\edet | $\varepsilon_{\mathrm{det}}$ | \erec | $\varepsilon_{\mathrm{rec/det}}$ | \esel | $\varepsilon_{\mathrm{sel/rec}}$
---|---|---|---|---|---
\etrg | $\varepsilon_{\mathrm{trg/sel}}$ | \etot | $\varepsilon_{\mathrm{tot}}$ | \mistag | $\omega$
\wcomb | $\omega^{\mathrm{comb}}$ | \etag | $\varepsilon_{\mathrm{tag}}$ | \etagcomb | $\varepsilon_{\mathrm{tag}}^{\mathrm{comb}}$
\effeff | $\varepsilon_{\mathrm{eff}}$ | \effeffcomb | $\varepsilon_{\mathrm{eff}}^{\mathrm{comb}}$ | \efftag | ${\varepsilon_{\mathrm{tag}}}(1-2\omega)^{2}$
\effD | ${\varepsilon_{\mathrm{tag}}}D^{2}$ | \etagprompt | $\varepsilon_{\mathrm{tag}}^{\mathrm{Pr}}$ | \etagLL | $\varepsilon_{\mathrm{tag}}^{\mathrm{LL}}$

####  C.3.9 Key decay channels

\BdToKstmm | ${{\mathrm{B}}^{0}}\\!\rightarrow{{\mathrm{K}}^{*0}}{\upmu^{+}}{\upmu^{-}}$ | \BdbToKstmm | ${{\kern 0.0pt\overline{\kern 0.0pt\mathrm{B}}}{}^{0}}\\!\rightarrow{{\kern 0.0pt\overline{\kern 0.0pt\mathrm{K}}}{}^{*0}}{\upmu^{+}}{\upmu^{-}}$ | \BsToJPsiPhi | ${{\mathrm{B}}^{0}_{\mathrm{s}}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}\phi$
---|---|---|---|---|---
\BdToJPsiKst | ${{\mathrm{B}}^{0}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{{\mathrm{K}}^{*0}}$ | \BdbToJPsiKst | ${{\kern 0.0pt\overline{\kern 0.0pt\mathrm{B}}}{}^{0}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi}}{{\kern 0.0pt\overline{\kern 0.0pt\mathrm{K}}}{}^{*0}}$ | \BsPhiGam | ${{\mathrm{B}}^{0}_{\mathrm{s}}}\\!\rightarrow\phi{\upgamma}$
\BdKstGam | ${{\mathrm{B}}^{0}}\\!\rightarrow{{\mathrm{K}}^{*0}}{\upgamma}$ | \BTohh | ${\mathrm{B}}\\!\rightarrow\mathrm{h}^{+}\mathrm{h}^{\prime-}$ | \BdTopipi | ${{\mathrm{B}}^{0}}\\!\rightarrow{{\uppi}^{+}}{{\uppi}^{-}}$
\BdToKpi | ${{\mathrm{B}}^{0}}\\!\rightarrow{{\mathrm{K}}^{+}}{{\uppi}^{-}}$ | \BsToKK | ${{\mathrm{B}}^{0}_{\mathrm{s}}}\\!\rightarrow{{\mathrm{K}}^{+}}{{\mathrm{K}}^{-}}$ | \BsTopiK | ${{\mathrm{B}}^{0}_{\mathrm{s}}}\\!\rightarrow{{\uppi}^{+}}{{\mathrm{K}}^{-}}$
\Cpipi | $C_{{{\uppi}^{+}}{{\uppi}^{-}}}$ | \Spipi | $S_{{{\uppi}^{+}}{{\uppi}^{-}}}$ | \CKK | $C_{{{\mathrm{K}}^{+}}{{\mathrm{K}}^{-}}}$
\SKK | $S_{{{\mathrm{K}}^{+}}{{\mathrm{K}}^{-}}}$ | \ADGKK | $A^{{\Delta\Gamma}}_{{{\mathrm{K}}^{+}}{{\mathrm{K}}^{-}}}$ |  |

####  C.3.10 Rare decays

\BdKstee | ${{\mathrm{B}}^{0}}\\!\rightarrow{{\mathrm{K}}^{*0}}{\mathrm{e}^{+}\mathrm{e}^{-}}$ | \BdbKstee | ${{\kern 0.0pt\overline{\kern 0.0pt\mathrm{B}}}{}^{0}}\\!\rightarrow{{\kern 0.0pt\overline{\kern 0.0pt\mathrm{K}}}{}^{*0}}{\mathrm{e}^{+}\mathrm{e}^{-}}$ | \bsll | ${\mathrm{b}}\\!\rightarrow{\mathrm{s}}\ell^{+}\ell^{-}$
---|---|---|---|---|---
\AFB | $A_{\mathrm{FB}}$ | \FL | $F_{\mathrm{L}}$ | \AT#1 \AT2 | $A_{\mathrm{T}}^{2}$
\btosgam | ${\mathrm{b}}\\!\rightarrow{\mathrm{s}}{\upgamma}$ | \btodgam | ${\mathrm{b}}\\!\rightarrow{\mathrm{d}}{\upgamma}$ | \Bsmm | ${{\mathrm{B}}^{0}_{\mathrm{s}}}\\!\rightarrow{\upmu^{+}}{\upmu^{-}}$
\Bdmm | ${{\mathrm{B}}^{0}}\\!\rightarrow{\upmu^{+}}{\upmu^{-}}$ | \Bsee | ${{\mathrm{B}}^{0}_{\mathrm{s}}}\\!\rightarrow{\mathrm{e}^{+}\mathrm{e}^{-}}$ | \Bdee | ${{\mathrm{B}}^{0}}\\!\rightarrow{\mathrm{e}^{+}\mathrm{e}^{-}}$
\ctl | $\cos{\theta_{\ell}}$ | \ctk | $\cos{\theta_{K}}$ |  |

####  C.3.11 Wilson coefficients and operators

\C#1 \C9 | $\mathcal{C}_{9}$ | \Cp#1 \Cp7 | $\mathcal{C}_{7}^{{}^{\prime}}$ | \Ceff#1 \Ceff9 | $\mathcal{C}_{9}^{\mathrm{(eff)}}$
---|---|---|---|---|---
\Cpeff#1 \Cpeff7 | $\mathcal{C}_{7}^{{}^{\prime}\mathrm{(eff)}}$ | \Ope#1 \Ope2 | $\mathcal{O}_{2}$ | \Opep#1 \Opep7 | $\mathcal{O}_{7}^{{}^{\prime}}$

####  C.3.12 Charm

\xprime | $x^{\prime}$ | \yprime | $y^{\prime}$ | \ycp | $y_{{C\\!P}}$
---|---|---|---|---|---
\agamma | $A_{\Gamma}$ | \dkpicf | ${{\mathrm{D}}^{0}}\\!\rightarrow{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ |  |

####  C.3.13 QM

\bra[1] \bra{a} | $\langle a|$ | \ket[1] \ket{b} | $|b\rangle$ | \braket[2] \braket{a}{b} | $\langle a|b\rangle$
---|---|---|---|---|---

###  C.4 Units (these macros add a small space in front)

\aunit[1] \aunit{kg} |  kg
---|---

####  C.4.1 Energy and momentum

\tev |  TeV | \gev |  GeV | \mev |  MeV
---|---|---|---|---|---
\kev |  keV | \ev |  eV | \gevgev | $\text{\,Ge\kern-1.00006ptV}^{2}$
\mevc | $\text{\,Me\kern-1.00006ptV\\!/}c$ | \gevc | $\text{\,Ge\kern-1.00006ptV\\!/}c$ | \mevcc | $\text{\,Me\kern-1.00006ptV\\!/}c^{2}$
\gevcc | $\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ | \gevgevcc | $\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{2}$ | \gevgevcccc | $\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$

####  C.4.2 Distance and area (these macros add a small space)

\km |  km | \m |  m | \ma | $\text{\,m}^{2}$
---|---|---|---|---|---
\cm |  cm | \cma | $\text{\,cm}^{2}$ | \mm |  mm
\mma | $\text{\,mm}^{2}$ | \mum | $\,\upmu\text{m}$ | \muma | $\,\upmu\text{m}^{2}$
\nm |  nm | \fm |  fm | \barn |  b
\mbarn |  mb | \mub | $\,\upmu\text{b}$ | \nb |  nb
\invnb | $\text{\,nb}^{-1}$ | \pb |  pb | \invpb | $\text{\,pb}^{-1}$
\fb |  fb | \invfb | $\text{\,fb}^{-1}$ | \ab |  ab
\invab | $\text{\,ab}^{-1}$ |  |  |  |

####  C.4.3 Time

\sec |  s | \ms |  ms | \mus | $\,\upmu\text{s}$
---|---|---|---|---|---
\ns |  ns | \ps |  ps | \fs |  fs
\mhz |  MHz | \khz |  kHz | \hz |  Hz
\invps | $\text{\,ps}^{-1}$ | \invns | $\text{\,ns}^{-1}$ | \yr |  yr
\hr |  hr |  |  |  |

####  C.4.4 Temperature

\degc |  ∘C | \degk |  K
---|---|---|---

####  C.4.5 Material lengths, radiation

\Xrad | $X_{0}$ | \NIL | $\lambda_{\rm int}$ | \mip | MIP
---|---|---|---|---|---
\neutroneq | $n_{\text{eq}}$ | \neqcmcm | $n_{\text{eq}}/\text{cm}^{2}$ | \kRad |  kRad
\MRad |  MRad | \ci |  Ci | \mci |  mCi

####  C.4.6 Uncertainties

\sx | $\sigma_{x}$ | \sy | $\sigma_{y}$ | \sz | $\sigma_{z}$
---|---|---|---|---|---
\stat |  (stat) | \syst |  (syst) | \lumi |  (lumi)

####  C.4.7 Maths

\order | $\mathcal{O}$ | \chisq | $\chi^{2}$ | \chisqndf | $\chi^{2}/\mathrm{ndf}$
---|---|---|---|---|---
\chisqip | $\chi^{2}_{\text{IP}}$ | \chisqvs | $\chi^{2}_{\text{VS}}$ | \chisqvtx | $\chi^{2}_{\text{vtx}}$
\chisqvtxndf | $\chi^{2}_{\text{vtx}}/\mathrm{ndf}$ | \chisqfd | $\chi^{2}_{\text{FD}}$ | \gsim |  $>$$\sim$
\lsim |  $<$$\sim$ | \mean[1] \mean{x} | $\left\langle x\right\rangle$ | \abs[1] \abs{x} | $\left\|x\right\|$
\Real | $\mathcal{R}e$ | \Imag | $\mathcal{I}m$ | \PDF | PDF
\sPlot | sPlot | \sFit | sFit | \deriv | $\mathrm{d}$

###  C.5 Kinematics

####  C.5.1 Energy, Momenta

\Ebeam | $E_{\mbox{\tiny BEAM}}$ | \sqs | $\sqrt{s}$ | \sqsnn | $\sqrt{s_{\scriptscriptstyle\text{NN}}}$
---|---|---|---|---|---
\pt | $p_{\mathrm{T}}$ | \ptsq | $p_{\mathrm{T}}^{2}$ | \ptot | $p$
\et | $E_{\mathrm{T}}$ | \mt | $M_{\mathrm{T}}$ | \dpp | $\Delta p/p$
\msq | $m^{2}$ | \dedx | $\mathrm{d}E/\mathrm{d}x$ |  |

