\newcommand{\cat}[1]{\mathsf{#1}}

\newcommand{\Sets}[0]{{\mathsf{Set}}}
\newcommand{\Set}[0]{{\mathsf{Set}}}
\newcommand{\sets}[0]{{\mathsf{Set}}}
\newcommand{\set}{{\mathsf{Set} }}
\newcommand{\Poset}[0]{\mathsf{Poset}}
\newcommand{\GSets}[0]{{G\dash\mathsf{Set}}}
\newcommand{\Groups}[0]{{\mathsf{Group}}}
\newcommand{\Grp}[0]{{\mathsf{Grp}}}

% Modifiers
\newcommand{\der}[0]{{\mathsf{d}}}
\newcommand{\dg}[0]{{\mathsf{dg}}}
\newcommand{\comm}[0]{{\mathsf{C}}}
\newcommand{\pre}[0]{{\mathsf{pre}}}
\newcommand{\fn}[0]{{\mathsf{fn}}}
\newcommand{\smooth}[0]{{\mathsf{sm}}}
\newcommand{\Aff}[0]{{\mathsf{Aff}}}
\newcommand{\Ab}[0]{{\mathsf{Ab}}}
\newcommand{\Add}[0]{{\mathsf{Add}}}
\newcommand{\Assoc}[0]{\mathsf{Assoc}}
\newcommand{\Ch}[0]{\mathsf{Ch}}
\newcommand{\Coh}[0]{{\mathsf{Coh}}}
\newcommand{\Comm}[0]{\mathsf{Comm}}
\newcommand{\Cor}[0]{\mathsf{Cor}}
\newcommand{\Corr}[0]{\mathsf{Cor}}
\newcommand{\Fin}[0]{{\mathsf{Fin}}}
\newcommand{\Free}[0]{\mathsf{Free}}
\newcommand{\Tors}[0]{\mathsf{Tors}}
\newcommand{\Perf}[0]{\mathsf{Perf}}
\newcommand{\Unital}[0]{\mathsf{Unital}}
\newcommand{\eff}[0]{\mathsf{eff}}
\newcommand{\derivedcat}[1]{\mathbf{D} {#1} }
\newcommand{\bderivedcat}[1]{\mathbf{D}^b {#1} }
\newcommand{\Cx}[0]{\mathsf{Ch}}
\newcommand{\Stable}[0]{\mathsf{Stab}}
\newcommand{\ChainCx}[1]{\mathsf{Ch}\qty{ #1 } }
\newcommand{\Vect}[0]{{ \mathsf{Vect} }}
\newcommand{\kvect}[0]{{ \mathsf{Vect}\slice{k} }}
\newcommand{\loc}[0]{{\mathsf{loc}}}
\newcommand{\locfree}[0]{{\mathsf{locfree}}}
\newcommand{\Bun}{{\mathsf{Bun}}}
\newcommand{\bung}{{\mathsf{Bun}_G}}

% Rings
\newcommand{\Local}[0]{\mathsf{Local}}
\newcommand{\Fieldsover}[1]{{ \mathsf{Fields}_{#1} }}
\newcommand{\Field}[0]{\mathsf{Field}}
\newcommand{\Number}[0]{\mathsf{Number}}
\newcommand{\Numberfield}[0]{\Field\slice{\QQ}}
\newcommand{\NF}[0]{\Numberfield}
\newcommand{\Art}[0]{\mathsf{Art}}
\newcommand{\Global}[0]{\mathsf{Global}}
\newcommand{\Ring}[0]{\mathsf{Ring}}
\newcommand{\Mon}[0]{\mathsf{Mon}}
\newcommand{\CMon}[0]{\mathsf{CMon}}
\newcommand{\CRing}[0]{\mathsf{CRing}}
\newcommand{\DedekindDomain}[0]{\mathsf{DedekindDom}}
\newcommand{\IntDomain}[0]{\mathsf{IntDom}}
\newcommand{\Domain}[0]{\mathsf{Domain}}
\newcommand{\DVR}[0]{\mathsf{DVR}}
\newcommand{\Dedekind}[0]{\mathsf{Dedekind}}

% Modules
\newcommand{\modr}[0]{{\mathsf{Mod}\dash\mathsf{R}}}
\newcommand{\modsleft}[1]{\mathsf{#1}\dash\mathsf{Mod}}
\newcommand{\modsright}[1]{\mathsf{Mod}\dash\mathsf{#1}}
\newcommand{\mods}[1]{{\mathsf{#1}\dash\mathsf{Mod}}}
\newcommand{\stmods}[1]{{\mathsf{#1}\dash\mathsf{stMod}}}
\newcommand{\grmods}[1]{{\mathsf{#1}\dash\mathsf{grMod}}}
\newcommand{\comods}[1]{{\mathsf{#1}\dash\mathsf{coMod}}}
\newcommand{\algs}[1]{{{#1}\dash\mathsf{Alg}}}
\newcommand{\Quat}[0]{{\mathsf{Quat}}}
\newcommand{\torsors}[1]{{\mathsf{#1}\dash\mathsf{Torsors}}}
\newcommand{\torsorsright}[1]{\mathsf{Torsors}\dash\mathsf{#1}}
\newcommand{\torsorsleft}[1]{\mathsf{#1}\dash\mathsf{Torsors}}
\newcommand{\bimod}[2]{({#1}, {#2})\dash\mathsf{biMod}}
\newcommand{\bimods}[2]{({#1}, {#2})\dash\mathsf{biMod}}
\newcommand{\Mod}[0]{{\mathsf{Mod}}}
\newcommand{\Dmod}[0]{{ \mathcal{D}\dash\mathsf{Mod} }}

\newcommand{\zmod}[0]{{\mathbb{Z}\dash\mathsf{Mod}}}
\newcommand{\rmod}[0]{{\mathsf{R}\dash\mathsf{Mod}}}
\newcommand{\amod}[0]{{\mathsf{A}\dash\mathsf{Mod}}}
\newcommand{\kmod}[0]{{\mathsf{k}\dash\mathsf{Mod}}}
\newcommand{\gmod}[0]{{\mathsf{G}\dash\mathsf{Mod}}}
\newcommand{\grMod}[0]{{\mathsf{grMod}}}
\newcommand{\gr}[0]{{\mathsf{gr}\,}}
\newcommand{\mmod}[0]{{\dash\mathsf{Mod}}}
\newcommand{\Rep}[0]{{\mathsf{Rep}}}
\newcommand{\Irr}[0]{{\mathsf{Irr}}}
\newcommand{\Adm}[0]{{\mathsf{Adm}}}
\newcommand{\semisimp}[0]{{\mathsf{ss}}}

% Vector Spaces and Bundles
\newcommand{\VectBundle}[0]{{ \Bun\qty{\GL_r} }}
\newcommand{\VectBundlerk}[1]{{ \Bun\qty{\GL_{#1}} }}
\newcommand{\VectSp}[0]{{ \VectSp }}
\newcommand{\VectBun}[0]{{ \VectBundle }}
\newcommand{\VectBunrk}[1]{{ \VectBundlerk{#1} }}
\newcommand{\Bung}[0]{{ \Bun\qty{G} }}

