$

such that $\varphi(0)=\varphi^{\prime}(0)=0$. Let us write
$\varphi=\sum_{m\geqslant 2}a_{m}T^{m}$ and $g_{\mathfrak{p}}=\sum_{m\geqslant
0}b_{m}Y_{2}^{m}$, where
$b_{m}\in\varpi_{\mathfrak{p}}R_{\mathfrak{p}}[[Y_{1}]]$. Then
$||c_{b}||_{\mathfrak{p},1}\leqslant|\omega_{\mathfrak{p}}|=\frac{1}{p^{[K\mathfrak{p}:\mathbb{Q}_{p}]}}$,
so that Proposition 3.1 applies to show that the size
$S_{\mathscr{X}_{R_{\mathfrak{p}}}}(\widehat{V}_{K_{\mathfrak{p}}})$ satisfies

$\log\frac{1}{S_{\mathscr{X}_{R_{\mathfrak{p}}}}(\widehat{V}_{K_{\mathfrak{p}}})}\leqslant
C(\alpha)[K_{\mathfrak{p}}:\mathbb{Q}_{p}]\frac{(\log p)^{2}}{p^{2}},$

where $C(\alpha)>0$ depends only on $\alpha$.

Step 3. In the general setting, let $Y$ be the blow-up of $X$ at $P$, and let
$M$ be the foliation on $Y$ induced by $L$. The exceptional divisor $E$ is
$M$-invariant and contains exactly two singularities $Q_{1}$ and $Q_{2}$
(defined over $K$) of $M$, both reduced and non-degenerate. In addition,
relabeling $Q_{1}$ and $Q_{2}$ if necessary, we may assume that there is a
smooth formal subscheme $\widehat{V}_{1}$ (resp. $\widehat{W}_{1}$) of
$\widehat{Y}_{Q_{1}}$ defined over $K$ (resp. $\widehat{Y}_{Q_{2}}$) which is
$\widehat{M}$-invariant and such that the morphism $Y\to X$ induces an
isomorphism $\widehat{V_{1}}\cong\widehat{V}$ (resp.
$\widehat{W_{1}}\cong\widehat{W}$). The blow-up $\mathscr{Y}$ of $\mathscr{X}$
along $\mathscr{P}$ is a smooth and quasi-projective model of $Y$ over
$R[1/N]$. In addition, $Q_{1}$ and $Q_{2}$ extends to points
$\mathscr{Q}_{1}\in\mathscr{Y}(R[1/N])$ and
$\mathscr{Q}_{2}\in\mathscr{Y}(R[1/N])$, and $M$ extends to a foliation
$\mathscr{M}\subset T_{\mathscr{Y}/\textup{Spec}\,R[1/N]}$. We have a
commutative diagram

${\mathscr{Y}}$${\textup{Bl}_{0}\mathbb{A}^{2}_{R[1/N]}}$${\mathscr{X}}$${\mathbb{A}^{2}_{R[1/N]},}$

where the horizontal arrows are étale morphims. Let
$U_{1}\cong\mathbb{A}^{2}_{R[1/N]}\subset\textup{Bl}_{0}\mathbb{A}^{2}_{R[1/N]}$
be the affine charts containing $Q_{1}$ with coordinates $(y_{1},y_{2})$
(centered at $Q_{1}$). Then the natural morphism
$U_{1}\to\mathbb{A}^{2}_{R[1/N]}$ maps $(y_{1},y_{2})$ to
$(y_{1}y_{2},y_{2})$. This immediately implies that the formal subscheme
$\widehat{V}_{1}$ is defined by $Y_{1}=\varphi(Y_{2})/Y_{2}$ in
$\widehat{Y}_{Q_{1}}=\textup{Spf}\,\widehat{\mathscr{O}}_{Y,Q_{1}}\cong\textup{Spf}\,K[[Y_{1},Y_{2}]]$.
Finally, recall that the size (of a smooth formal scheme) is invariant by
étale localization. Let again $\mathfrak{p}$ be a maximal ideal of $R$, and
suppose that $\mathfrak{p}\nmid N$, and that $K_{\mathfrak{p}}$ is absolutely
unramified. Then Lemma 2.3 and Step 2 imply that
$S_{\mathscr{X}_{R_{\mathfrak{p}}}}(\widehat{V}_{K_{\mathfrak{p}}})$ satisfies

$\log\frac{1}{S_{\mathscr{X}_{R_{\mathfrak{p}}}}(\widehat{V}_{K_{\mathfrak{p}}})}\leqslant\log\frac{1}{S_{\mathscr{Y}_{R_{\mathfrak{p}}}}(\widehat{W}_{K_{\mathfrak{p}}})}\leqslant
C^{\prime}(\alpha)[K_{\mathfrak{p}}:\mathbb{Q}_{p}]\frac{(\log p)^{2}}{p^{2}}$

where $C^{\prime}(\alpha)>0$ depends only on $\alpha$. This immediately
implies that $\widehat{V}$ is $A$-analytic, completing the proof of the
theorem. ∎

Let $L$ be a foliation on a smooth complex quasi-projective surface $X$, and
let $P$ be a singular point of $L$. By [CS82, Theorem], there exists a
(possibly singular) analytic curve passing through $P$. The following is an
easy consequence of Theorem 4.1 above.

######  Corollary 4.2.

Let $X$ be a smooth quasi-projective surface over a number field $K$, and let
$L$ be a foliation on $X$. Suppose that $L$ is closed under $p$-th powers for
almost all primes $p$. Let $P\in X(K)$ be a singular point of $L$, and let
$\widehat{V}$ be an $\widehat{L}$-invariant smooth formal subcheme of the
completion $\widehat{X}_{P}$ of $X$ at $P$ defined over $K$, where
$\widehat{L}$ denotes the $($formal$)$ foliation induced by $L$ on
$\widehat{X}_{P}$. Then $\widehat{V}$ is $A$-analytic.

###### Proof.

Let $\overline{K}$ be an algebraic closure of $K$. By a result proved by
Seidenberg ([Sei68]), there exists a composition of a finite number of blow-
ups $Y_{\overline{K}}\to X_{\overline{K}}$ of $\overline{K}$-rational points
such the foliation $M_{\overline{K}}$ induced by $L$ on $Y_{\overline{K}}$ has
reduced singularities. The variety $Y_{\overline{K}}$ is defined over a finite
extension $F$ of $K$. Let $M$ be the foliation on $Y$ indiced by $L$. Notice
that $M$ is closed under $p$-th powers for almost all primes $p$. Then Theorem
4.1 applied to the proper transform of $\widehat{V}_{F}$ in $Y$ together with
[BCL09, Proposition 3.4] and Lemma 2.3 imply that $\widehat{V}$ is A-analytic.
∎

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