####  C.5.2 PID

\dllkpi | $\mathrm{DLL}_{{\mathrm{K}}{\uppi}}$ | \dllppi | $\mathrm{DLL}_{{\mathrm{p}}{\uppi}}$ | \dllepi | $\mathrm{DLL}_{{\mathrm{e}}{\uppi}}$
---|---|---|---|---|---
\dllmupi | $\mathrm{DLL}_{{\upmu}\pi}$ |  |  |  |

####  C.5.3 Geometry

\degrees | ∘ | \murad | $\,\upmu\text{rad}$ | \mrad |  mrad
---|---|---|---|---|---
\rad |  rad |  |  |  |

####  C.5.4 Accelerator

\betastar | $\beta^{*}$ | \lum | $\mathcal{L}$ | \intlum[1] \intlum{2 $\text{\,fb}^{-1}$ } | $\int\mathcal{L}=2\,\text{\,fb}^{-1}$
---|---|---|---|---|---

###  C.6 Software

####  C.6.1 Programs

\bcvegpy | Bcvegpy | \boole | Boole | \brunel | Brunel
---|---|---|---|---|---
\davinci | DaVinci | \dirac | Dirac | \evtgen | EvtGen
\fewz | Fewz | \fluka | Fluka | \ganga | Ganga
\gaudi | Gaudi | \gauss | Gauss | \geant | Geant4
\hepmc | HepMC | \herwig | Herwig | \moore | Moore
\neurobayes | NeuroBayes | \photos | Photos | \powheg | Powheg
\pythia | Pythia | \resbos | ResBos | \roofit | RooFit
\root | Root | \spice | Spice | \tensorflow | TensorFlow
\urania | Urania |  |  |  |

####  C.6.2 Languages

\cpp | C++ | \ruby | Ruby | \fortran | Fortran
---|---|---|---|---|---
\svn | svn | \git | git | \latex | LaTeX

####  C.6.3 Data processing

\kbit |  kbit | \kbps |  kbit/s | \kbytes |  kB
---|---|---|---|---|---
\kbyps |  kB/s | \mbit |  Mbit | \mbps |  Mbit/s
\mbytes |  MB | \mbyps |  MB/s | \gbit |  Gbit
\gbps |  Gbit/s | \gbytes |  GB | \gbyps |  GB/s
\tbit |  Tbit | \tbps |  Tbit/s | \tbytes |  TB
\tbyps |  TB/s | \dst | DST |  |

###  C.7 Detector related

####  C.7.1 Detector technologies

\nonn | $\mathrm{{\mathit{n^{+}}}\mbox{-}on\mbox{-}{\mathit{n}}}$ | \ponn | $\mathrm{{\mathit{p^{+}}}\mbox{-}on\mbox{-}{\mathit{n}}}$ | \nonp | $\mathrm{{\mathit{n^{+}}}\mbox{-}on\mbox{-}{\mathit{p}}}$
---|---|---|---|---|---
\cvd | CVD | \mwpc | MWPC | \gem | GEM

####  C.7.2 Detector components, electronics

\tell1 | TELL1 | \ukl1 | UKL1 | \beetle | Beetle
---|---|---|---|---|---
\otis | OTIS | \croc | CROC | \carioca | CARIOCA
\dialog | DIALOG | \sync | SYNC | \cardiac | CARDIAC
\gol | GOL | \vcsel | VCSEL | \ttc | TTC
\ttcrx | TTCrx | \hpd | HPD | \pmt | PMT
\specs | SPECS | \elmb | ELMB | \fpga | FPGA
\plc | PLC | \rasnik | RASNIK | \elmb | ELMB
\can | CAN | \lvds | LVDS | \ntc | NTC
\adc | ADC | \led | LED | \ccd | CCD
\hv | HV | \lv | LV | \pvss | PVSS
\cmos | CMOS | \fifo | FIFO | \ccpc | CCPC

####  C.7.3 Chemical symbols

\cfourften | $\mathrm{C_{4}F_{10}}$ | \cffour | $\mathrm{CF_{4}}$ | \cotwo | $\mathrm{CO_{2}}$
---|---|---|---|---|---
\csixffouteen | $\mathrm{C_{6}F_{14}}$ | \mgftwo | $\mathrm{MgF_{2}}$ | \siotwo | $\mathrm{SiO_{2}}$

###  C.8 Special Text

\eg | e.g. | \ie | i.e. | \etal | et al.
---|---|---|---|---|---
\etc | etc. | \cf | cf. | \ffp | ff.
\vs | vs. |  |  |  |

####  C.8.1 Helpful to align numbers in tables

\phz |
---|---

##  Appendix D Correlation matrices

The statistical correlation matrices of the determined $C\\!P$-violating
observables are shown in Table 10. The systematic correlations are shown in
Table 11.

Table 10: Statistical correlation matrix of $B^{0}$ observables, in percent. | $\mathcal{A}_{K\pi}$ | $\mathcal{R}_{\pi K}^{+}$ | $\mathcal{R}_{\pi K}^{-}$ | $\mathcal{A}_{K\pi\pi\pi}$ | $\mathcal{R}_{\pi K\pi\pi}^{+}$ | $\mathcal{R}_{\pi K\pi\pi}^{-}$ | $\mathcal{A}_{CP}^{KK}$ | $\mathcal{R}_{CP}^{KK}$ | $\mathcal{A}_{CP}^{\pi\pi}$ | $\mathcal{R}_{CP}^{\pi\pi}$ | $\mathcal{A}_{CP}^{\pi\pi\pi\pi}$ | $\mathcal{R}_{CP}^{\pi\pi\pi\pi}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\mathcal{A}_{K\pi}$ | $100$ | $9$ | $-12$ | $0$ | $0$ | $0$ | $0$ | $-1$ | $0$ | $0$ | $0$ | $0$
$\mathcal{R}_{\pi K}^{+}$ |  | $100$ | $6$ | $0$ | $0$ | $0$ | $0$ | $3$ | $0$ | $2$ | $0$ | $0$
$\mathcal{R}_{\pi K}^{-}$ |  |  | $100$ | $0$ | $0$ | $0$ | $0$ | $3$ | $0$ | $2$ | $0$ | $0$
$\mathcal{A}_{K\pi\pi\pi}$ |  |  |  | $100$ | $7$ | $-5$ | $0$ | $0$ | $0$ | $0$ | $0$ | $-1$
$\mathcal{R}_{\pi K\pi\pi}^{+}$ |  |  |  |  | $100$ | $8$ | $0$ | $0$ | $0$ | $0$ | $0$ | $2$
$\mathcal{R}_{\pi K\pi\pi}^{-}$ |  |  |  |  |  | $100$ | $0$ | $0$ | $0$ | $0$ | $0$ | $1$
$\mathcal{A}_{CP}^{KK}$ |  |  |  |  |  |  | $100$ | $4$ | $0$ | $0$ | $0$ | $0$
$\mathcal{R}_{CP}^{KK}$ |  |  |  |  |  |  |  | $100$ | $0$ | $5$ | $0$ | $0$
$\mathcal{A}_{CP}^{\pi\pi}$ |  |  |  |  |  |  |  |  | $100$ | $4$ | $0$ | $0$
$\mathcal{R}_{CP}^{\pi\pi}$ |  |  |  |  |  |  |  |  |  | $100$ | $0$ | $0$
$\mathcal{A}_{CP}^{\pi\pi\pi\pi}$ |  |  |  |  |  |  |  |  |  |  | $100$ | $0$
$\mathcal{R}_{CP}^{\pi\pi\pi\pi}$ |  |  |  |  |  |  |  |  |  |  |  | $100$

Table 11: Systematic correlation matrix of $B^{0}$ observables, in percent. | $\mathcal{A}_{K\pi}$ | $\mathcal{R}_{\pi K}^{+}$ | $\mathcal{R}_{\pi K}^{-}$ | $\mathcal{A}_{K\pi\pi\pi}$ | $\mathcal{R}_{\pi K\pi\pi}^{+}$ | $\mathcal{R}_{\pi K\pi\pi}^{-}$ | $\mathcal{A}_{CP}^{KK}$ | $\mathcal{R}_{CP}^{KK}$ | $\mathcal{A}_{CP}^{\pi\pi}$ | $\mathcal{R}_{CP}^{\pi\pi}$ | $\mathcal{A}_{CP}^{\pi\pi\pi\pi}$ | $\mathcal{R}_{CP}^{\pi\pi\pi\pi}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\mathcal{A}_{K\pi}$ | $100$ | $0$ | $-5$ | $81$ | $15$ | $-19$ | $78$ | $-3$ | $79$ | $-3$ | $78$ | $-3$
$\mathcal{R}_{\pi K}^{+}$ |  | $100$ | $15$ | $14$ | $7$ | $6$ | $12$ | $5$ | $11$ | $3$ | $5$ | $-1$
$\mathcal{R}_{\pi K}^{-}$ |  |  | $100$ | $-19$ | $7$ | $4$ | $-26$ | $14$ | $-25$ | $4$ | $-26$ | $-1$
$\mathcal{A}_{K\pi\pi\pi}$ |  |  |  | $100$ | $-1$ | $-9$ | $77$ | $0$ | $77$ | $-1$ | $76$ | $-30$
$\mathcal{R}_{\pi K\pi\pi}^{+}$ |  |  |  |  | $100$ | $15$ | $18$ | $0$ | $17$ | $1$ | $19$ | $4$
$\mathcal{R}_{\pi K\pi\pi}^{-}$ |  |  |  |  |  | $100$ | $-24$ | $0$ | $-24$ | $1$ | $-23$ | $4$
$\mathcal{A}_{CP}^{KK}$ |  |  |  |  |  |  | $100$ | $-3$ | $94$ | $-2$ | $93$ | $0$
$\mathcal{R}_{CP}^{KK}$ |  |  |  |  |  |  |  | $100$ | $-2$ | $15$ | $-1$ | $-2$
$\mathcal{A}_{CP}^{\pi\pi}$ |  |  |  |  |  |  |  |  | $100$ | $-3$ | $95$ | $0$
$\mathcal{R}_{CP}^{\pi\pi}$ |  |  |  |  |  |  |  |  |  | $100$ | $-2$ | $0$
$\mathcal{A}_{CP}^{\pi\pi\pi\pi}$ |  |  |  |  |  |  |  |  |  |  | $100$ | $-3$
$\mathcal{R}_{CP}^{\pi\pi\pi\pi}$ |  |  |  |  |  |  |  |  |  |  |  | $100$

LHCb collaboration

A. N. Other1.

1University of nowhere

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# A Deployment-First Methodology to Mechanism Design and Refinement in
Distributed Systems

Martijn de Vos, Georgy Ishmaev, Johan Pouwelse, Stefanie Roos
Delft University of Technology, The Netherlands

###### Abstract

Catalyzed by the popularity of blockchain technology, there has recently been
a renewed interest in the design, implementation and evaluation of
decentralized systems. Most of these systems are intended to be deployed at
scale and in heterogeneous environments with real users and unpredictable
workloads. Nevertheless, most research in this field evaluates such systems in
controlled environments that poorly reflect the complex conditions of real-
world environments. In this work, we argue that deployment is crucial to
understanding decentralized mechanisms in a real-world environment and an
enabler to building more robust and sustainable systems. We highlight the
merits of deployment by comparing this approach with other experimental setups
and show how our lab applied a _deployment-first methodology_. We then outline
how we use Tribler, our peer-to-peer file-sharing application, to deploy and
monitor decentralized mechanisms at scale. We illustrate the application of
our methodology by describing a deployment trial in experimental tokenomics.
Finally, we summarize four lessons learned from multiple deployment trials
where we applied our methodology.

###### Index Terms:

Decentralized Systems, Research Methodology, Experimental Setups, System
Failures.