% Algebras
\newcommand{\Hopf}[0]{\mathsf{Hopf}}
\newcommand{\alg}[0]{\mathsf{Alg}}
\newcommand{\Alg}[0]{{\mathsf{Alg}}}
\newcommand{\scalg}[0]{\mathsf{sCAlg}}
\newcommand{\cAlg}[0]{{\mathsf{cAlg}}}
\newcommand{\calg}[0]{\mathsf{CAlg}}
\newcommand{\liegmod}[0]{{\mathfrak{g}\dash\mathsf{Mod}}}
\newcommand{\liealg}[0]{{\mathsf{Lie}\dash\mathsf{Alg}}}
\newcommand{\Lie}[0]{\mathsf{Lie}}
\newcommand{\kalg}[0]{{\mathsf{Alg}_{/k} }}
\newcommand{\kAlg}[0]{{\mathsf{Alg}_{/k} }}
\newcommand{\kSch}[0]{{\mathsf{Sch}_{/k}}}
\newcommand{\rAlg}[0]{{\mathsf{Alg}_{/R}}}
\newcommand{\ralg}[0]{{\mathsf{Alg}_{/R}}}
\newcommand{\zalg}[0]{{\mathsf{Alg}_{/\ZZ}}}
\newcommand{\CCalg}[0]{{\mathsf{Alg}_{\mathbb{C}} }}
\newcommand{\dga}[0]{{\mathsf{dg\Alg} }}
\newcommand{\cdga}[0]{{ \mathsf{c}\dga }}
\newcommand{\dgla}[0]{{\dg\Lie\Alg }}
\newcommand{\Poly}[0]{{\mathsf{Poly} }}
\newcommand{\Hk}[0]{{\mathsf{Hk} }}

\newcommand{\Grpd}[0]{{\mathsf{Grpd}}}
\newcommand{\inftyGrpd}[0]{{ \underset{\infty}{ \Grpd } }}
\newcommand{\Algebroid}[0]{{\mathsf{Algd}}}

% Schemes and Sheaves
\newcommand{\Loc}[0]{\mathsf{Loc}}
\newcommand{\Locsys}[0]{\mathsf{LocSys}}
\newcommand{\Ringedspace}[0]{\mathsf{RingSp}}
\newcommand{\RingedSpace}[0]{\mathsf{RingSp}}
\newcommand{\LRS}[0]{\Loc\RingedSpace}
\newcommand{\IndCoh}[0]{{\mathsf{IndCoh}}}
\newcommand{\Ind}[0]{{\mathsf{Ind}}}
\newcommand{\Pro}[0]{{\mathsf{Pro}}}
\newcommand{\DCoh}[0]{{\mathsf{DCoh}}}
\newcommand{\QCoh}[0]{{\mathsf{QCoh}}}
\newcommand{\Cov}[0]{{\mathsf{Cov}}}
\newcommand{\sch}[0]{{\mathsf{Sch}}}
\newcommand{\presh}[0]{ \underset{ \mathsf{pre} } {\mathsf{Sh} }}
\newcommand{\prest}[0]{ {\underset{ \mathsf{pre} } {\mathsf{St} } } }
\newcommand{\Descent}[0]{{\mathsf{Descent}}}
\newcommand{\Desc}[0]{{\mathsf{Desc}}}
\newcommand{\FFlat}[0]{{\mathsf{FFlat}}}
\newcommand{\Perv}[0]{\mathsf{Perv}}
\newcommand{\smsch}[0]{{ \smooth\Sch }}
\newcommand{\Sch}[0]{{\mathsf{Sch}}}
\newcommand{\Schf}[0]{{\mathsf{Schf}}}
\newcommand{\Sh}[0]{{\mathsf{Sh}}}
\newcommand{\St}[0]{{\mathsf{St}}}
\newcommand{\Stacks}[0]{{\mathsf{St}}}
\newcommand{\Vark}[0]{{\mathsf{Var}_{/k} }}
\newcommand{\Var}[0]{{\mathsf{Var}}}
\newcommand{\Open}[0]{{\mathsf{Open}}}

% Homotopy
\newcommand{\CW}[0]{{\mathsf{CW}}}
\newcommand{\sset}[0]{{\mathsf{sSet}}}
\newcommand{\sSet}[0]{{\mathsf{sSet}}}
\newcommand{\ssets}[0]{\mathsf{sSet}}
\newcommand{\hoTop}[0]{{\mathsf{hoTop}}}
\newcommand{\hoType}[0]{{\mathsf{hoType}}}
\newcommand{\ho}[0]{{\mathsf{ho}}}
\newcommand{\SHC}[0]{{\mathsf{SHC}}}
\newcommand{\SH}[0]{{\mathsf{SH}}}
\newcommand{\Spaces}[0]{{\mathsf{Spaces}}}
\newcommand{\GSpaces}[1]{{G\dash\mathsf{Spaces}}}
\newcommand{\Spectra}[0]{{\mathsf{Sp}}}
\newcommand{\Sp}[0]{{\mathsf{Sp}}}
\newcommand{\Top}[0]{{\mathsf{Top}}}
\newcommand{\Bord}[0]{{\mathsf{Bord}}}
\newcommand{\TQFT}[0]{{\mathsf{TQFT}}}
\newcommand{\Kc}[0]{{\mathsf{K^c}}}
\newcommand{\triang}[0]{{\mathsf{triang}}}
\newcommand{\TTC}[0]{{\mathsf{TTC}}}

% Infty Cats
\newcommand{\Finset}[0]{{\mathsf{FinSet}}}
\newcommand{\Cat}[0]{\mathsf{Cat}}
\newcommand{\Fun}[0]{{\mathsf{Fun}}}
\newcommand{\Kan}[0]{{\mathsf{Kan}}}
\newcommand{\Monoid}[0]{\mathsf{Mon}}
\newcommand{\Arrow}[0]{\mathsf{Arrow}}
\newcommand{\quasiCat}[0]{{ \mathsf{quasiCat} } }
\newcommand{\inftycat}[0]{{ \underset{\infty}{ \Cat}  }}
\newcommand{\inftycatn}[1]{{ \underset{(\infty, {#1})}{ \Cat}  }}
\newcommand{\core}[0]{{ \mathsf{core} }}

% New?
\newcommand{\Prism}[0]{\mathsf{Prism}}
\newcommand{\Solid}[0]{\mathsf{Solid}}
\newcommand{\WCart}[0]{\mathsf{WCart}}

% Motivic
\newcommand{\Torsor}[1]{{\mathsf{#1}\dash\mathsf{Torsor}}}
\newcommand{\Torsorleft}[1]{{\mathsf{#1}\dash\mathsf{Torsor}}}
\newcommand{\Torsorright}[1]{{\mathsf{Torsor}\dash\mathsf{#1} }}
\newcommand{\Quadform}[0]{{\mathsf{QuadForm}}}
\newcommand{\HI}[0]{{\mathsf{HI}}}
\newcommand{\DM}[0]{{\mathsf{DM}}}
\newcommand{\hoA}[0]{{\mathsf{ho}_*^{\scriptstyle \AA^1}}}
\newcommand\Tw[0]{\mathsf{Tw}}
\newcommand\SB[0]{\mathsf{SB}}
\newcommand\CSA[0]{\mathsf{CSA}}
\newcommand{\CSS}[0]{{ \mathsf{CSS} } }

% Unsorted
\newcommand{\FGL}[0]{\mathsf{FGL}}
\newcommand{\FI}[0]{{\mathsf{FI}}}
\newcommand{\CE}[0]{{\mathsf{CE}}}
\newcommand{\Fuk}[0]{{\mathsf{Fuk}}}
\newcommand{\Lag}[0]{{\mathsf{Lag}}}
\newcommand{\Mfd}[0]{{\mathsf{Mfd}}}
\newcommand{\Riem}[0]{\mathsf{Riem}}
\newcommand{\Wein}[0]{{\mathsf{Wein}}}
\newcommand{\gspaces}[1]{{#1}\dash{\mathsf{Spaces}}}
\newcommand{\deltaring}[0]{{\delta\dash\mathsf{Ring}}}
\newcommand{\terminal}[0]{{ \mathscr{1}_{\scriptscriptstyle \uparrow} }}
\newcommand{\initial}[0]{{ \mathscr \emptyset^{\scriptscriptstyle \downarrow} }}

