Modular Stacks : Generalizing dihedral group-modular curve connections(Moduli spaces, Galois representations and L-functions)

Modular

Stacks:

Generalizing dih edral

group-modular curve connections

MICHAEL D. FRIED

1991 Ma thematicsSubjectClassification. Primary llF32, llG18, llH58; Secondary $20B05$,

$20C25,20D25,20E18,20F34$.

Supportedby NSF

_{#DMS-99305590}

We thank Y. Ihara, M. Matsumoto and H. Nakamura for listening to the author justify and

improve his presentation of these topics. W. Feit, R. Howe, J. G. Thompson$md$H. Voelklein

contributedtohis understanding of modular representationtheoryappliedtothe$\iota\dot{r}$versal

frat-tini cover. An unpublished 1987 preprint included the construction of the universal exponent

$p$ frattini covers of$A_{5}$. Others may have gone over that territory since. Serre, in particular,

gavean$Ext$ interpretationto simplifyotir arguments to efficient use of Lewy decompositions

of projective indecomposables $($see

\S II.B

数理解析研究所講究録

M. FRIED

ABSTRACT. To each finite group $G$ we can attach a projective profinite

group, $\overline{G}$

: tlie universal frattini $coler$ of $G[FrJ$, \S 20.6$]$

.

collection of$r$ conjugacy classes $C$ of $G$, there is a natural moduli space.

Its points are equivalence classes of covers of the Rieinannsphere $\mathbb{P}^{1}$ with

geometricmonodroinygroup$G$having$C$ asthe conjugacy classes of branch

cyclesofthe cover. We conjointhese two constructions, applying the latter

toa naturalcoflnal collection offiiute quotients of$\overline{G}$

.

This produces

invari-ants for the arithmetic theory of curve covers. We consider here aspecial

case that uses a prime $p$ dividing $|G|$ and conjugacy classes of $C$ of

or-der relati$\backslash \nu ely$prime to$p$. Thisp-uiiraiiiiIied Iifting$in1’\cdot al\cdot iant,$ $\iota/(G, p, C)$, is

compatiblewith tcrniinology of [Se3], to wliicli tlie author contributed. We

call the tower of $(G, p, C)$-nioduli spaces that arise from this construction

a modular stack. Arithmetic geometers know a specialcase: the tower of

covers$JY’0(p)arrow X_{0}(p^{2})arrow X0(p^{3})\cdots$ of modularcurves. Points on$X_{0}(p^{n})$

correspond topairs of eiliptic curves with acyclic $p^{n}$-powerisogeny. Here

$G$ is tlie dihedral group $D_{p}$ of order $2p,$ $r=4$ and four repetitions of the

iii$\iota’olution$ conjugacy class in $D_{p}$ comprise $C$ [DFr,

\S 5.1-5.2].

stackarises from the Deligne-Mufflordpaper $[DeMu]$. Itssubtle, yet

com-patible,use here has a modular representation interpretation.

What we understand of any$mod\iota ilar$ stack comes from this lifting

in-variant. This lives in the$p’$-primequotient of$\tilde{G}$. It is arare, yetsignificant,

event. that the modular stack attached to $(G, p, C)$ may have finite length

(unlikethe to$\iota ver$of modularcurves). The materialfroin \S II on tlie

univer-sal frattiiiicover(especially applied to$A_{n}$ andconjugacy cla.ssesof 3-cycles)

provides examples. $I\nwarrow’c$eping 10 the p-miramified case allows these

deffiii-tions an elementary elegance. To show how to generalize beyond this we

coinpute alow level quotient of the 2-ramified invariant of L. Schnepp’s

3-branch pointcover ofdegree 20. The arithmetic applicationsof $v(G, p, C)$

$i_{I1}$ \S IV point toa test for the Drinfeld-Grothendieck-Ihara relationson $G_{\mathbb{Q}}$

[I] applied todetecting felds ofdefinition of curve covers. A plan for that

project coiicludes tliispaper.

REFERENCES

[A] J. L. Alperin,Local representation theory, Cambridgestudies in advanced

mathemat-ics, vol. 11, CambridgeUniv. Press, 1986.

[AJL] H.Andersen, J. Jorgensenand P. Landrock, The projectiveindecomposablemodules

of$SL_{2}(p^{n})$, Proc. Lonclon Math. Soc. 46 (1983),38-52.

[Be] D. Benson, The Loewy structures for the projecti ve indecomposable modules for$A_{8}$

an$dA_{9}$ in characteris tic2, Comni. in Alg. 11 (1983), 1395-1451.

[DFr] P. Debes and M. Fried, Nonrigid constructions in the Inverse Galois Problem, PJM

163

#1

$[DeI\backslash Iu]$ P. Deligne and D. Muffiord, The irreducibily of tfie space of curves ofgiven genus,

Inst. Hautes Etudes Math.36 (1967), 75-100.

[Frl] M. Fried, GIobal construction of general xxceptionaI Covers: lvith motivation for

applications to encoding, AMS Contemporary Mathseries (1994).

[Fr2] M. Fried, Orl atheoi.em ofMacCluer, Acta Arith. 25 (1974), 122-127.

[Fr3] bI. Fried, $F\lrcorner_{\wedge}\backslash position$ _oil an $Arithmetic- Gr\cdot onp$ Theoretic Coiinection via$Riem\partial nn$’s

Existence Tlieorem, A.M.S. Publications, Proceedings of Symposia in Pure Math:

Santa Cmz Conference on Finite Groups,, vol. 37, 1980, pp. 571-601.

[Fr4] $h^{1}I$. Fried, Galois groupsand ComplexMultiplication,TAMS 235 (1978), 141-162.

[FJ] M.D. Fried and M. Jarden, Field Arithinetic, Ergebnisse der Mathematik III, vol.11,

Springer Verlag, Heidelberg, 1986.

MODULAR STACKS

[IlahIu] J. ltarris andD..Muniford, On the Kodairadimension of tliemoduli space of$cur\tau’ es$,

$hu\iota\cdot ent$. math. 67 (1982), 23-86.

[I] Y. niara, Proceedings of the

_{Internationa.I}

1

Springer-Verlag, Hong$I\langle ong$, 1991.

$[l|[]$ B. Mazur. Lect$uI^{\cdot}e$Notes in$\Lambda/fathematics$, vol. 601, Springer-Verlag) 1977.

[N] D. G. Northcott, An in troductioii to liomological algebra, Canibridge Univ. Press,

Great Britain, 1962.

[Sel] J.-P.Serre, L’in$l’ari$aiit de $it^{\gamma}itt$ de la foi.me$T(x^{2})$,Math. Helvetici59 (1984),651-676.

[Se2] J.-P. Serre, Letter on Ext, Private Correspondence (1988).

[Se3] J.-P. Serre Comptes Rendus (1989).

[V] P. Vojta, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, 1987.

UC IRVINE, IRVINE, CA 92717, USA

E-mail a$dd\iota\cdot ess$: [email protected]