##  I Introduction

The scale and complexity of distributed systems have increased tremendously
since the field’s inception. We live in an era of complex ultra-large-scale
networks characterized by the number of participating nodes and by high
heterogeneity, flexibility, non-trivial social dependencies, and emergent
properties [25, 11, 3]. Ensuring the proper functioning, deployment,
monitoring and maintenance of such systems requires system designers to obtain
insights into their performance and correctness. Given the scale and
unpredictability of such systems, obtaining engineering insights presents
unique challenges for researchers and developers.

Decentralized blockchain applications embody the characteristics of complex
ultra-large-scale networks [24, 2]. At the same time, the costs of failures in
these applications are very high due to built-in financial mechanisms [29].
Though realistic experimental setups such as testnets aim to address these
challenges to a degree, properly testing and evaluating such systems remains a
challenging endeavour [5].

The mainstream paradigm in distributed systems research is a top-down design
that focuses on predicting performance, failures and limitations through
experimentation in controlled environments. These experiments usually are
carried out as simulations or emulations informing researchers’ design choices
[4]. An empirical study of failures can provide invaluable insights into
different types of distributed systems [13, 12, 26]. Detection of events that
make a system fail to operate according to its specifications - detection of
failures - is often a critical task in distributed systems given complex
interdependencies[17].

However, empirical experimentation, guided by workloads extracted from a
deployed system, is uncommon in academic research [4]. Mature research
methodologies with an emphasis on deployment are still missing even in
empirically-driven fields of distributed systems research, for example,
blockchain applications [18].

We address this gap by presenting our deployment-first methodology for
designing and evaluating decentralized systems.111The scope of this paper is
primarily limited to decentralized systems. However, some of these insights
could be relevant to a wider field of distributed systems. We specifically
focus on findings that can be obtained from the study of failures after the
deployment. Our methodology is based on nearly two decades of experience
developing the Tribler software [22, 28, 8], serving as infrastructure for
deploying and evaluating decentralized mechanisms at scale. We demonstrate
that while the deployment-first research methodology can be demanding in terms
of time investment, additional insights obtained are worth this time
investment. A nuanced evaluation of trade-offs between different experimental
setups is also instrumental in designing research methodologies.

Figure 1: The standard research methodology to design, build, evaluate and
deploy decentralized systems.

In summary, this work makes the following contributions:

  1. 1.

We compare different experimental setups and highlight the merits of
deployment as a critical step in the research methodology when building and
evaluating distributed systems (Section II).

  2. 2.

We formulate our _deployment-first research methodology_ to evaluate
decentralized mechanisms at scale (Section III). We also describe how we use
Tribler, our decentralized file-sharing application and research vehicle for
conducting large-scale deployment trials.

  3. 3.

We illustrate how we applied our research methodology to obtain unique
insights in a complex use case on tokenomics in decentralized networks
(Section III-B).

  4. 4.

Based on our experiences with our methodology, we summarize four lessons we
learned (Section III-C).

##  II The Status Quo of Decentralized Systems Experiments

This work argues that deployment experiments are essential to build robust
decentralized mechanisms. We first describe the standard methodology to
research distributed systems and then outline the merits of deployment as part
of the research methodology.

###  II-A Standard Research Methodology

Figure 1 shows the standard research methodology to design, build, evaluate,
and deploy decentralized systems. This research methodology is based on
reports in the fields’ literature [15, 14, 4], on discussions with other
researchers, and our own experiences. The following five steps describe this
methodology:

  1. 1.

Mechanism Design. A researcher starts by designing a particular mechanism.
Research with an exclusive focus on understanding theoretical models is
usually limited to this step [4].

  2. 2.

Software Implementation. The researcher then works on a software
implementation of the designed mechanism. Assuming that the original design
was well-executed, the implementation phase should not result in significant
changes to the original models. As such, we left out this feedback loop from
Figure 1.

  3. 3.

Experiments. With the implementation, the researcher conducts experiments in
an environment controlled by the analyst, e.g., on a local computer or a
compute cluster. These experiments usually aim to verify the implementation’s
correctness and quantify system metrics, e.g., scalability and fault
tolerance.

  4. 4.

Refining The Design Using Experimental Results. Based on the experimental
results obtained in the previous step, the researcher updates the mechanism
design, updates the accompanying implementation and re-runs experiments. For
instance, an experiment revealing that a particular design has low scalability
(e.g., in the number of participants the system can support) might require the
researcher to identify bottlenecks in the mechanism and resolve them.

  5. 5.

Deployment and Monitoring. The researcher can deploy the algorithm in a real-
world setting after the experiments are finished. The deployed software will
likely be continuously monitored to detect failures or anomalies.

Most academic research does not further test and evaluate their mechanisms
using deployment and ends at step (4) [4]. This is not unexpected since local
experiments usually suffice to prove the mechanism’s trade-offs, correctness
or performance to a scientific community. As such, the time investment and
resource costs do not justify the need for deployment.222In contrast to that,
the deployment of novel system designs is a common practice in the blockchain
industry[18]. However, the quality of experimental evaluations is lacking
compared to academic research, as industry whitepapers often present biased
and inflated results [20]. We argue, however, that trade-offs and limitations
associated with the usage of experimental setups to evaluate decentralized
systems are more nuanced.

Property | Simulation | Emulation | Testnet | Real-world Deployment
---|---|---|---|---
Cost of experiment | Medium | Medium | Low/High | Low/High
Scalability | Medium/High | Resource constrained | Resource constrained | Resource constrained
Environmental realism | Low | Low/Medium | High | Very High
Failures discoverability | Impossible | Low | Medium | High
Reproducibility | High | Medium | Low | Low
Control | High | High | Medium | Low
Speed of change | Fast | Fast | Medium | Slow
Debugability | High | Medium | Medium | Low
TABLE I: A comparison between four different experimental setups: simulation,
emulation, testnet, and real-world deployment.

###  II-B The Merits of Deployment

We start by comparing the trade-offs between different experimental setups
used to evaluate decentralized systems. Based on our literature research, we
choose to compare the following four experimental setups:

  1. 1.

A simulation is a model of an application tested on a model of an environment;

  2. 2.

An emulation is an application that runs in an environment where some parts of
it are modelled;

  3. 3.

A testnet is an application that is deployed on multiple machines run by
researchers or volunteer testers;

  4. 4.

A real-world deployment is an application that end users run on their
machines.

We compare in Table I eight different properties of these experimental setups
and briefly discuss them below.

Costs of experiments. It is not always feasible to represent the costs of
experiments in commensurable scales. Most often, the costs of experiments can
be described by the monetary costs of purchasing necessary computation
resources. Both in simulation and emulation, these costs are relatively
manageable. However, in the case of a deployed system, experimentation costs
can be much higher or lower, depending on the application context. Experiments
on a deployed system that require changes in system parameters or
functionality can cause failures and loss of users or market share [5].
However, experiments on a deployed system can have meager costs in some
situations, e.g. if volunteer users provide their resources.333See
https://github.com/ethereum/ropsten/blob/master/revival.md. In testnets, if
most of the resources are provided by volunteer users, the experiment costs
can be low for researchers. However, attracting a sufficient amount of users
for a testnet can also require some initial investments and upfront costs to
get traction, as can be the case with marketing costs for blockchain testnets
[10].

Scalability. of experiments is strongly correlated with the costs of
experiments. In the case of simulations, it is relatively cheap to scale up
the simulation size. In emulation, the upper bound for scalability is
typically limited by the available computational resources of a testbed. In
the case of a testnet, the scalability is limited to the computational
resources available to the researcher. In a real-world deployment, the scale
of the experiment is usually limited by the number of end-users and the
resources they contribute.

Environmental realism. This is one of the two key features that set testnets
and deployments apart from other experimental setups. Environmental realism is
comprised of three different parameters: (1) client heterogeneity, e.g.,
differences in hardware capabilities; (2) variability in external parameters
such as network conditions; (3) the effect of user behaviour. A real-world
deployment captures all these three parameters. The key difference with the
testnet is the effect of user behaviour; e.g., a testnet can be exclusive to
expert users, reducing realism. User incentives in testnets are also sometimes
simplified, e.g., in blockchain testnets, there usually are no financial
incentives. Both in simulation and emulation, all three parameters of
environmental realism are lower compared to other experimental setups.
External parameters such as network conditions can be more realistic with
emulation. Realistic client heterogeneity is challenging to represent
realistically in an emulation conducted with heterogeneous hardware. One
observation is that environmental realism can be improved for simulation and
emulation if these setups are designed with values known from measurements of
deployed systems. Two key factors limit such measurements: first, the measured
system should have a very similar application context; Second, it is not
certain if the results of the measurements still hold as the system evolves.

There has been some work that aims to bring environmental realism to
simulation or emulation setups. Sarzyniec et al. present Distem, a
virtualisation platform to enable resource heterogeneity in a homogeneous
compute cluster [23]. While this is a step to make experiments more accurate,
the failure model and user-generated workloads are not carried over from the
real-world environment. Recent work on scalability experiments with BFT
consensus protocols proposes a simulator [1]. Understanding the _realism gap_
between simulators and real-world environments is a key part of this work.

Discoverability of new failures. This is another key property that
distinguishes controlled and uncontrolled experiment setups. Certain types of
failures can only be discovered in a real-world environment, particularly
emergent and user-caused failures [13]. Testnets can reveal certain types of
emergent and partial failures [5]. Relatively fewer novel types of failures
can be revealed by experiments using emulation setups. In principle,
simulations do not allow for discovering new types of failures.

Reproducibility. This parameter is tied to the availability of the same setup
to different researchers. Simulation, at least in theory, allows for the
highest level of reproducibility, given that all artefacts can easily be
published. With emulations, the evaluated software can be made available, but
access to an identical experimental testbed is not always available. It could
be argued that while reproducibility is low for both testnets and deployed
systems, a testnet allows for a relatively easier replication of experimental
conditions, which is almost impossible with deployed systems.

Control. Simulation and emulation are the experimental setups that give
researchers the most control over the flow of their experiments. With testnets
and real-world deployed systems, researchers usually have little to no control
over the system while it is running since there is a dependency on the
volunteers or end users that are running the software.

Speed of change. The speed at which an experiment can be modified is a
distinguishing factor between different experimental setups. This speed of
change is usually the lowest in a deployed real-world system, given the delays
caused by the propagation of software updates. Testnets can allow for somewhat
quicker deployment cycles. In both cases, deployment and the collection of
results can be time-consuming. Simulation and emulation setups have relatively
low external constraints and quickly be changed.

Debugability. Discovering, analyzing and reproducing software bugs in deployed
systems require dedicated infrastructure and can be a time-consuming process.
In comparison, testnets and emulation provide more debugability since the
researcher has a higher level of control. Simulation can provide the highest
discoverability of bugs as long as the scale of the simulation is not too
large.

This analysis shows that we can not have a comprehensive evaluation without
deployment. We need to account for environmental realism and failures, which
are detectable only in a real-world scenario.

Figure 2: Our deployment-first research methodology. The key difference with
Figure 1 is that we directly apply new insights or real-world workload traces
to the mechanism design and experiments, respectively (step 6).