% Universal guys

\newcommand{\coeq}[0]{\operatorname{coeq}}
\newcommand{\cocoeq}[0]{\operatorname{eq}}
\newcommand{\dgens}[1]{\gens{\gens{ #1 }}}
\newcommand{\ctz}[1]{\, {\converges{{#1} \to\infty}\longrightarrow 0} \, }
\newcommand{\conj}[1]{{\overline{{#1}}}}
\newcommand{\complex}[1]{{ {#1}_{\scriptscriptstyle \bullet}} }
\newcommand{\cocomplex}[1]{ { {#1}^{\scriptscriptstyle \bullet}} }
\newcommand{\bicomplex}[1]{{ {#1}_{\scriptscriptstyle \bullet, \bullet}} }
\newcommand{\cobicomplex}[1]{ { {#1}^{\scriptscriptstyle \bullet, \bullet}} }

\newcommand{\floor}[1]{{\left\lfloor #1 \right\rfloor}}
\newcommand{\ceiling}[1]{{\left\lceil #1 \right\rceil}}
\newcommand{\fourier}[1]{\widehat{#1}}
\newcommand{\embedsvia}[1]{\xhookrightarrow{#1}}
\newcommand{\openimmerse}[0]{\underset{\scriptscriptstyle O}{\hookrightarrow}}
\newcommand{\weakeq}[0]{\underset{\scriptscriptstyle W}{\rightarrow}}

\newcommand{\fromvia}[1]{\xleftarrow{#1}}
\newcommand{\generators}[1]{\left\langle{#1}\right\rangle}
\newcommand{\gens}[1]{\left\langle{#1}\right\rangle}
\newcommand{\globsec}[1]{{{\Gamma}\qty{#1} }}
\newcommand{\Globsec}[1]{{{\Gamma}\qty{#1} }}
\newcommand{\langL}[1]{ {}^{L}{#1} }

\newcommand{\equalsbecause}[1]{\overset{#1}{=}}
\newcommand{\congbecause}[1]{\overset{#1}{\cong}}
\newcommand{\congas}[1]{\underset{#1}{\cong}}
\newcommand{\isoas}[1]{\underset{#1}{\cong}}
\newcommand{\addbase}[1]{{ {}_{\pt} }}
\newcommand{\ideal}[1]{\mathcal{#1}}
\newcommand{\adjoin}[1]{ { \left[ \scriptstyle {#1} \right] } }
\newcommand{\polynomialring}[1]{ { \left[ {#1} \right] } }
\newcommand{\htyclass}[1]{ { \left[ {#1} \right] } }
\newcommand{\qtext}[1]{{\quad \operatorname{#1} \quad}}
\newcommand{\abs}[1]{{\left\lvert {#1} \right\rvert}}
\newcommand{\stack}[1]{\mathclap{\substack{ #1 }}} 

\newcommand{\powerseries}[1]{ { \left[ {#1} \right] } }
\newcommand{\functionfield}[1]{ { \left( {#1} \right) } }
\newcommand{\rff}[1]{ \functionfield{#1} }

\newcommand{\fps}[1]{{\left[\left[ #1 \right]\right]  }}
\newcommand{\formalseries}[1]{ \fps{#1} }
\newcommand{\formalpowerseries}[1]{ \fps{#1} }
\newcommand\fls[1]{{\left(\left( #1 \right)\right)  }}

\newcommand\lshriek[0]{{}_{!}}
\newcommand\pushf[0]{{}^{*}}

\newcommand{\nilrad}[1]{{\sqrt{0_{#1}} }}
\newcommand{\jacobsonrad}[1]{{J ({#1}) }}
\newcommand{\localize}[1]{ \left[ { \scriptstyle { {#1}\inv}  } \right]}
\newcommand{\primelocalize}[1]{ \left[ { \scriptstyle { { ({#1}^c) }\inv}  } \right]}
\newcommand{\plocalize}[1]{\primelocalize{#1}}
\newcommand{\sheafify}[1]{ \left( #1 \right)^{\scriptscriptstyle \mathrm{sh}} }
\newcommand{\complete}[1]{{ {}_{ \hat{#1} } }}
\newcommand{\takecompletion}[1]{{ \overbrace{#1}^{\widehat{\hspace{4em}}}  }}
\newcommand{\pcomplete}[0]{{ {}^{ \wedge }_{p} }}
\newcommand{\kv}[0]{{ k_{\hat{v}} }}
\newcommand{\Lv}[0]{{ L_{\hat{v}} }}

\newcommand{\twistleft}[2]{{ {}^{#1} #2 }}
\newcommand{\twistright}[2]{{ #2 {}^{#1} }}
\newcommand{\liesover}[1]{{ {}_{/ {#1}} }}
\newcommand{\liesabove}[1]{{ {}_{/ {#1}} }}
\newcommand{\slice}[1]{_{/ {#1}} }
\newcommand{\coslice}[1]{_{{#1/}} }
\newcommand{\quotright}[2]{ {}^{#1}\mkern-2mu/\mkern-2mu_{#2} }
\newcommand{\quotleft}[2]{ {}_{#2}\mkern-.5mu\backslash\mkern-2mu^{#1} }
\newcommand{\invert}[1]{{ \left[ { \scriptstyle \frac{1}{#1} } \right] }}
\newcommand{\symb}[2]{{ \qty{ #1 \over #2 } }}
\newcommand{\squares}[1]{{ {#1}_{\scriptscriptstyle \square} }} 
\newcommand{\shift}[2]{{ \Sigma^{\scriptstyle[#2]} #1 }}

\newcommand\cartpower[1]{{ {}^{ \scriptscriptstyle\times^{#1} }  }}
\newcommand\disjointpower[1]{{ {}^{ \scriptscriptstyle\coprod^{#1} }  }}
\newcommand\sumpower[1]{{ {}^{ \scriptscriptstyle\oplus^{#1} }  }}
\newcommand\prodpower[1]{{ {}^{ \scriptscriptstyle\times^{#1} }  }}
\newcommand\tensorpower[2]{{ {}^{ \scriptstyle\otimes_{#1}^{#2} }  }}
\newcommand\tensorpowerk[1]{{ {}^{ \scriptscriptstyle\otimes_{k}^{#1} }  }}
\newcommand\derivedtensorpower[3]{{ {}^{ \scriptstyle {}_{#1} {\otimes_{#2}^{#3}} }  }}
\newcommand\smashpower[1]{{ {}^{ \scriptscriptstyle\smashprod^{#1} }  }}
\newcommand\wedgepower[1]{{ {}^{ \scriptscriptstyle\smashprod^{#1} }  }}
\newcommand\fiberpower[2]{{ {}^{ \scriptscriptstyle\fiberprod{#1}^{#2} }  }}
\newcommand\powers[1]{{ {}^{\cdot #1} }}
\newcommand\skel[1]{{ {}^{ (#1) } }}
\newcommand\transp[1]{{ \, {}^{t}{ \left( #1 \right) } }}

\newcommand{\inner}[2]{{\left\langle {#1},~{#2} \right\rangle}}
\newcommand{\inp}[2]{{\left\langle {#1},~{#2} \right\rangle}}
\newcommand{\poisbrack}[2]{{\left\{ {#1},~{#2} \right\} }}
\newcommand\tmf{ \mathrm{tmf} }
\newcommand\taf{ \mathrm{taf} }
\newcommand\TAF{ \mathrm{TAF} }
\newcommand\TMF{ \mathrm{TMF} }
\newcommand\String{ \mathrm{String} }

\newcommand{\BO}[0]{{\B \Orth}}
\newcommand{\EO}[0]{{\mathsf{E} \Orth}}
\newcommand{\BSO}[0]{{\B\SO}}
\newcommand{\ESO}[0]{{\mathsf{E}\SO}}
\newcommand{\BG}[0]{{\B G}}
\newcommand{\EG}[0]{{\mathsf{E} G}}
\newcommand{\BP}[0]{{\operatorname{BP}}}
\newcommand{\BU}[0]{\B{\operatorname{U}}}