##  III Tribler: Deploying and Monitoring Decentralized Mechanisms at Scale

Tribler is our lab’s peer-to-peer file-sharing software and research vehicle
to deploy and evaluate decentralized mechanisms at scale [8]. We have used the
Tribler software for almost two decades to obtain unique insights into the
complex interactions and dynamics in live peer-to-peer networks [22]. Over 30
PhD researchers and BSc/MSc students have used our software to evaluate their
mechanisms. Tribler also has a stable user base that enables longitudinal
deployment experiments. Over 1.8 million users have downloaded the Tribler
software, and at the time of writing, Tribler has 40’000 unique monthly
users.444See https://release.tribler.org

Tribler was initially designed as a file-sharing application that allows users
to download torrent files anonymously using a custom onion-routing protocol
[16]. Tribler uses the IPv8 networking library that supports authenticated
messaging and enables the construction and maintenance of decentralized
overlays. Over the years, however, Tribler has evolved from a BitTorrent
download client to a versatile application with features such as keyword
search, bundling torrents into channels, and reputation mechanisms to address
free-riding behaviour [19].

###  III-A Deployment-First Research Methodology

Tribler is a vital part of our labs’ research methodology since it enables us
to deploy and evaluate decentralized mechanisms at scale. Tribler also allowed
us to try out a different research methodology with an increased focus on
deployment. Originating from our experiences with Tribler and deployment
efforts, we now present our _deployment-first_ methodology of decentralized
systems design and refinement based on a continuous experimentation cycle.
Figure 2 visualizes an updated approach to the traditional research
methodology shown in Figure 1. We argue that in continuous experimentation,
the system’s deployment stage does not take place after experiments (step 3 in
Figure 2). Instead, we treat deployment as a critical next step in our
research methodology that happens after experiments. Potential findings from
deployment studies include discovering new types of failures that do not occur
in a controlled environment and novel insights or performance issues caused by
the unpredictability of real-world environments. These insights feed directly
into the refinement of the design and experiments in the following two ways
(step 6 in Figure 2). First, we leverage our new insights to update and
improve the decentralized mechanism, similar to how the standard research
methodology uses experimental results for refinement. Second, we use
information obtained from deployment to refine our experiments in a controlled
environment. This can be done, for example, by replaying a workload trace
obtained from the live network during in-house experiments to evaluate
mechanisms under a more realistic workload. A key focus during deployment is
on monitoring the mechanism to detect failure or anomalies. This is further
discussed in Section III-C.

Our methodology does not substitute the need for experiments in controlled
environments. On the opposite, data obtained from a deployment trial, such as
network characteristics, the performance of clients, and user behaviour,
should be used to address the limitations of other experimental setups, such
as simulation and emulation. Therefore, this data increases the realism of
local experiments and helps in further validating mechanisms before
deployment.

###  III-B Motivating Use-Case: Experimental Tokenomics

We now describe how we have applied our deployment-first methodology during a
recent deployment trial. This trial uses tokenomics to address free-riding
behaviour while downloading content with Tribler.

Mechanism Design and Objectives. A fundamental issue in peer-to-peer networks
is free-riding behaviour, where one peer takes more resources from the
community than it contributes [9]. In Tribler, this manifests as a user
downloading more data from others than contributing back (seeding). Earlier
work established that free-riding behaviour in Tribler is typical, resulting
in fewer uploaders and degradation of download speed [19]. Since our anonymous
downloading mechanism increases resource usage even further, addressing free-
riding behaviour became an important issue as the Tribler network grew.

Our solution to free-riding combines three complementary mechanisms, each
designed, evaluated and deployed in Tribler using our deployment-first
methodology (also see Figure 3). The first mechanism is a lightweight,
decentralized ledger named TrustChain, which stores all pair-wise bandwidth
transfers between users in the network in the form of records [21]. TrustChain
is designed explicitly for lightweight accounting in decentralized networks
and is highly scalable in the number of participants because it avoids a
global consensus mechanism. Users share these records with other users using a
simple gossiping mechanism. Our second mechanism is a reputation mechanism
that, based on received records, computes a trustworthiness score for other
users. The third mechanism is a resource allocation mechanism that determines
for each user to which other users it will upload data. The combined working
of these mechanisms allows users to identify free-riders themselves and
consequentially refuse them services while giving honest users preferential
treatment. Additional details and experimental results can be found in our
other work [7].

Figure 3: Three complementary mechanisms we used to address free-riding
behaviour in Tribler [7]. We evaluated each mechanism using our deployment-
first methodology.

Applying our Deployment-First Methodology. We deployed each of the three
mechanisms and went through various deployment cycles to improve and fine-tune
them. As a first step, we designed TrustChain, implemented it and conducted
correctness and validation experiments on our compute cluster (steps 1-3 in
Figure 2). We then integrated TrustChain into the Tribler software,
implemented a crawler to gather created records, and published a new software
release. Due to the lack of real-world traces and insights, we could not
adequately set some parameters, for example, the interval at which TrustChain
records are shared with other users. Only after a few deployment cycles did we
have insights on setting such parameters.

We continuously monitored the created TrustChain records in the Tribler
network, and we were able to detect various failures and design shortcomings
that were not discovered during our local experiments. For example, our
deployment revealed that our initial design of TrustChain was falling short
because a user can only be engaged in recording one transaction at the same
time. This shortcoming significantly limited the speed at which records could
be created and is an essential limitation since the Tribler software
frequently communicates with other users simultaneously. It bootstrapped a
redesign of the format of TrustChain records with support for concurrent
transactions (see [7]). At the same time, we used the collected TrustChain
records to start designing our reputation mechanism (see [21]). We also
discovered various bugs in the deployment stage, for example, one bug was
related to database corruption that occasionally occurs on a particular
version of Windows.

###  III-C Lessons Learned

We have conducted multiple deployment trials with Tribler. Due to space
constraints, we cannot discuss all insights obtained when applying our
deployment-first research methodology. However, we will summarize four lessons
we learned when working on the previously described use case and our other
deployment trials.

Lesson I: Plan for Mechanism Upgrades and Maintaining Backwards Compatibility.
Tribler consists of various mechanisms that we continuously monitor and
improve. Upgrading these mechanisms sometimes required us to make changes that
break compatibility with prior versions. This compatibility break results in
the fragmentation of the network since users with different versions of a
particular mechanism can no longer communicate with each other. Additionally,
such breaking changes often require software logic that updates locally stored
data (e.g., in a database) to be compatible with the new mechanism.

We aim to minimize the number of breaking mechanism changes to avoid too much
fragmentation of our network and to ensure sufficient usage of newly deployed
mechanisms. This aim is also motivated by our observation that users are
relatively slow in updating their Tribler software when a new release is
published, especially if the benefits of the software update are
unclear.555See https://release.tribler.org. During our deployment trials, we
learned that we should plan for mechanism upgrades already while designing a
particular mechanism. We note that this problem is not exclusive to Tribler
since many blockchain systems occasionally have to upgrade their network
protocol by releasing a new software version or forking the network, e.g., to
fix security issues or improve performance [27].

Lesson II: The Importance of Monitoring. Continuously monitoring the behaviour
of new mechanisms is critical to detect failures and anomalies in deployment
[28]. We engineer a crawler during every deployment trial and provision it
when a new Tribler release is published. This crawler joins a particular
overlay network as a peer, sends data queries to other Tribler instances and
persists the retrieved information in a local database. This practice is
comparable to collecting, analyzing and visualizing the transactions made in
blockchain networks.

We have deployed multiple crawlers to gather data from our live network. For
example, alongside the TrustChain ledger, we also deployed a crawler that
collects records created by users. However, due to churn, the crawler
sometimes is unable to collect particular data points. Because a user could
have gone offline before the crawler sent a request, our datasets did not
always contain all the data points we required. Despite this, the data
collected during deployment revealed a large-scale outage due to a software
bug since the number of created TrustChain records dropped significantly.
These experiences taught us that monitoring infrastructure is crucial to
planning a deployment.

Lesson III: Document all Design Decisions and Changes. Successfully applying
our deployment-first methodology requires adequate planning and introduces
unique challenges for developers and researchers. In early deployment trials,
we could have documented our design and deployment decisions better and,
therefore, would have avoided repeating prior mistakes. Over the years, we
adopted the _open science_ approach [6] to publicly record all our source
code, design decisions and meeting minutes. We also carefully report our
observations from the deployment environment and document failures to avoid
repeating particular mistakes in future iterations of a mechanism. This open
science approach is now an essential aspect of our Tribler development cycle
and research methodology. Open science also helps other researchers understand
and replicate our prior results. It is also beneficial for users interested in
understanding how the Tribler software behaves, what data is being collected,
and what mechanisms are being executed on their devices. All this information
is publicly available on our GitHub repository.666See
https://github.com/tribler/tribler/issues.

Lesson IV: Do not Deploy Too Much at Once. A common mistake we made during
early deployment trials was that we tended to include multiple new features or
mechanisms in a single release. Not only did this prolong the time between
software releases, but it also increased the risk of breaking the Tribler
software when there was a defect in one of the newly-deployed mechanisms. It
also made it impossible to isolate the effects of specific changes. To avoid
these risks, we currently aim to include at most one new feature per release
and aim for short release cycles. For example, we shipped each of the
mechanisms described in the use case in Section III-B with separate releases
with a few months between them.

We also learned that mechanism design is an incremental process that requires
multiple iterations to grow and become fruitful. For example, when designing a
socio-economic mechanism, it is often impossible to adequately parameterize
the mechanism since the dynamics of the deployment environment are not known
apriori by the researcher. Only in response to data collected from a real-
world environment the mechanism can be made robust and optimized for a
particular application domain.

##  IV The Road Ahead

We have argued that the increasing complexity and dependencies on
decentralized systems such as blockchain applications require more robust and
mature experimentation methodologies. Such methodologies are needed to
identify new types of failures in realistic environments. We argued that
deployment should be explicitly integrated as a key step in the research
methodology to improve the evaluation of decentralized mechanisms.

We have presented our _deployment-first approach_ that goes beyond the
standard research methodologies. We also presented Tribler, our research
vehicle for deploying decentralized mechanisms. We showed how we use insights
from deployment trials to improve the design of decentralized mechanisms and
their experiments. We have shown that experimental setups based on deployment
provide (1) insights into new types of failures; and (2) a foundation for the
design of realistic experiments in controlled environments. By describing a
tokenomics use case, we demonstrated the feasibility of our deployment-first
approach in practice.

Our deployment-first approach is a continuously evolving methodology. One
possible extension is the addition of infrastructure and approaches for A/B
testing decentralized mechanisms. This approach would serve different
algorithms and parameters to distinct subsets of users.

## References

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  * [22] Johan Pouwelse et al.  Tribler: a social-based peer-to-peer system.  Concurrency and computation: Practice and experience, 2008.
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setuptheorem setup questiontheorem question lemmatheorem lemma
assumptiontheorem assumption conjecturetheorem conjecture propositiontheorem
proposition corollarytheorem corollary problemtheorem problem claimtheorem
claim definitiontheorem definition exampletheorem example remarktheorem remark
remarkstheorem remarks

# Constructing algebraic solutions of Painleve VI equation from $p$-adic Hodge
theory and Langlands Correspondence

Jinbang Yang<EMAIL_ADDRESS>School of Mathematical Sciences, University
of Science and Technology of China, Hefei, Anhui 230026, PR China  and  Kang
Zuo<EMAIL_ADDRESS>Institut f ür Mathematik, Universit ät Mainz, Mainz
55099, Germany

###### Abstract.

We construct infinitely many non-isotrivial families of abelian varieties over
given four punctured projective lines. These families lead to algebraic
solutions of Painleve VI equation. Finally, based on a recent paper by Lin-
Sheng-Wang, we prove a complete characterization for the locus of motivic
Higgs bundles in the moduli space as fixed points of an “additive” self-map.
This is a note based on the lecture given by the second named author on
04.Nov.2022 at Tsinghua University.