\newcommand{\MO}[0]{{\operatorname{MO}}}
\newcommand{\MSO}[0]{{\operatorname{MSO}}}
\newcommand{\MSpin}[0]{{\operatorname{MSpin}}}
\newcommand{\MSp}[0]{{\operatorname{MSpin}}}
\newcommand{\MString}[0]{{\operatorname{MString}}}
\newcommand{\MStr}[0]{{\operatorname{MString}}}
\newcommand{\MU}[0]{{\operatorname{MU}}}


\newcommand{\KO}[0]{{\operatorname{KO}}}
\newcommand{\KU}[0]{{\operatorname{KU}}}

\newcommand{\smashprod}[0]{\wedge}

\newcommand{\ku}[0]{{\operatorname{ku}}}
\newcommand{\hofib}[0]{{\operatorname{hofib}}}
\newcommand{\hocofib}[0]{{\operatorname{hocofib}}}
\DeclareMathOperator{\Suspendpinf}{{\Sigma_+^\infty}}


\newcommand{\Loop}[0]{{\Omega}}
\newcommand{\Loopinf}[0]{{\Omega}^\infty}
\newcommand{\Suspend}[0]{{\Sigma}}
\newcommand*\dif{\mathop{}\!\operatorname{d}}
\newcommand*{\horzbar}{\rule[.5ex]{2.5ex}{0.5pt}}
\newcommand*{\vertbar}{\rule[-1ex]{0.5pt}{2.5ex}}
\newcommand\Fix{ \mathrm{Fix} }
\newcommand\CS{ \mathrm{CS} }
\newcommand\FP{ \mathrm{FP} }
\newcommand\places[1]{ \mathrm{Pl}\qty{#1} }
\newcommand\Ell{ \mathrm{Ell} }
\newcommand\homog{ { \mathrm{homog} } }
\newcommand\Kahler[0]{\operatorname{Kähler}}
\newcommand\Prinbun{\mathrm{Bun}^{\mathrm{prin}}}
\newcommand\aug{\fboxsep=-\fboxrule\!\!\!\fbox{\strut}\!\!\!}
\newcommand\compact[0]{\operatorname{cpt}}
\newcommand\hyp[0]{{\operatorname{hyp}}}
\newcommand\jan{\operatorname{Jan}}
\newcommand\curl{\operatorname{curl}}
\newcommand\kbar{ { \bar{k} } }
\newcommand\ksep{ { k\sep } }
\newcommand\mypound{\scalebox{0.8}{\raisebox{0.4ex}{\#}}}
\newcommand\rref{\operatorname{RREF}}
\newcommand\RREF{\operatorname{RREF}}
\newcommand{\Tatesymbol}{\operatorname{TateSymb}}
\newcommand\tilt[0]{ {}^{ \flat } }
\newcommand\vecc[2]{\textcolor{#1}{\textbf{#2}}}
\newcommand{\Af}[0]{{\mathbb{A}}}
\newcommand{\Ag}[0]{{\mathcal{A}_g}}
\newcommand{\Mg}[0]{{\mathcal{M}_g}}
\newcommand{\Ahat}[0]{\hat{ \operatorname{A}}_g }
\newcommand{\Ann}[0]{\operatorname{Ann}}
\newcommand{\sinc}[0]{\operatorname{sinc}}
\newcommand{\Banach}[0]{\mathcal{B}}
\newcommand{\Arg}[0]{\operatorname{Arg}}
\newcommand{\BB}[0]{{\mathbb{B}}}
\newcommand{\Betti}[0]{{\operatorname{Betti}}}
\newcommand{\CC}[0]{{\mathbb{C}}}
\newcommand{\CF}[0]{\operatorname{CF}}
\newcommand{\CH}[0]{{\operatorname{CH}}}
\newcommand{\CP}[0]{{\mathbb{CP}}}
\newcommand{\CY}{{ \text{CY} }}
\newcommand{\Cl}[0]{{ \operatorname{Cl}} }
\newcommand{\Crit}[0]{\operatorname{Crit}}
\newcommand{\DD}[0]{{\mathbb{D}}}
\newcommand{\DSt}[0]{{ \operatorname{DSt}}}
\newcommand{\Def}{\operatorname{Def} }
\newcommand{\Diffeo}[0]{{\operatorname{Diffeo}}}
\newcommand{\Diff}[0]{\operatorname{Diff}}
\newcommand{\Disjoint}[0]{\displaystyle\coprod}
\newcommand{\resprod}[0]{\prod^{\res}}
\newcommand{\restensor}[0]{\bigotimes^{\res}}
\newcommand{\Disk}[0]{{\operatorname{Disk}}}
\newcommand{\Dist}[0]{\operatorname{Dist}}
\newcommand{\EE}[0]{{\mathbb{E}}}
\newcommand{\EKL}[0]{{\mathrm{EKL}}}
\newcommand{\QH}[0]{{\mathrm{QH}}}
\newcommand{\AMGM}[0]{{\mathrm{AMGM}}}
\newcommand{\resultant}[0]{{\mathrm{res}}}
\newcommand{\tame}[0]{{\mathrm{tame}}}
\newcommand{\primetop}[0]{{\scriptscriptstyle \mathrm{prime-to-}p}}
\newcommand{\VHS}[0]{{\mathrm{VHS} }}
\newcommand{\ZVHS}[0]{{ \ZZ\mathrm{VHS} }}
\newcommand{\CR}[0]{{\mathrm{CR}}}
\newcommand{\unram}[0]{{\scriptscriptstyle\mathrm{un}}}
\newcommand{\Emb}[0]{{\operatorname{Emb}}}
\newcommand{\minor}[0]{{\operatorname{minor}}}
\newcommand{\Et}{\text{Ét}}
\newcommand{\trace}{\operatorname{tr}}
\newcommand{\Trace}{\operatorname{Trace}}
\newcommand{\Kl}{\operatorname{Kl}}
\newcommand{\Rel}{\operatorname{Rel}}
\newcommand{\Norm}{\operatorname{Nm}}
\newcommand{\Extpower}[0]{\bigwedge\nolimits}
\newcommand{\Extalgebra}[0]{\bigwedge\nolimits}
\newcommand{\Extalg}[0]{\Extalgebra}
\newcommand{\Extcomplex}[0]{\cocomplex{ \Extalgebra} }
\newcommand{\Extprod}[0]{\bigwedge\nolimits}
\newcommand{\Ext}{\operatorname{Ext} }
\newcommand{\FFbar}[0]{{ \bar{ \mathbb{F}} }}
\newcommand{\FFpn}[0]{{\mathbb{F}_{p^n}}}
\newcommand{\FFp}[0]{{\mathbb{F}_p}}
\newcommand{\FF}[0]{{\mathbb{F}}}
\newcommand{\FS}{{ \text{FS} }}
\newcommand{\Fil}[0]{{\operatorname{Fil}}}
\newcommand{\Flat}[0]{{\operatorname{Flat}}}
\newcommand{\Fpbar}[0]{\bar{\mathbb{F}_p}}
\newcommand{\Fpn}[0]{{\mathbb{F}_{p^n} }}
\newcommand{\Fppf}[0]{\mathrm{\operatorname{Fppf}}}
\newcommand{\Fp}[0]{{\mathbb{F}_p}}
\newcommand{\Frac}[0]{\operatorname{Frac}}
\newcommand{\GF}[0]{{\mathbb{GF}}}
\newcommand{\GG}[0]{{\mathbb{G}}}
\newcommand{\GL}[0]{\operatorname{GL}}
\newcommand{\GW}[0]{{\operatorname{GW}}}
\newcommand{\Gal}[0]{{ \mathsf{Gal}} }
\newcommand{\bigo}[0]{{ \mathsf{O}} }
\newcommand{\Gl}[0]{\operatorname{GL}}
\newcommand{\Gr}[0]{{\operatorname{Gr}}}
\newcommand{\HC}[0]{{\operatorname{HC}}}
\newcommand{\HFK}[0]{\operatorname{HFK}}
\newcommand{\HF}[0]{\operatorname{HF}}
\newcommand{\HHom}{\mathscr{H}\kern-2pt\operatorname{om}}
\newcommand{\HH}[0]{{\mathbb{H}}}
\newcommand{\HP}[0]{{\operatorname{HP}}}
\newcommand{\HT}[0]{{\operatorname{HT}}}
\newcommand{\HZ}[0]{{H\mathbb{Z}}}
\newcommand{\Hilb}[0]{\operatorname{Hilb}}
\newcommand{\Homeo}[0]{{\operatorname{Homeo}}}
\newcommand{\Honda}[0]{\mathrm{\operatorname{Honda}}}
\newcommand{\Hsh}{{ \mathcal{H} }}
\newcommand{\Id}[0]{\operatorname{Id}}
\newcommand{\Intersect}[0]{\displaystyle\bigcap}
\newcommand{\JCF}[0]{\operatorname{JCF}}
\newcommand{\RCF}[0]{\operatorname{RCF}}
\newcommand{\Jac}[0]{\operatorname{Jac}}
\newcommand{\II}[0]{{\mathbb{I}}}
\newcommand{\KK}[0]{{\mathbb{K}}}
\newcommand{\KH}[0]{ \K^{\scriptscriptstyle \mathrm{H}} }
\newcommand{\KMW}[0]{ \K^{\scriptscriptstyle \mathrm{MW}} }
\newcommand{\KMimp}[0]{ \hat{\K}^{\scriptscriptstyle \mathrm{M}} }
\newcommand{\KM}[0]{ \K^{\scriptstyle\mathrm{M}} }
\newcommand{\Kah}[0]{{ \operatorname{Kähler} } }
\newcommand{\LC}[0]{{\mathrm{LC}}}
\newcommand{\LL}[0]{{\mathbb{L}}}
\newcommand{\Log}[0]{\operatorname{Log}}
\newcommand{\MCG}[0]{{\operatorname{MCG}}}
\newcommand{\MM}[0]{{\mathcal{M}}}
\newcommand{\mbar}[0]{\bar{\mathcal{M}}}
\newcommand{\MW}[0]{\operatorname{MW}}
\newcommand{\Mat}[0]{\operatorname{Mat}}
\newcommand{\NN}[0]{{\mathbb{N}}}
\newcommand{\NS}[0]{{\operatorname{NS}}}
\newcommand{\OO}[0]{{\mathcal{O}}}
\newcommand{\OP}[0]{{\mathbb{OP}}}
\newcommand{\OX}[0]{{\mathcal{O}_X}}
\newcommand{\Obs}{\operatorname{Obs} }
\newcommand{\obs}{\operatorname{obs} }
\newcommand{\Ob}[0]{{\operatorname{Ob}}}
\newcommand{\Op}[0]{{\operatorname{Op}}}
\newcommand{\Orb}[0]{{\mathrm{Orb}}}
\newcommand{\Conj}[0]{{\mathrm{Conj}}}
\newcommand{\Orth}[0]{{\operatorname{O}}}
\newcommand{\PD}[0]{\mathrm{PD}}
\newcommand{\PGL}[0]{\operatorname{PGL}}
\newcommand{\GU}[0]{\operatorname{GU}}
\newcommand{\PP}[0]{{\mathbb{P}}}
\newcommand{\PSL}[0]{{\operatorname{PSL}}}
\newcommand{\Pic}[0]{{\operatorname{Pic}}}
\newcommand{\Pin}[0]{{\operatorname{Pin}}}
\newcommand{\Places}[0]{{\operatorname{Places}}}
\newcommand{\Presh}[0]{\presh}
\newcommand{\QHB}[0]{\operatorname{QHB}}
\newcommand{\QHS}[0]{\mathbb{Q}\kern-0.5pt\operatorname{HS}}
\newcommand{\QQpadic}[0]{{ \QQ_p }}
\newcommand{\ZZelladic}[0]{{ \ZZ_\ell }}
\newcommand{\QQ}[0]{{\mathbb{Q}}}
\newcommand{\QQbar}[0]{{ \bar{ \mathbb{Q} } }}
\newcommand{\Quot}[0]{\operatorname{Quot}}
\newcommand{\RP}[0]{{\mathbb{RP}}}
\newcommand{\RR}[0]{{\mathbb{R}}}
\newcommand{\Rat}[0]{\operatorname{Rat}}
\newcommand{\Reg}[0]{\operatorname{Reg}}
\newcommand{\Ric}[0]{\operatorname{Ric}}
\newcommand{\SF}[0]{\operatorname{SF}}
\newcommand{\SL}[0]{{\operatorname{SL}}}
\newcommand{\SNF}[0]{\mathrm{SNF}}
\newcommand{\SO}[0]{{\operatorname{SO}}}
\newcommand{\SP}[0]{{\operatorname{SP}}}
\newcommand{\SU}[0]{{\operatorname{SU}}}
\newcommand{\F}[0]{{\operatorname{F}}}
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# Talk 1: Jordan Ellenberg, Sparsity of rational points on moduli spaces