##  0\. Introduction

Let $R$ be a commutative ring with identity and $X$ be a scheme over $R$. An
abelian scheme $A$ over $X$ together with a polarization $\mu$ is called of
_of $\mathrm{GL}_{2}$-type_, if there exists a number field $K$ of degree
$\dim_{X}A$ such that the ring of integers $\mathcal{O}_{K}$ can be embedded
into the endomorphism ring $\mathrm{End}_{\mu}(A/X)$. We will call the abelian
scheme _of $\mathrm{GL}_{2}(K)$-type_, if we want to emphasize the role of
$K$.

Let $f\colon A\rightarrow X$ be an abelian scheme of
$\mathrm{GL}_{2}(K)$-type. Let $D$ denote the discriminant locus and let
$X^{0}$ denote the complement of $D$ in $X$. Let $\Delta\subset A$ denote the
inverse image of $D$ under the structure morphism $f$ and let $A^{0}$ denote
the complement of $\Delta$ in $A$. Then we obtain the smooth abelian scheme
$f^{0}\colon A^{0}\rightarrow X^{0}$.

(0.1)
$\textstyle{A^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{0}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\Delta\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

For $R=\mathbb{C}$ we consider the Betti-local system

$\mathbb{V}=R^{1}_{\mathrm{B}}f^{0}_{*}\mathbb{Z}_{A^{0}}$

attached to $f^{0}$, which is a $\mathbb{Z}$-local system over the base
$X^{0}$. Since $f$ is of $\mathrm{GL}_{2}(K)$-type, the action of
$\mathcal{O}_{K}$ on $f$ induces an action of $K$ on the
$\overline{\mathbb{Q}}$-local system $\mathbb{V}\otimes\overline{\mathbb{Q}}$.
Taking the $K$-eigen sheaves decomposition

$\mathbb{V}\otimes\overline{\mathbb{Q}}=\bigoplus_{i=1}^{g}\mathbb{L}_{i}.$

Then there $\mathbb{L}_{i}$’s are of rank-$2$ over $X^{0}$ and defined over
the ring of integers of some number field. On the other hand, consider the
logarithmic de Rham bundle attached to the abelian scheme $f$ and denote

$(V,\nabla)=R^{1}_{\mathrm{dR}}f_{*}\Big{(}\Omega^{*}_{A/X}(\log\Delta),\mathrm{d}\Big{)}.$

On this de Rham bundle, there is a canonical filtration satisfying Griffiths
transversality given by

$E^{1,0}:=R^{0}f_{*}\Omega^{1}_{A/X}(\log\Delta))\subset V.$

Taking the grading with respect to this filtration, one gets a logarithmic
graded Higgs bundle, which is so-called Kodaira-Spencer map attached to $f$

(0.2) $(E,\theta):=(E^{1,0}\oplus
E^{0,1},\theta):=\mathrm{Gr}_{E^{1,0}}(V,\nabla)=\Big{(}R^{0}f_{*}\Omega^{1}_{A/X}(\log\Delta)\oplus
R^{1}f_{*}\mathcal{O}_{A},\mathrm{Gr}(\nabla)\Big{)}.$

Since $f$ is of $\mathrm{GL}_{2}(K)$-type, one also gets a $K$-eigen
decomposition of the Higgs bundle

(0.3) $(E,\theta)=\bigoplus_{i=1}^{g}(E,\theta)_{i}.$

Under Hitchin-Simpson’s non-abelian Hodge theory, these eigensheaves
$\\{(E,\theta)_{i}\\}_{i=1,\cdots,g}$ are just those Higgs bundles correspond
to the local systems $\\{\mathbb{L}_{i}\\}_{i=1,\cdots,g}$.

Local systems and Higgs bundles arising as subquotient of local systems and
Higgs bundles of some families of varieties are called _motivic_. Sometimes
they are also called _coming from geometry origin_. Simpson had found a
characterization for a rank-2 local system being motivic:

######  Theorem 0.1 (Simpson).

A rank-2 local system $\mathbb{L}$ is motivic if and only if the following two
conditions hold:

  1. (1)

$\mathbb{L}$ is defined over the ring of integers of some number field, and

  2. (2)

for each element $\sigma\in\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ the
Higgs bundle corresponding to the Galois conjugation $\mathbb{L}^{\sigma}$ is
again graded.

######  Conjecture  (Simpson).

A rigid local system is motivic.

Simpson’s conjecture has been proved for the case of rank-2 by Corlette-
Simpson [CS08] and rank-3 by Langer-Simpson [LS18] for cohomologically rigid
local systems. The conjecture predicts that any rigid local system
$\mathbb{L}$ shall enjoy all properties of motivic local systems. For example,

  * •

its corresponding filtered de Rham bundle is isomorphic to the underlying
filtered de Rham bundle of some Fontaine-Faltings modules at almost all
places, and

  * •

if $\mathbb{L}$ is in addtion cohomologically rigid, then it is defined over
the ring of integers of some number field.

Those two properties have been verified by Esnault-Groechenig recently [EG20],
[EG18].

In this note, we take $X$ as the complex projective line $\mathbb{P}^{1}$ and
$D$ as the $4$ punctures $\\{0,1,\infty,\lambda\\}$. Our goal is find some
motivic rank-$2$ logarithmic graded Higgs bundles over
$(\mathbb{P}^{1},\\{0,1,\infty,\lambda\\})$, which are coming from
$\mathrm{GL}_{2}$-type abelian schemes over $\mathbb{P}^{1}$ with discriminant
locus contained in $\\{0,1,\infty,\lambda\\}$.

Beavuille has shown that there exist exactly 6 non-isotrivial families of
elliptic curves over $\mathbb{P}^{1}$ with semistable reductions over
$\\{0,1,\infty,\lambda_{i}\\}$ for ${1\leq i\leq 6}$. All of them are modular
curves with respect to certain mixed level structures. Based on Beavuille’s
result, Viehweg-Zuo have shown that there no more non-isotrivial abelian
schemes over $\mathbb{P}^{1}$ of $\mathrm{GL}_{2}(K)$-type with
$[K:\mathbb{Q}]\geq 2$ and with semistable reductions over
$\\{0,1,\infty,\lambda\\}$. So except Beavuille’s example any non-isotrivial
smooth abelian schemes over $\mathbb{P}^{1}\setminus\\{0,1,\infty,\lambda\\}$
of $\mathrm{GL}_{2}(E)$-type must have non-semistable reduction at some point
in $\\{0,1,\infty,\lambda\\}$. In this case, the some eigenvalues of the local
monodromies of motivic local system must be roots of unity other than $1$.

In this note, we consider the simplest situation: motivic rank-2 local systems
whose local monodromies around $\\{0,1,\infty,\lambda\\}$ are unipotent and
around $\infty$ is quasiunipotent with eigenvalues $\\{-1,-1\\}$. We call such
type local monodromy to be _of type- $(1/2)_{\infty}$_, or just _of type-
$(1/2)$_.

######  Theorem 0.2.

For given $\lambda\in\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$, there exists
infinitely many non-isotrivial abelian schemes of $GL_{2}$-type over
$\mathbb{P}^{1}\setminus\\{0,1,\infty,\lambda\\}$ with the associated rank-$2$
eigen local systems being of type-$(1/2)_{\infty}$.

Let $M_{0,n}$ denote the moduli space of $n$-punctured projective lines and
let $S_{0,n}$ denote the total space of the universal family of $n$-punctured
projective lines with structure morphism

$p_{n}\colon S_{0,n}\rightarrow M_{0,n}.$

Then $M_{0,4}\simeq\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$, and
$S_{0,4}=\bigcup\limits_{\lambda\in
M_{0,4}}\big{(}\mathbb{P}^{1}\setminus\\{0,1,\infty,\lambda\\}\big{)}$ is an
algebraic surface. Once we vary the parameter $\lambda$, Theorem 0.2 implies
the following result:

######  Theorem 0.3.

There exist infinitely many non-isotrivial abelian schemes

$f:A\to\widetilde{S}_{0,4}$

of $\mathrm{GL}_{2}$-type over fiber products of finite étale base changes

$\textstyle{\widetilde{S}_{0,4}:=\widetilde{M}_{0,4}\times_{M_{0,4}}S_{0,4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{S}_{0,4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{4}}$$\textstyle{\widetilde{M}_{0,4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$finite
étale$\textstyle{M_{0,4}}$

and such that the local monodromies of $f$ around $\\{0,1,\lambda\\}$ are
unipotent and around $\\{\infty\\}$ is quasi-unipotent with all eigenvalues
being $-1$.

######  Corollary  (Corollary 4.).

Let $f:A\to\widetilde{S}_{0,4}$ be a family given in Theorem 0.3. Then all
rank-$2$ eigen local systems associated to the family $f$ are algebraic
solutions of Painleve VI equation of the type-$(1/2)_{\infty}$.

For given $\lambda\in\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$, any family
$f_{\lambda}\colon A_{\lambda}\rightarrow\mathbb{P}^{1}$ in Theorem 0.2 has
semistable reduction over $\\{0,1,\lambda\\}$ and potentially semistable
reduction over $\infty$. Thus the eigen Higgs bundles $(E,\theta)_{i}$
associated to this family (constructed in 0.3) have the following form

(0.4)
$E_{i}=\mathcal{O}\oplus\mathcal{O}(-1),\qquad\theta_{i}\colon\mathcal{O}\xrightarrow{\neq
0}\mathcal{O}(-1)\otimes\Omega^{1}_{\mathbb{P}^{1}}(\log\\{0,1,\infty,\lambda\\})$

and are endowed with natural parabolic structures on the punctures
$\\{0,1,\infty,\lambda\\}$ of type-$(1/2)_{\infty}$. Here
type-$(1/2)_{\infty}$ parabolic structures means that the parabolic structures
at $0$, $1$ and $\lambda$ are trivial and the parabolic filtration at $\infty$
is

$\left(E_{i}\mid_{\infty}\right)_{\alpha}=\left\\{\begin{array}[]{cc}E_{i}\mid_{\infty}&0\leq\alpha\leq
1/2,\\\ 0&1/2<\alpha<1.\\\ \end{array}\right.$

Let $M^{gr{1\over 2}}_{Hig\lambda}$ denote the moduli space of rank-2 semi-
stable graded Higgs bundles over $\mathbb{P}^{1}$ with the parabolic structure
on $\\{0,1,\infty,\lambda\\}$ of type-$(1/2)_{\infty}$. Then any Higgs bundle
$(E,\theta)\in M^{gr{1\over 2}}_{Hig\lambda}$ is parabolic stable and has the
form as in (0.4).

In view of $p$-adic Hodge theory, a Higgs bundle $(E,\theta)$ over the Witt
ring $W(\mathbb{F}_{q})$ realized by an abelian scheme over
$W(\mathbb{F}_{q})$ of $\text{GL}(E)_{2}$-type has to be the grading of an
$K$-eigen sheaf of the Fontaine-Faltings module attached to the abelian
scheme. Hence, by Lan-Sheng-Zuo functor $(E,\theta)$ is _periodic_ on
$M^{gr{1\over 2}}_{Hig\lambda}$ over $W(\mathbb{F}_{q})$.

One identifies

$M^{gr{1\over 2}}_{Hig\lambda}=\mathbb{P}^{1}$

by sending $(E,\theta)$ to the zero locus of the Higgs map
$(\theta)_{0}\in\mathbb{P}^{1}$, and takes then the elliptic curve
$C_{\lambda}$ of the Weierstrass form $y^{2}=z(z-1)(z-\lambda)$ as the double
cover

$\pi:C_{\lambda}\to\mathbb{P}^{1}$

ramified on $\\{0,1,\infty,\lambda\\}$.