[ADDING Conference](https://www.daniellitt.com/adding)

> [Reference for this talk](https://arxiv.org/abs/2109.01043)

:::{.question}
How many homogeneous forms in $\ZZ[x_0,\cdots, x_n]$ are there with discriminant equal to 1?
More accurately, there is a $\PGL\dash$action, so how many $\PGL$ orbits are there?
More generally: how many are there with discriminant divisible only by primes in some finite set $S$?
:::

:::{.conjecture title="Shafarevich conjecture-esque"}
There are finitely many.
:::

:::{.remark}
The general philosophy is that there should be only finitely many classes of $X$ with good reduction outside of $S$ -- this is known e.g. for abelian varieties of dimension $g$, see Faltings' proof of Mordell.
:::

:::{.remark}
In terms of rational points, there is a trichotomy:

- $g=0$,
- $g=1$,
- $g > 1$.

This in fact breaks into a dichotomy:

- $g\leq 1$,
- $g > 1$.

:::

:::{.definition title="Sparse points"}
There are bounds on rational point counts on $X$ with height at most $B$:

- $g=0 \leadsto \ll_X B^2$
- $g=1\leadsto \ll_X \log B$ (Mordell)
- $g\geq 2\leadsto \ll_X C_X$ where $C$ is a constant depending on $X$ (Faltings)

Here we'll dichotomize this in a different way: $g=0$ and $g > 0$.
We say $X$ has **sparse points** if $\size X(\QQ) \ll B^\eps$.
:::


:::{.theorem title="E, Lawrence, Venkatesh 2021"}
Let $K\in \NF, S\in \Places(K)$ finite, $U_K \injects \PP^N$ a quasiprojective variety with a geometric variation of Hodge structure with finite-to-one period map, making $U_X$ an interesting moduli space of something.
Then the periods of $U(\OO_K\invert{S})$ are sparse.
:::

:::{.remark}
The anabelian part: $U$ has large $\pi_1$, so lots of etale covers.
E.g. if $U$ is a moduli of hypersurfaces, take the universal curve $\mch\to U$, then $H^n(\mch_t; C_m)$ varies and this can be interpreted as a moduli space with level structure.
We try to show that large $\pi_1$ implies sparseness. 
This isn't quite true, since e.g. blowing up introduces lots of rational points, so a stronger condition is needed: $\pi_1(U)$ is infinite and for every finite-dimensional $V \subseteq U$, $\pi_1(V)\to \pi_1(U)$ has infinite image.
:::

:::{.remark}
Why is it useful to have lots of etale covers?
Consider $x-y=1$ for $x,y\in \ZZ\invert{S}\units$ and $S = \ts{2,3}$.
The $S\dash$units theorem of Siegel guarantees there are only finitely many solutions.
One way to approach this: if $x=2^a3^b$, we know $x$ up to squares: $x=s,2s,3s,6s$ for $s$ a square, as is $y$.
This yields a system 
\[
m^2-n^2 &=1 \\
m^2-2n^2 &= 1 \\
2m^2-n^2 &= 1 \\
\vdots & \qquad 
,\]
which e.g. some can be solved using techniques for Pell equations.
Having higher degree rigidifies the situation, so perhaps there are more techniques to solve them.
Strategy: trade one hard equation for a finite list of higher degree easier equations.
:::

:::{.fact}
A partitioning trick for integral points: $Y\to U$ is a finite etale cover, then there are a finite number of twists $\ts{Y_1,\cdots, Y_m}$ (all isomorphic over $\QQbar$) with $Y_i\to U$ such that every point in $U(\ZZ\invert{S})$is in the image of $Y(\ZZ\invert{S})$ for some $i$.
So 
\[
\disjoint_i Y_i(\ZZ\invert{S}) \covers U(\ZZ\invert{S})
.\]
:::

:::{.theorem title="Heath-Brown 2004 (determinantal methods)"}
Let $X \subset \PP^2$ be a plane curve of degree $d$, then there is a uniform bound:
\[
\size \ts{ p\in X(\QQ) \st \height(p)\leq B} \leq C_{d, \eps}B^{2d+\eps}
.\]

> Note: missed the exponent on $B$, need to fix.

:::

:::{.remark}
Useful to control numbers of points for curves you know nothing about.
Walsh removes the $\eps$ in all terms, Selberg, Brolog generalized to higher dimensions, CCDN make the constant effective.
Pitch: these theorems are useful for other theorems which are not ostensibly about uniformity!
:::

:::{.remark}
All we can control in this situation is the degree of $Y_i\to U$ and $U\subseteq \PP^N$ has a degree, so we can control the degree of the $Y_i$.
Broberg gives a bound $\ll B^{n+1\over d^{1\over n}}$ where $n=\dim U$. 
It doesn't actually matter what this is, just that it decreases in $d$, and we can take higher degree covers.
:::

:::{.remark}
"Anabelian": $\pi_1$ somehow tells the entire story.
:::

:::{.remark}
Heath-Brown's technique uses $p\dash$adic repulsion of points for $X(\QQ)\to X(\QQpadic)$ where low-height points do not end up nearby.
Recall that there is a SES
\[
1\to \pi_1^\et(X_{\QQbar}) \to \pi_1^\et(X_\QQ) \to G_\QQ\to 1
,\]
and any point $p\in X(\QQ)$ gives a section, thought of as $X(\QQ) \to H^1(G_\QQ; \pi_1^\et(X_{\QQbar}))$.
Anabelian-ness: embed into some large interesting geometric space like this.

This cohomology group has a topology where $p, q\in X(\QQ)$ are nearby iff there exists a higher degree etale cover $Y\to X$ (small subsets correspond to large index subgroups in the profinite topology) such that $p, q$ both lift to $Y(\QQ)$.
:::

:::{.remark}
How this goes for curves: $C(\QQ) \to \Jac(\QQ)$, and one can tensor up to $\Jac(\QQ) \tensor_{\ZZ} \ZZpadic$.
Modern take: points are close if they differ by a power of $p$ in the Mordell-Weil group.
Interpretation of the main theorem: Heath-Brown in more general profinite topologies.
:::


# Talk 2: Wanlin Li, Ceresa cycle and hyperellipticity

:::{.remark}
A hyperbolic curve is determined by its $\pi_1^\et$.
:::

:::{.remark}
Recall $y^2=f(x)$ defines a hyperelliptic curve $C$, which admits an involution $(x,y)\to(x,-y)$ and produces a degree 2 map
\[
C &\to \PP^1 \\
(x,y) &\mapsto x
.\]

Let $\kbar$ be a separable closure of $k$.
There is a fibration induced by taking a geometric point of $\spec k$ and pulling back:

\begin{tikzcd}
	{C_{\kbar}} && C \\
	\\
	{\spec \kbar} && {\spec k}
	\arrow[from=3-1, to=3-3]
	\arrow[from=1-3, to=3-3]
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-1, to=3-1]
	\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJDX3tcXGtiYXJ9Il0sWzIsMCwiQyJdLFsyLDIsIlxcc3BlYyBrIl0sWzAsMiwiXFxzcGVjIFxca2JhciJdLFszLDJdLFsxLDJdLFswLDFdLFswLDNdLFswLDIsIiIsMSx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dXQ==)

As in topology, this induces a LES in homotopy, which here splits into SESs.
In particular, 
\[
1\to \pi_1(C_{\kbar})\to \pi_1(C_k) \to \Gal(\kbar/k)
.\]
A point induces a section and thus a map $\Gal(\kbar/k) \to \Aut \pi_1(C_{\kbar})$.
Take the lower central series of $\pi \da \pi_1(C_{\kbar})$, this induces
\[
1\to L^2\pi/L^3\pi \to \pi/L^3\pi \to \pi/L^1 \pi = \pi^\ab \to 1
\]
where the first term is abelian.

> See Davis-Pries-Wickelgren for applications to Fermat curves.

:::

:::{.question}
This extension corresponds to an element in $\mu(C) \in H^1(G_{\kbar}; \Hom(\pi^\ab, L^2\pi/L^3\pi ) )$, and when $C$ is hyperelliptic $\mu(C) = 0$.
Does the converse hold?
:::

:::{.theorem title="Bisogno-L-Litt-Srinivasan"}
There exist non-hyperelliptic curves  $C$ over $k$ such that $\mu(C)$ is torsion -- in particular, the *Fricke-Macbeath* curve, which is genus 7 Hurwitz.
Moreover if $C_1\to C_2$ then $\mu(C_1)$ torsion implies $\mu(C_2)$ torsion.

> See <https://arxiv.org/abs/2004.06146>?

:::

:::{.theorem title="Harris-Pulte (Hain-Matsumoto)"}
$\mu(C)$ is the $\ell\dash$adic cycles class associated to the **Ceresa cycle**.

> See [Hain-Matsumoto](https://arxiv.org/abs/math/0306037) and [Pulte](https://projecteuclid.org/journals/duke-mathematical-journal/volume-57/issue-3/The-fundamental-group-of-a-Riemann-surface--Mixed-Hodge/10.1215/S0012-7094-88-05732-8.short)

:::

:::{.remark}
For $C\slice k$ and $p\in C(K)$, the Abel-Jacobi map yields
\[
\AJ: C &\injects \Jac(C) \\
q &\mapsto [q-p]
.\]
So define the **Ceresa cycle** as 
\[
\tilde c \da \AJ(C) - \bar{ \AJ(C) } \da [q-p] - [p-q]
.\]
Note that $\tilde c$ is homologically trivial in Chow, but algebraically *nontrivial* for a very general $C\slice \CC$ with $g\geq 3$.
:::

:::{.theorem title="Beauville"}
There is an explicit non-hyperelliptic curve $C$ with $\tilde c$ torsion:
\[
x^4 + xz^3 + y^3z = 0 \subseteq \PP^2, \qquad p = \tv{0,0,1}
.\]

> See <https://arxiv.org/abs/2105.07160>?

:::

:::{.remark}
Consider curves over the local field $K = \CC\fps{t}$.
Note $\Gal(\bar{\CC\fps t}/\CC\fps{t}) = \hat \ZZ$, so this resembles a circle, and one can degenerate a family over the punctured disc.
Apply nonabelian Picard-Lefschetz due to Asada-Matsumoto-Oda: if $C$ has semistable reduction then the monodromy of $C/\CC\fps{t}$ is given by a multi-twist, i.e. a product of Dehn twists about simple closed curves.
One can explicitly compute the Ceresa class in this situation.
The degeneration data can be encoded as a tropical curve (essentially the dual graph of the special fiber).