######  Conjecture  (Sun-Yang-Zuo [SYZ17]).

The self-map induced by Higgs-de Rham flow on ${M^{gr{1\over
2}}_{Hig\lambda}}$ over ${\mathbb{F}_{q}}$ comes from multiplication map by
$p$ on the associated elliptic curve over $\mathbb{F}_{q}$

$\textstyle{C_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[p]}$$\textstyle{C_{\lambda}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{Higg,\lambda}^{gr}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{Higg,\lambda}^{gr}}$

The conjecture implies two things:

  1. (1)

a Higgs bundle $(E,\theta)$ is periodic if and only if $\pi^{-1}(\theta)_{0}$
is a torsion point in $C_{\lambda}$ and of order $p^{f}-1.$

  2. (2)

for a prime $p>2$ and assume $C_{\lambda}$ is supersingular then
$\phi_{\lambda}(z)=z^{p^{2}}$. Hence, any Higgs bundle
$(E,\theta)\in{M^{gr{1\over 2}}_{Hig\lambda}}(\overline{\mathbb{F}}_{q})$ is
periodic.

The Conjecture has been checked by Sun-Yang-Zuo for $p<50.$ Very recently it
has been proved by Lin-Sheng-Wang and becomes a theorem.

######  Theorem 0.4 (Lin-Sheng-Wang [LSW22]).

Section 0 holds true.

The technique used in Theorem 0.3 combined with Theorem 0.4 lead us to prove
the following result on a characterization for motivic Higgs bundles contained
in $M^{gr{1\over 2}}_{Hig\lambda}$.

######  Theorem 0.5.

A Higgs bundle $(E,\theta)\in M^{gr{1\over 2}}_{Hig\lambda}$ is an eigensheaf
of the Kodaira-Spencer map attached to an abelian scheme of
$\text{GL}_{2}$-type if and only if $\pi^{-1}(\theta)_{0}$ is a torsion point
in $C_{\lambda}$.

##  1\.  Discussion on Theorem 0.2

The underlying principle behind Theorem 0.2 is very simple, the so-called
isomonodromy deformation for motivic local systems over mixed characteristic.
To illustrate the idea, let’s first look at the situation over complex
numbers. We assume, there exists an abelian scheme
$f_{\lambda_{0}}:A_{\lambda_{0}}\to\mathbb{P}^{1}$ of $\text{GL}_{2}(K)$-type
over complex numbers with bad reduction on
$\\{0,\,1,\,\lambda_{0},\,\infty\\}$ of type-(1/2). Then the filtered
logarithmic de Rham bundle decomposes as $K$-eigen sheaves

$(V,\nabla,E^{1,0})=:(R^{1}_{\mathrm{dr}}f_{*}\Omega^{*}_{A_{\lambda_{0}}/\mathbb{P}^{1}}(\log\Delta),d),\,R^{0}f_{*}\Omega^{1}_{A_{\lambda_{0}}/\mathbb{P}^{1}}(\log\Delta))=\bigoplus_{i=1}^{g}(V,\nabla,E^{1,0})_{i},$

where each eigen sheaf has the form

$(V,\nabla,E^{1,0})_{i}\simeq(\mathcal{O}\oplus\mathcal{O}(-1),\nabla_{i},\mathcal{O}).$

Consider a family of 4-punctured projective line

$(\mathbb{P}^{1},\\{0,\,1,\,\lambda,\,\infty\\})_{\hat{U}_{\lambda_{0}}}\to\hat{U}_{\lambda_{0}}$

over a formal neighborhood $\hat{U}_{\lambda_{0}}\subset M_{0\,4}$ of
$\lambda_{0}.$ Then by forgetting the Hodge filtration the de Rham bundle
extends to a de Rham bundle $(V,\nabla)_{\hat{U}_{\lambda_{0}}}$ over
$(\mathbb{P}^{1},\\{0,\,1,\,\lambda,\,\infty\\})_{\hat{U}_{\lambda_{0}}}$. It
is known the abelian scheme extends over
$(\mathbb{P}^{1},\\{0,\,1,\,\lambda,\,\infty\\})_{\hat{U}_{\lambda_{0}}}$ if
and only if the Hodge filtration $E^{1,0}$ extends to a sub bundle in the de
Rham bundle $(V,\nabla)_{\hat{U}_{\lambda_{0}}}$. Using the $K$-eigen sheave
decomposition we see that the obstruction for extending the Hodge filtration
$E^{1,0}=\bigoplus_{i=1}^{g}\mathcal{O}$ lies in
$\bigoplus_{i=1}^{g}H^{1}(\mathbb{P}^{1},\mathcal{O}(-1))=0$. Hence, the
abelian scheme $f_{\lambda_{0}}$ extends over the base
$(\mathbb{P}^{1},\\{0,\,1,\,\lambda,\,\infty\\})_{\hat{U}_{\lambda_{0}}}$.

Back to the situation over mixed characteristic, along the diagram below. One

  * •

starts with moduli space ${M^{gr{1\over 2}}_{Hig\lambda}}$ of rank-$2$ stable
graded Higgs bundles on $\mathbb{P}^{1}_{\mathbb{Z}_{q}}$ of parabolic
type-$(1/2)_{\infty}$ on $\\{0,1,\lambda,\infty\\}$,

  * •

finds periodic Higgs bundles over $\mathbb{P}^{1}_{\mathbb{F}_{q}}$(i.e. fixed
points of iterations of the self map on ${M^{gr{1\over
2}}_{Hig\lambda}}\otimes\mathbb{F}_{q}$ induced by Higgs-de Rham flow),

  * •

gets Fontaine-Faltings modules via Lan-Sheng-Zuo functor from those periodic
Higgs bundles and lifts these modules to $\mathbb{P}^{1}_{\mathbb{Z}_{q}}$,

  * •

obtains rank-2 $\ell$-adic local systems on
$\mathbb{P}^{1}-\\{0,\,1,\,\lambda,\,\infty\\}$ over $\mathbb{F}_{q}$ by
forgetting Hodge filtration in the Fontaine-Faltings module, tensoring with
$\mathbb{Q}_{p}$, and applying Deligne’s $p$-$\ell$ companion conjecture
proven by Abe [Abe13],

  * •

finds $\text{GL}_{2}$-type abelian schemes over
$\mathbb{P}^{1}_{\mathbb{F}_{q}}$ with bad reductions of type-$(1/2)_{\infty}$
realizing those $\ell$-adic local systems via Drinfeld’s theorem on Langlands
correspondence.

  * •

modifies those $\mathrm{GL}_{2}$-type abelian schemes up to some $p$-isogeny
and lifts these modified $\mathrm{GL}_{2}$-type abelian schemes to
$\mathbb{P}^{1}_{\mathbb{Z}_{q}}$.

$\textstyle{M_{\rm p-adic}^{\rm
cris}/\chi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$Fontaine-
Faltings$\scriptstyle{p\text{\rm-adic RH}}$$\textstyle{M_{\rm
dR}^{FF}/\chi\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$forgetting
Hodge$\scriptstyle{\text{filtration, }\otimes\mathbb{Q}_{p}}$Higgs-de Rham
flowby Lan-Sheng-Zuo$\textstyle{M^{\rm
F-isoc}/\chi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{Deligne's
}p-\ell}$companion by Abe$\textstyle{M_{\ell-\text{adic}}}$$\textstyle{M_{\rm
Higg}^{\rm
per}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{M^{\rm
gr{1\over 2}}_{\rm Hig\,\lambda}}}$

And a type of boundedness and rigidity arguments of classifying maps from the
log base curve $(\mathbb{P}^{1},\\{0,1,\lambda,\infty\\})$ into the fine
moduli space of polarized abelian varieties shows that the abelian scheme
lifts over complex numbers for any $\lambda\in M_{0,4}(\mathbb{C}).$

Below we give more detailed explanations on the technique issue in each step:

###  1.1. Constructing Fontaine-Faltings modules over $\mathbb{Z}_{q}$ from
semistable Higgs bundles over $\mathbb{F}_{q}$ via Higgs de Rham flow.

Take a prime $p$ and an element $\lambda\in\mathbb{Z}_{q^{2}}$ such that the
modulo $p$ reduction of $C_{\lambda}$ is supersingular. Consider the moduli
space $M^{gr{1\over 2}}_{Hig\lambda}\otimes\mathbb{F}_{q^{2}}$ of rank-2
stable logarithmic graded Higgs bundles over
$(\mathbb{P}^{1},D)_{\mathbb{F}_{q^{2}}}$ with parabolic structures on $D$ of
type-$(1/2)$. Identifying $M^{gr{1\over
2}}_{Hig\lambda}\otimes\mathbb{F}_{q^{2}}$ with the projective line
$\mathbb{P}^{1}_{\mathbb{F}_{q^{2}}}$, the self-map induced by Higgs-de Rham
flow is given by $z\mapsto z^{p^{2}}$. We show that, for any $n\geq 1$, each
$(\overline{E},\overline{\theta})\in M^{gr{1\over
2}}_{Hig\lambda}(\overline{\mathbb{F}}_{q^{2n}})$ is automatically periodic
and lifts uniquely to a periodic Higgs bundle $(E,\theta)\in M^{gr{1\over
2}}_{Hig\lambda}(\mathbb{Z}_{q^{2n}}).$ Hence, the Lan-Sheng-Zuo functor
induces following bijection

$M^{gr{1\over 2}}_{Hig\lambda}(\mathbb{F}_{q^{2n}})\simeq M^{FF\,{1\over
2}}_{dR\,\lambda}(\mathbb{Z}_{q^{2n}})/\chi,$

where the right hand side is $M^{FF\,{1\over
2}}_{dR\,\lambda}(\mathbb{Z}_{q^{2n}})$ modulo an equivalent relation. The
$M^{FF\,{1\over 2}}_{dR\,\lambda}(\mathbb{Z}_{q^{2n}})$ is the set of rank-$2$
Fontaine-Faltings module over $(\mathbb{P}^{1},D)_{\mathbb{Z}_{q^{2n}}}$ such
that the monodromy is of type-$(1/2)$, and two Fontaine-Faltings modules in
$M^{FF\,{1\over 2}}_{dR\,\lambda}(\mathbb{Z}_{q^{2n}})$ are called equivalent
if they are differed by a constant rank-$1$ Fontaine-Faltings modules.

###  1.2. Deligne’s $p$-$\ell$ companion conjecture solved by Abe.

Consider the functors defined by forgetting Hodge filtration and tensoring
with $\mathbb{Q}_{p}$, which induce an injective map between a set of
logarithmic Fontaine-Faltings modules with parabolic structure-(${1\over 2}$)
and a set of logarithmic $F$-isocrystals. These functors keeps the same type
parabolic structure, and the result logarithmic $F$-isocrystals have a fixed
constant determinant, the $F$-isocrystal $\mathcal{E}_{\rm cy}$ associated to
the cyclotomic character. The injectivity follows the fact that the underlying
log-parabolic de Rham bundles are parabolic stable mod $p$.

We apply furtherly the forgetful functor by sending logarithmic
$F$-isocrystals to overconvergent F-isocrystals, which is known injective by
work of Kedlaya [Ked07, Proposition 6.3.2]. Hence, by putting every thing
together we obtain an injective map

$M^{gr{1\over 2}}_{Hig\lambda}(\mathbb{F}_{q^{2n}})\simeq M^{FF\,{1\over
2}}_{dR\,\lambda}(\mathbb{Z}_{q^{2n}})/\chi\hookrightarrow M^{F-iso\,{1\over
2}\,\dagger}_{\lambda}(\mathbb{Q}_{q^{2n}})/\chi,$

where $M^{F-iso\,{1\over 2}\dagger}_{\lambda}(\mathbb{Q}_{q^{2n}})$ is the set
of overconvergent F-isocrystals with parabolic structure of type-$({1/2})$ and
determinant $\mathcal{E}_{\rm cy}$.

We choose a prime $\ell\not=p$ and fix an isomorphism
$\phi:\overline{\mathbb{Q}}_{p}\simeq\overline{\mathbb{Q}}_{\ell}$. Given an
overconvergent $F$-isocrystal $(V,\nabla,\Phi)^{\dagger}\in M^{F-iso\,{1\over
2}\dagger}_{\lambda}(\mathbb{Q}_{q^{2n}})$, by applying Deligne’s $p-\ell$
companion proved by Abe to $(V,\nabla,\Phi)^{\dagger}$ we find a rank-2
$\ell$-adic irreducible local system $\mathbb{L}$ with cyclotomic determinant
corresponding to $(V,\nabla,\Phi)^{\dagger}$ in the sense the characteristic
polynomial of $\Phi_{x}$ on $V_{x}$ and the characteristic polynomial of
$\sigma_{x}$ on $\mathbb{L}_{x}$ via $\phi$ at all close point $x$ in the
close fiber. By the compatibility of local-global Langlands correspondence we
find the local monodromies of $\mathbb{L}$ around $\\{0,1,\lambda\\}$ are
unipotent and around $\infty$ is quasi-unipotent with eigenvalues
$\\{-1,\,-1\\}.$ Hence, we obtain a bijective functor

$M^{F-iso\,{1\over 2}\dagger}_{\lambda}(\mathbb{Q}_{q^{2n}})/\chi\simeq
M^{{1\over 2}}_{\ell-\text{adic}\,\lambda}(\mathbb{F}_{q^{2n}})/\sim,$

where $M^{{1\over 2}}_{\ell-\text{adic}\,\lambda}(\mathbb{F}_{q^{2n}})$ is the
set of equivalent classes of rank-2 $\ell$-adic local systems over
$(\mathbb{P}^{1}-D)_{\mathbb{F}_{q}^{2n}}$ with cyclotomic determinant,
unipotent local monodromies around $\\{0,1,\lambda\\}$ and quasi-unipotent
local monodromy around $\infty$ of eigenvalues $\\{-1,-1\\}$. The $\sim$
stands for an equivalent relation, two local systems $\mathbb{L}$ and
$\mathbb{L}^{\prime}$ are called equivalent if the restriction of them on
geometric fundamental group are isomorphic. As we show that the restriction of
$\mathbb{L}$ to the geometric fundamental group is automatically irreducible,
two local systems $\mathbb{L}$ and $\mathbb{L}^{\prime}$ are equivalent if and
only if there are differed by a character of the absolute Galois group of
$\mathbb{F}_{q^{2n}}$.

Composing the above functors together we obtain an injective functor

$M^{gr{1\over 2}}_{Hig\lambda}(\mathbb{F}_{q^{2n}})\simeq M^{FF\,{1\over
2}}_{dR\,\lambda}(\mathbb{Z}_{q^{2n}})/\chi\hookrightarrow M^{{1\over
2}}_{\ell-\text{adic}\,\lambda}(\mathbb{F}_{q^{2n}})/\sim.$

Over any field $E$ we have

$M^{gr{1\over 2}}_{Hig\lambda}\otimes E\simeq\mathbb{P}^{1}_{E},$

in particular,

$\\#M^{gr{1\over
2}}_{Hig\lambda}(\mathbb{F}_{q^{2n}})=\\#\mathbb{P}^{1}(\mathbb{F}_{q^{2n}})=q^{2n}+1.$

On the other hand by applying a formula due to Hongjie Yu [Y22], which solved
a conjecture by Deligne on counting the number of $\ell$-adic local systems on
a punctured smooth projective curve $(C,D)/\mathbb{F}_{q}$ in terms of the
number of semistable parabolic graded Higgs bundles on $(C,D)/\mathbb{F}_{q}$,
we find

$\\#M^{{1\over 2}}_{\ell-\text{adic}\,\lambda}(\mathbb{F}_{q^{2n}})=q^{2n}+1.$

This equality implies that the above injective functor is, in fact, bijective

(1.1) $M^{gr{1\over 2}}_{Hig\lambda}(\mathbb{F}_{q^{2n}})\simeq M^{FF\,{1\over
2}}_{dR\,\lambda}(\mathbb{Z}_{q^{2n}})/\chi\simeq M^{{1\over
2}}_{\ell-\text{adic}\,\lambda}(\mathbb{F}_{q^{2n}})/\sim.$

###  1.3. Constructing abelian scheme over $\mathbb{F}_{q}$ via Langlands
correspondence and lifting Hodge filtration of relative differential 1-forms
to characteristic zero.

Given a local system $\mathbb{L}\in M^{{1\over 2}}_{\ell-
adic\,\lambda}(\mathbb{F}_{q^{2n}})$ with cyclotomic determinant. Then one
shows that the restriction of $\mathbb{L}$ to the geometric fundamental group
is irreducible with infinite local monodromy at least on one puncture. By
applying Drinfeld’s theorem [Dri81] to $\mathbb{L}$ there exists an abelian
scheme

$f:A\to\mathbb{P}^{1}_{\mathbb{F}_{q^{2n}}}$

with the bad reduction $\Delta$ over $\\{0,1,\infty,\lambda\\}$ and such that:

  1. (1)

The abelian scheme is of $\mathrm{GL}_{2}(K)$-type, where $K$ is the number
field generated by traces of Frobenius on $(\mathbb{L})_{x}$ at all close
points $x\in\mathbb{P}^{1}-D$.

  2. (2)

Let

$\mathbb{V}:=R^{1}_{\mathrm{et}}f_{*}\overline{\mathbb{Q}}_{\ell\,A^{0}}=\bigoplus_{i=1}^{g}\mathbb{L}_{i}$

be the $K$-eigen decomposition of the $\ell$-adic local system attached to the
family. Then the local system $\mathbb{L}$ is isomorphic to a $K$-eigen sheaf,
say $\mathbb{L}_{1}$, and all eigen sheaves have the cyclotomic character.
Moreover, the local monodromies matrices of eigen sheaves at any puncture
$x\in\\{0,1,\infty,\lambda\\}$ are of the same type.

  3. (3)

Consider the realization of the $\mathcal{O}_{K}$-log Dieudonné crystal
attached to $f$ over
$(\mathbb{P}^{1},\\{0,1,\infty,\lambda\\})_{\mathbb{Z}_{q^{2n}}}$, one gets

$(V,\nabla,\Phi,\mathcal{V})=\Big{(}R^{1}_{\mathrm{cry}}f_{*}\mathcal{O}_{A,crys}\Big{)}(\mathbb{P}^{1}_{\mathbb{Z}_{q^{2n}}}).$

The $\mathrm{GL}_{2}$-structure induces a $K$-eigen sheaves decomposition of
$F$-isocrystals

$(V,\nabla,\Phi)\otimes\mathbb{Q}_{p}=\bigoplus_{i=1}^{g}(V,\nabla,\Phi)_{i\,\mathbb{Q}_{p}}$

where $(V,\nabla,\Phi)_{i,\mathbb{Q}_{p}}$ is a $\sigma^{f}$-log
$F$-isocrystal with determinant $\mathcal{E}_{cy}$. Under $p$-$\ell$
companion, the eigen sheaves correspond to local systems with cyclotomic
determinant. The parabolic structure induced by the residue of the connection
is of type-$(1/2)$.

According the bijection (1.1), each $(V,\nabla,\Phi)_{i\mathbb{Q}_{p}}$ has an
integral lattice, which underlies a Fontaine-Faltings module
$(V,\nabla,Fil,\Phi)^{FF}_{i}$ over $\mathbb{Z}_{q^{2n}}.$ Consequently the
multiplication field $K$ is unramified at each place above $p$. This shows
that the above decomposition can be extended over $\mathbb{Z}_{q^{2n}}$. In
other words, there exists $(V,\nabla,\Phi)_{i}$ such that

$(V,\nabla,\Phi)=\bigoplus_{i=1}^{g}(V,\nabla,\Phi)_{i}$

and

$(V,\nabla,\Phi)_{i}\otimes\mathbb{Q}_{p}\simeq(V,\nabla,\Phi)^{FF}_{i}\otimes\mathbb{Q}_{p}.$

Consider the new $\mathcal{O}_{K}$-lattice

$(V,\nabla,\Phi)^{FF}:=\bigoplus_{i=1}^{g}(V,\nabla,\Phi)_{i}^{FF}$

of $(V,\nabla,\Phi)\otimes\mathbb{Q}_{p}$ defined from the Fontaine-Faltings
module

$(V,\nabla,Fil,\Phi)^{FF}:=\bigoplus_{i=1}^{g}(V,\nabla,Fil,\Phi)_{i}^{FF}.$

By extending the coefficient, one gets a Verschiebung on
$(V,\nabla,\Phi)\otimes\mathbb{Q}_{p}$. By restricting onto the new lattice,
one gets a Verschiebung structure $\mathcal{V}$ on $(V,\nabla,Fil,\Phi)^{FF}$.

According equivalence of the Dieudonné functor, from the new modified
$\mathcal{O}_{K}$-log Dieudonné crystal $(V,\nabla,\Phi,\mathcal{V})^{FF}$,
one gets a $p$-isogeny $f^{\prime}:A^{\prime}\to\mathbb{P}^{1}$ of the
original abelian scheme such that $(V,\nabla,\Phi,\mathcal{V})^{FF}$ is the
Dieudonné crystal attached to $f^{\prime}$.

From the family $f^{\prime}$, by taking relative differential $1$-forms one
gets the natural Hodge filtration on
$(V,\nabla)^{FF}\otimes\mathbb{F}_{q^{2n}}$ given by

$E^{1,0}:=R^{0}f^{\prime}_{*}\Omega^{1}_{A^{\prime}/\mathbb{P}^{1}}(\log\Delta)\subset(V,\nabla)^{FF}\otimes\mathbb{F}_{q^{2n}}=R^{1}_{dR}f^{\prime}_{*}(\Omega^{\bullet}_{A^{\prime}/\mathbb{P}^{1}}(\log\Delta),d),$

which is a rank-$g$ sub bundle. Since the relative Frobenius $\Phi$ in the
Fontaine-Faltings module satisfies the strong $p$-divisible condition with
respect to the filtration $Fil$, the Hodge filtration is coincide with the
modulo $p$ reduction of the filtration in the Fontaine-Faltings module. In
other words, $Fil$ is a filtration lifts the Hodge filtration relative
differential $1$-forms attached to $f^{\prime}$.