> See [Asada-Matsumoto-Oda](https://www.sciencedirect.com/science/article/pii/002240499400112V)

:::

:::{.theorem title="?"}
$\mu(C)$ is always torsion for $C$ defined over $\CC\fps{t}$.
:::

:::{.remark}
There is a notion of "hyperelliptic" for tropical curves: quotienting by the involution yields a tree.
:::

# Misc Notes

:::{.remark}
See Iwasawa group.

:::

:::{.conjecture}
Section conjecture: $\Sec(Y/K) \cong Y(K)$, i.e. every section comes from a rational point.
:::

:::{.remark}
See the recent Lawrence-Venkatesh proof of Mordell.
See *Selmer section set* and *adelic sections*.
:::

:::{.remark}
Hyperbolic curves:

- $g=0 \leadsto \PP^1\sm Z$ where $\size Z \geq 3$
- $g=1 \leadsto$ affine
- $g\geq 2$: anything.

For $X\slice K$ for $K\in \NF$ smooth hyperbolic with good reduction away from $S$, $\size X(\OO_{K, S}) < \infty$ by Faltings.
See Bloch-Kato and Fontaine-Mazur conjectures.
:::

# Sunday, May 01

:::{.remark}
I missed the first two talks 😦
:::

# Kiran Kedlaya: Crystalline companions as an anabelian phenomenon

:::{.remark}
Setup: $k = \FF_q, q=p^n, X\in \Sch\slice k$ smooth geometrically connected, $\ell\neq p$ arbitrary.
Recall
\[
1\to \pi_1 X_{\kbar} \to \pi_1 X \to G_k = \ZZhat \to 1
.\]
Anabelian philosophy: everything you'd want to know about $X$ is contained in $\pi_1 X$.
:::

:::{.theorem title="Tamagawa"}
If $X$ is an affine curve, then $\pi_1^{\tame} X$ determines $X$.
:::

:::{.remark}
Problem: if $X$ is an affine genus $g$ curve with $m$ punctures, $\pi_1^{\primetop} X_{\kbar}$ is the prime-to-$p$ completion of $\Free(2g+m-1)$, independently of $X$.
Since sections induce $G_k\to \Out(\pi_1 X_{\kbar})$, we have lots of tame continuous $\bar{\QQladic}$ reps of $\pi_1 X_{\kbar}$, but very few are fixed by Frobenius.
:::

:::{.conjecture title="Deligne"}
Such representations only have "geometric origins", i.e. if $\mce$ is a lisse $\QQladic\dash$sheaf, i.e. a lisse $F\dash$sheaf with $[F:\QQladic] < \infty$, which is irreducible with determinant of finite order, then it appears on relative etale cohomology of $\pi:Y\to X$ for some $Y$.
:::

:::{.remark}
This is known in $\dim X = 1$, due to Deligne, around the same time Drinfeld proved Langlands for $\GL_2(k(X))$ for $k(X)$ a function field (or really the adeles).
So all arithmetic reps of $\pi_1$ *come from geometry*.
:::

:::{.remark}
Note that $Y$ will eventually not even be a scheme.
The determinant condition rules out transcendental twists.
Galois side: lisse sheaves on $X$; automorphic side: reps values in $\GL_n(\AA_K)$ for $K$ a field.
The proof above involves exhibiting the Galois objects as coming from relative etale cohomology in moduli of shtukas.
A priori one only knows Frobenius traces, but this turns out to be enough to uniquely characterize things in this situation.
:::

:::{.conjecture}
Later Lafforgue did this for $\GL_n$, but the corresponding statement about arithmetic reps is wide open.
:::

:::{.remark}
If $\mce$ as above, the Frobenius traces at all closed points $x\in \abs{X}$ are algebraic over $\QQ$.
:::

:::{.definition title="Companion sheaves"}
Fix an algebraic closure $\QQbar$ and two primes $\ell, \ell'\neq p$ and fix embeddings $\QQbar\injects \QQbar_\ell$ and $\QQbar \injects \QQbar_{\ell'}$.
Let $\mce,\mce'$ be lisse $\QQbar_\ell, \QQbar_{\ell'}$ sheaves (resp.) on $X$.
Say $\mce,\mce'$ are **companions** iff for every $x\in\abs{X}$ the Frobenius traces at $x$ are equal in $\QQbar$.
:::

:::{.remark}
Note that $\mce$ determines $\mce'$ up to semisimplification, using $L\dash$function techniques.
Moreover properties like being irreducible or having finite determinant hold simultaneously for them.
:::

:::{.theorem title="Drinfeld, L. Lafforgue, Deligne"}
With this setup, a lisse $\QQbar_{\ell}$ sheaf $\mce_\ell$ admits a compantion $\mce'$ which is a $\QQbar_{\ell'}$ for chosen embeddings of $\QQ$ for which all traces are in $\QQbar$, i.e. irreducible and finite determinant.
This is true in arbitrary dimension.
:::

:::{.remark}
What about when $\ell=p$?
There are somehow too many and too few lisse $\QQbar_p$ sheaves!
E.g. the Lefschetz trace formula doesn't hold.
Instead use the Riemann-Hilbert correspondence -- the \(p\dash \)adic analogues of lisse sheaves are certain \(p\dash \)adic integrable connections.
How to construct: start with $X$ affine and glue.
Choose a smooth affine formal scheme over $W(k)$ with special fiber $P_k \cong X$.
Let $K = \ff W(k)$ and $P_K$ be the Raynaud generic fiber.
See Tate model, Berkovich model, etc for rigid analytic geometry.
Let $\AA_K^n \leadsto \hat{\AA^n}_{W(k)}$, a closed unit disc over $K$.
:::

:::{.definition title="Convergent isocrystal"}
Some definitions:

- A **convergent isocrystal** is a vector bundle with an integrable connection on $P_K$.
- A **convergent $F\dash$isocrystal** is this and a compatible action of (a lift of) Frobenius.
- An **overconvergent $F\dash$isocrystal** is this on some structural enlargment of $P_K$ which takes the closed unit disc to the disc of radius $1+\eps$.
:::

:::{.remark}
These are similar to $\pi_1(X_{\kbar})\dash$representations.
Issue: \(p\dash \)adic antidifferentiation is hard; the integral of a formal power series converging on the closed disc only converges on the open disc.
:::

:::{.remark}
If $X$ is smooth this recovers Berthelot's rigid cohomology, which is a refinement of crystalline cohomology if $X$ is proper.
This yields a 6 functor formalism, and has the same moving parts as etale cohomology.
:::

:::{.theorem title="Abe"}
The Langlands correspondence extends to these when $\dim X = 1$.

The content here: Lafforgue's original proof can now be run with \(p\dash \)adic coefficients instead of just \(\ell\dash \)adic coefficients.
:::

:::{.theorem title="Abe-?, Drinfeld-Abe-Esmult, K"}
The companion theorem extends to both $\ell=p$ and $\ell'=p$.
:::

:::{.remark}
Deligne posited the existence of a *petit camarade crystalline*, a little crystalline friend. ❇🙂
:::

## Applications

:::{.remark}
A partial result toward a conjecture of Simpson.
Let $X\slice \CC$ be a smooth cohomologically rigid local system (so no nontrivial deformations) which is irreducible with finite determinant.
Are these of geometric origin?
In particular, is there a $\ZVHS$?
Esnault-Groecherig show that the monodromy representation factors through $\GL_n(\OO_K) \to \GL_n(\CC)$ for some $K\in \NF$.

Idea: start from complex geometry, go to \(p\dash \)adic geometry, yields an overconvergent $F\dash$crystal.
This yields integrality at $p$; use companions to go back to a lisse \(\ell\dash \)adic sheaf, then back to $\CC$ to get integrality at $\ell$.
:::

:::{.remark}
Going the other way, $\ell\to p$: one can prove "of geometric origin" results when $\rank \mce = 2$ (Krishnmorthy-Pal).
Idea: go from $\ell\to p$, make a candidate for the crystalline Dieudonne module for some family of AVs.
One will have a bound on the motivic weight, which is at most $\rank \mce - 1$.

A word on the proof: define a moduli stack $M_n$ of mod $p^n$ $F\dash$crystals, which is a horrendous algebraic stack.
These are roughly coherent sheaves with extra data.
Study some finite-type pieces using slops, and is universally closed since one can take flat limits along curves.
Take the Zariski closure of companion points, then take stable images to get some $M_n''$.

Show that every point in each component of it is a companion point using horizontal companions (as opposed to vertical in the fiber direction).
Then show each component maps isomorphically to $S$, which is a pointwise condition on $S$.
This only uses the companion on the fiber, which is easier to study.
:::


# Alex Smith:  Simple abelian varieties over finite fields with extreme point counts

:::{.theorem title="Howe-Kedlaya"}
Given $n > 0$, there is an $A\in \Ab\Var\slice{\FF_2}$ with $\size A(\FF_2) = n$.
:::

:::{.remark}
Recall the Weil bounds: given $A\slice{\FF_q}$,
\[
(q-2q^{1\over 2} + 1)^g \leq \size A(\FF_q) \leq (q+2q^{1\over 2} +1)^g
.\]

Let $\ts{\alpha_i}_{1\leq i\leq 2g}$ be the eigenvalues of $\Frob\actson H_{\et}^1(A \fiberprod{\FF_q} \bar{\FF_q}; \ZZladic )$.
Recall the Weil conjectures:

- All embeddings $\QQ( \alpha_i) \injects \CC$ satisfy $\abs{ \alpha_i} = q^{1\over 2}$.
- Lefschetz trace: $\size A(\FF_q) = \prod_{1\leq i\leq 2g}( \alpha_i - 1) = \prod(q+1 - \alpha_i - \bar{ \alpha_i} )^{1\over 2}$
  Note that these real numbers sit in $[-2q^{1\over 2}, 2q^{1\over 2}] \subseteq \RR$.

:::

:::{.definition title="Totally $\Sigma$"}
Given $\Sigma \subseteq \CC$ we say $\alpha$ is **totally $\Sigma$** iff all conjugates of $\alpha$ are contained in $\Sigma$.
:::

:::{.remark}
Remarkably for AVs, the $\alpha_i$ tell the entire story!
Honda-Tate theory gives a correspondence
\[
\correspond{
  \text{Abelian varieties over $\FF_q$}
}/\sim_{\scriptscriptstyle \text{isogeny}}
&\mapstofrom
\correspond{
  \text{Totally $I_q$ algebraic integers}
}/\sim_{\scriptscriptstyle\text{conjgacy}}
\]
where $I = [-2\sqrt q, 2\sqrt q]$.
Tate showed injectivity, Honda used CM theory to show surjectivity.
:::

:::{.theorem title="von Bomel-Costa-Li-Poonen-S."}
Given any $q$ and any $n \gg_q 0$, there is an $A\in\Ab\Var\slice{\FF_q}$ with $\size A(\FF_q) = n$.
:::

:::{.remark}
Idea: Howe-Kedlaya works infinitely often.
One can't attain *every* integer in the Weil bound interval, but you can get pretty close:
:::

:::{.theorem title="?"}
Given $g \gg_q 0$ and
\[
n\in [(q-2\sqrt q + 3)^g, (q+2\sqrt q - 1 - q\inv)^g]
,\]
there exists $A$ with $\size A(\FF_q) = n$.
:::

:::{.proof title="Sketch"}
Given $\alpha\in \RR$ and $\eps$, one can produce totally $I$ algebraic integers for $I = [ \alpha, 4+ \alpha+\eps]$ fitting the arcsin distribution:

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![](figures/2022-05-01_15-53-12.png)

For $\eps = \alpha = 0$, choose a large cyclotomic polynomial, whose roots are roughly equidistributed in $S^1$, then map to $[-2, 2]$.
Solving this for non-rational $\alpha$ and $\eps=0$ is a big open problem.
:::

## Schur-Segal-Smyth trace problem

:::{.problem title="?"}
Find the minimal $t$ such that for any $\eps > 0$, infinitely many totally positive algebraic integers satisfy
\[
\trace(\alpha)/\deg(\alpha) < t + \eps
.\]

:::

:::{.remark}
Idea: all conjugates greater than zero, how can you minimize the average trace?
The cyclotomic method above infinitely many whose trace is at most 2.
So using the arcsin distribution yields $t< 2$, open question: is $t=2$?
Progress has been slow and revolves around an old trick.
:::

:::{.proposition title="?"}
$t\geq 1$.
:::

:::{.proof title="?"}
Take a totally positive algebraic integer with conjugates \( \ts{\alpha_i}_{i\leq n} \) with minimal polynomial $p$.
Now apply AMGM:
\[
1\leq \abs{p(0)} = \qty{\prod a_i}^{n\over n}\leq_{\AMGM}\qty{\sum a_i \over n}^n
.\]
:::

:::{.remark}
We can do slightly better.
If $\abs{p(1)}\geq 1$, then $t\geq 1.05$.
If $\abs{p(a)p(\bar a)} \geq 1$ for $a\da {3+\sqrt 5\over 2}$, then $t\geq 1.1$.
More generally this is written as a resultant, i.e. $\resultant(p, x^3+3x+1)$.

Smyth shows that $t = \trace(\alpha)/\deg( \alpha)\geq 1.771$ with 14 exceptions; Wang-Wu-Wu shows $t\geq 1.793$.
Serre showed this argument can never show $t\geq 1.899$, Alex showed it can not show $t\geq 1.81$, so we're approaching the limit of Smyth's method.
:::

:::{.theorem title="Alex"}
Smyth's method limits to the right answer, and thus $t\leq 1.81$.
In particular, $t\neq 2$.
:::

:::{.remark}
Consequence: there are things that work better than the arcsin distribution.
:::

:::{.theorem title="?"}
Given sufficiently large square $q$, there are infinitely many $A\slice{\FF_q}$ with
\[
\size A(\FF_q) \geq (q + 2\sqrt q - 0.81)^{\dim A}
,\]
but only finitely many with
\[
\size A(\FF_q) \geq (q+2\sqrt q - 0.8)^{\dim A}
.\]
:::

:::{.definition title="?"}
Given algebraic integers $\alpha$ with conjugates $\ts{\alpha_i}_{i\leq n}$, let
\[
\mu_{\alpha}= {1\over n}\sum \delta_{ \alpha_i}
.\]
:::

:::{.remark}
There is a weak-$*$ topology on the space of such measures.
:::

:::{.theorem title="?"}
Choose $\Sigma \subseteq \RR$ with countably many components (e.g. excluding Cantor sets) of *capacity* $c > 1$.
TFAE are equivalent for a probability measure $\mu$ on $\Sigma$:

- There are totally $\Sigma$ algebraic integers $\alpha_i$ whose distributions $\mu_{\alpha_i}$ as above conver to $\mu$.
- For any integer polynomial $Q\neq 0$,
\[
\int_{\Sigma} \log\abs{Q} \dmu \geq 0
.\]

:::

:::{.remark}
Idea of proof: apply Minkowski's 2nd theorem as a source of promising polynomials.
Use an optimized distribution that avoids the 14 exceptions, whose average traces beat the previous averages:

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![](figures/2022-05-01_16-31-56.png)

:::