###  1.4. Lifting abelian scheme from characteristic $p$ to characteristic
zero by Grothendieck-Messing-Kato logarithmic deformation theorem

By Zarhin’s trick, the fiber product

$f^{{}^{\prime}(4,4)}:A^{{}^{\prime}4,4}=(A^{\prime}\times
A^{{}^{\prime}\,t})^{4}\to\mathbb{P}^{1}$

with the induced $\mathcal{O}^{(4,4)}_{K}$-multiplication carries an
$\mathcal{O}^{(4,4)}_{K}$-principle polarization

$\iota:A^{{}^{\prime}4,4}\simeq A^{{}^{\prime}4,4\,t}.$

This $\mathcal{O}^{4,4}_{K}$-polarization induces an isomorphism between the
$\mathcal{O}^{4,4}_{K}$-eigen sheaves of the Dieudonné module and its dual

$\iota^{*}:(V,\nabla,\Phi,\mathcal{V})^{FF}=\bigoplus_{i=1}^{8g}(V,\nabla,\Phi,\mathcal{V})_{i}^{FF}\simeq\bigoplus_{i=1}^{8g}(V,\nabla,\Phi,\mathcal{V})_{i}^{FF\vee}=(V,\nabla,\Phi,\mathcal{V})^{FF\vee},$

which carries the Hodge filtration $Fil$ and $Fil^{\vee}$ as liftings of the
Hodge bundles $E^{1,0}_{f^{{}^{\prime}4,4}}$ and
$E^{1,0}_{f^{{}^{\prime}4,4t}}$. One checks that
$\text{Gr}_{E^{1,0}_{f^{{}^{\prime}4,4}}}(V,\nabla)^{FF}\otimes\mathbb{F}_{q^{2n}}$
and
$\text{Gr}_{E^{1,0}_{f^{{}^{\prime}4,4t}}}(V,\nabla)^{FF\vee}\otimes\mathbb{F}_{q^{2n}}$
are parabolic stable with respect to the $\mathcal{O}^{4,4}_{K}$-eigen sheaves
decomposition. Hence, we obtain

(1.2) $\iota^{*}Fil=Fil^{\vee}$

By Faltings-Chai Theorem [FC90] on the fine arithmetic moduli space
$\mathcal{A}_{8g,1,3}=:\mathcal{X}^{0}$ of principle polarized abelian
varieties with level-3 exists over $\mathbb{Z}[e^{{2i\pi\over 3}},\,1/3]$,
which is smooth and carries an universal abelian scheme

$\mathcal{A}^{0}\to\mathcal{X}^{0}.$

Further more, there exists a smooth Toroidal compactification
$\mathcal{X}\supset\mathcal{X}^{0}$ over $\mathbb{Z}[e^{{2i\pi\over
3}},\,1/3]$ with a smooth compactification of the universal abelian scheme

$f:\mathcal{A}\to\mathcal{X}$

such that $\mathcal{A}\setminus\mathcal{A}^{0}$ is a relative normal crossing
divisor over $\mathcal{X}\setminus\mathcal{X}^{0}=:\infty$. For an
$\mathcal{O}^{(4,4)}_{E}$-principle polarized abelian scheme
$f^{{}^{\prime}(4,4)}$ together with a lifting of the base
$(\mathbb{P}^{1},D)_{\mathbb{Z}_{q^{2n}}}$
In order to obtain a period map into the fine moduli space $\mathcal{X}^{0}$
for $f^{{}^{\prime}(4,4)}$ we take the base change

$\chi:(\mathbb{P}^{1},D)_{\chi}\to(\mathbb{P}^{1},D)$

defined by torsion-6 subgroup of $f^{s(4,4)}$. The fiber product of the base
change

$f^{{}^{\prime}4,4}_{\chi}:A^{{}^{\prime}4,4}_{\chi}\to\mathbb{P}^{1}_{\chi}$

has semistable reduction on $D_{\chi}$ and carries a level-6 structure. Hence,
$f^{s(\lambda 4,4)0}_{\chi}$ induces a log period map

$\psi:(\mathbb{P}^{1},D)_{\chi}\to(\mathcal{X},\infty)\otimes k.$

By Kato’s log deformation theorem [Kat96], the local liftings of $\psi$ on
$W_{n}(k)$ defines an obstruction-cocycle in
$H^{1}(\mathbb{P}^{1}_{\chi}\otimes
k,\psi^{*}\Theta_{\mathcal{X}/k}(\log\infty))$. The local liftings can be glue
if and only if the obstruction class vanishes.

By Faltings-Chai [FC90], the universal Kodaira-Spencer map induces an
identification

$\Theta_{\mathcal{X}}(\log\infty)\simeq S^{2}E_{\mathcal{X}}^{0,1}$

and we may identify the obstruction for lifting of $\psi$ with the obstruction
for lifting the Hodge bundle $E^{1,0}_{f^{{}^{\prime}4,4}}$ satisfying (1.2).
As $E^{1,0}_{f^{{}^{\prime}4,4}}$ lifts we show $\psi$ lifts. Which
corresponds to a lifting of the abelian scheme

$(f^{{}^{\prime}4,4}:A^{{}^{\prime}4,4}_{\chi}\to\mathbb{P}^{1}_{\chi})_{\mathbb{Z}_{q^{2n}}}$

with a lifting of $\mathcal{O}^{4,4}_{K}$-action, as the the Dieudonné crystal
with $Fil$ carries an $\mathcal{O}^{4,4}_{K}$-action.

As $f^{{}^{\prime}4,4}$ over $\mathbb{P}^{1}_{\chi}/\mathbb{F}_{q^{2n}}$
descends to $\mathbb{P}^{1}/\mathbb{F}_{q^{2n}}$ and the Fontaine-Faltings
module attached to $f^{{}^{\prime}4,4}$ over
$\mathbb{P}^{1}_{\chi}/W(\mathbb{F}_{q^{2n}})$ descends to
$\mathbb{P}^{1}/W(\mathbb{F}_{q^{2n}})$ we obtain $f^{{}^{\prime}4,4}$
descends to $\mathbb{P}^{1}/W(\mathbb{F}_{q^{2n}})$.

##  2\. Sketch the proof of Theorem 0.5.

Let $g:C\to\mathbb{P}^{1}$ be the Legendre family. Then one may identify the
smooth locus of $g$ with $M_{0,4}$, the moduli space of projective line with
$4$-punctures, which sends $\lambda$ to the projective line with punctures at
$\\{0,1,\lambda,\infty\\}$. For any $\lambda\neq 0,1,\infty$, the fiber of $g$
at $\lambda$ is just the elliptic curve given by the double cover
$\pi_{\lambda}:C_{\lambda}\to\mathbb{P}^{1}$ ramified on
$\\{0,1,\lambda,\infty\\}$

For a $\lambda_{0}\in M_{0,4}$, take a Higgs bundle $(E,\theta)$ in
$M^{gr{1\over 2}}_{Hig\,\lambda_{0}}.$ Then Theorem 0.5 claims that
$(E,\theta)$ is a motivic Higgs bundle if and only if
$(\theta)_{0}\in\mathbb{P}^{1}$ is a torsion point with respect to
$\lambda_{0}$ (i.e. the preimage in $\pi_{\lambda_{0}}^{-1}(\theta)_{0}\in
C_{\lambda_{0}}$ are torsion points). In the following, we give a sketch of
the proof of Theorem 0.5.

Assume $(E,\theta)$ is motivic. Then the modulo $\mathfrak{p}$ reduction of
$(E,\theta)$ is periodic for almost all places $\mathfrak{p}$. According
Theorem 0.4, the modulo $\mathfrak{p}$ reduction of $(\theta)_{0}$ is torsion.
By a theorem of Pink [Pin04], it itself is torsion.

Conversely, assume $(\theta)_{0}$ is a torsion point with order $m$, in the
following we show $(E,\theta)$ is motivic. We first show a very special case:

Case 1. Assume $\lambda_{0}$ takes a value in $\mathcal{O}_{K}$, the ring of
integers of some number field, such that $C_{\lambda_{0}}$ is an elliptic
curve with complex multiplication. Choose a sufficient large place
$\mathfrak{p}$ such that the reduction of $C_{\lambda_{0}}$ at $\mathfrak{p}$
is supersingular and $\mathfrak{p}\nmid m$.

Since the modulo $\mathfrak{p}$ reduction $(\bar{E},\bar{\theta})$ of the
Higgs bundle $(E,\theta)$ is also torsion and of order $m$ with
$\mathfrak{p}\nmid m$. By Theorem 0.4, the reduction $(\bar{E},\bar{\theta})$
is periodic. According the bijection in (1.1), the modulo $p$ reduction
$(\bar{E},\bar{\theta})$ lifts to a periodic Higgs bundle $(E,\theta)^{per}$
on $(\mathbb{P}^{1},\\{0,1,\lambda_{0},\infty\\})/W(\mathbb{F}_{q_{0}^{2n}})$.
In Section 1.4 we show that there exists an abelian scheme
$f_{\lambda_{0}}:A\to\mathbb{P}^{1}$ of $\text{GL}_{2}(K)$-type over
characteristic $0$ with bad reduction on $\\{0,\,1,\,\lambda_{0},\,\infty\\})$
of type-(1/2) such that $(E,\theta)^{per}$ is an $K$-eigen Higgs bundle
attached to $f_{\lambda_{0}}$. Hence, $(E,\theta)^{per}$ is motivic. In
particular $(\theta^{per})_{0}$ is also torsion and has the some modulo
$\mathfrak{p}$ reduction as $\theta_{0}$.

We claim that $(E,\theta)\simeq(E,\theta)^{per}$. In particular, $(E,\theta)$
is motivic. Since $C_{\lambda_{0}}$ has complex multiplication, the field
generated by $p$-torsion point must be ramified above $p$. In particular, the
order of $(\theta^{per})_{0}$ does not divided by $p$. Together with the fact
that $(\theta)_{0}\equiv(\theta^{per})_{0}\pmod{\mathfrak{p}}$, one gets
$(E,\theta)\simeq(E,\theta)^{per}$.

Case 2. Let $\Sigma_{m}\subset C$ be the $m$-torsion (multiple) section,
$T_{m}=\pi(\Sigma_{m})\subset M_{0,4}.$ Then $T_{m}$ is étale over $M_{0,4}$.
Let $T^{\prime}_{m}$ be the irreducible component of $T_{m}$ containing
$(\theta)_{0}$.

Recall the fact that the set of elliptic curves with complex multiplication
are dense in the moduli space. We can find a subset of infinitely many
$\\{\lambda_{i}\\}$ with the some modulo $\mathfrak{p}$-reduction such that
all $C_{\lambda_{i}}$ are elliptic curve with complex multiplication. Choose a
point $z_{i}$ in the intersection of $T^{\prime}_{m}$ and the fiber above
$\lambda_{i}$ for each $i$.

With the method in case 1, we find abelian schemes
$f_{\lambda_{i}}:A_{\lambda_{i}}\to\mathbb{P}^{1}$ of $\text{GL}_{2}(K)$-type
with bad reduction on $\\{0,\,1,\,\lambda_{i},\,\infty\\},$ such that
$(E,\theta)_{z_{i}}$ is an $K$-eigen Higgs bundle attached to
$f_{\lambda_{i}}$. One shows that infinitely many of $\\{f_{\lambda_{i}}\\}$
glue together into an abelian scheme $f:A\to M_{0,4}$. Consequently, the
subset $\\{z_{i}\\}$, where $z_{i}=(\theta_{z_{i}})_{0}$, glue into a
component $Z_{m}$ of $\Sigma_{m}$ such that for any $z_{\lambda}\in
Z_{m}\cap(\mathbb{P}^{1},\\{0,\,1,\,\lambda,\,\infty\\})$ the Higgs bundle
$(E,\theta)_{z_{\lambda}}$ with $(\theta_{z_{\lambda}})_{0}=z_{\lambda}$ is a
motivic Higgs bundle on $(\mathbb{P}^{1},\\{0,\,1,\,\lambda,\,\infty\\}).$

By the construction of $Z_{m}$ and $T^{\prime}_{m}$, Both of then are
irreducible, relative dimensional one over the base and their intersect set is
infinite. Hence $Z_{m}=T_{m}$. In particular $(E,\theta)$ is motivic.

######  Acknowledgement.

The authors warmly thank Raju Krishnamoorthy, Mao Sheng,
Carlos Simpson, Rui-Ran Sun, Hong-Jie Yu and Shing-Tung Yau for helpful
discussions.

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