BOUNDEDNESS OF TRACE FIELDS OF RANK TWO LOCAL SYSTEMS YEUK HAY JOSHUA LAM

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BOUNDEDNESS OF TRACE FIELDS OF RANK TWO

LOCAL SYSTEMS

YEUK HAY JOSHUA LAM

Abstract. Let pbe a ﬁxed prime number, and qa power of p. For

any curve over Fqand any local system on it, we have a number ﬁeld

generated by the traces of Frobenii at closed points, known as the trace

ﬁeld. We show that as we range over all pointed curves of type (g, n) in

characteristic pand rank two local systems with inﬁnite monodromy at

inﬁnity, the set of trace ﬁelds which are unramiﬁed at pand of bounded

degree is ﬁnite. This proves observations of Kontsevich obtained via

numerical computations, which are in turn closely related to the ana-

logue of Maeda’s conjecture over function ﬁelds. We also prove a similar

ﬁniteness result across all primes p. The key ingredients of the proofs

are Chin’s theorem on independence of ℓof monodromy groups, and

the boundedness of abelian schemes of GL2-type over curves in posi-

tive characteristics, obtained using partial Hasse invariants; the latter

is an analogue of Faltings’ Arakelov theorem for abelian varieties in our

setting.

Contents

1. Introduction 1

2. Drinfeld’s work on function ﬁeld Langlands 4

3. Partial Hasse invariants 6

4. Frobenius untwisting 8

5. Mapping spaces 10

6. Proofs of main results 11

References 15

1. Introduction

Let pbe a prime number, and qsome power of p. Let ¯

C/Fqbe a smooth,

projective curve of genus g, and Za non-empty subset of npoints of ¯

C; we

write C:=¯

C−Zfor the open curve. We will refer to ( ¯

C, Z) as a pointed

curve of type (g, n), and to the points in Zas the cusps of C.

Deﬁnition 1.1. For ( ¯

C, Z)/Fqas above, let L(¯

C, Z) be the set of isomor-

phism classes of rank two local systems Lon Cwith inﬁnite monodromy

around each point in Z, and such that det L∼

=Qℓ(−1).

Date: November 28, 2024.

arXiv:2210.13563v3 [math.NT] 27 Nov 2024

2 YEUK HAY JOSHUA LAM

For such a local system L, the work of Drinfeld, which was later general-

ized to arbitrary rank by Laﬀorgue, implies that there is a unique number

ﬁeld Fand an embedding σ:F −→ Qℓsuch that σ(F) is the ﬁeld generated

by the traces of Frobenii at closed points of C. We refer to Fas the trace

ﬁeld of L, and denote by F(L(¯

C, Z)) the set of trace ﬁelds of local systems

in L(¯

C, Z). It is natural to wonder about the distribution of such trace

ﬁelds; our ﬁrst main result is the following boundedness statement.

Theorem 1.2. Fix a pair (g, n). Let Fg,n :=S¯

C,Z,q F(L(¯

C, Z)) be the set

of trace ﬁelds of local systems in L(¯

C, Z), as (¯

C, Z)and qvary over all

pointed curves of type (g, n)and powers of p, respectively. Then, for any

integer d, there are only ﬁnitely many ﬁelds in Fg,n with degree ≤dand

which are unramiﬁed at p.

Remark 1.3. It is also straightforward to see that, if we ﬁx the trace ﬁeld

Fand the pointed curve ( ¯

C, Z)/Fqas in Theorem 1.2, then only ﬁnitely

many elements of Sq′L(¯

CFq′, ZFq′) have trace ﬁeld F, where ( ¯

CFq′, ZFq′)

denotes the basechange to Spec(Fq′). On the other hand, it is possible to

have a positive dimensional family of curves, all of which admit rank two

local systems with the same trace ﬁeld; this can be ruled out in certain

special cases, as is done in [Lam22].

We now state a result where the characteristic pis allowed to vary. For

this purpose, for a pointed curve ( ¯

C, Z)/Fqin characteristic p, we write

Lp(¯

C, Z) for L(¯

C, Z) to emphasize the characteristic of the base ﬁeld.

Theorem 1.4. There are only ﬁnitely many ﬁelds in Sp, ¯

C,Z F(Lp(¯

C, Z)) of

degree dand completely split at p.

The heuristic in the next section leads us to expect that Theorem 1.2

should hold without the unramiﬁed-at-passumption, and Theorem 1.4 should

hold without the splitness condition, though we are not able to prove these:

indeed, our method to bound the degree of the Hodge bundle uses partial

Hasse invariants, and the obtained bounds depend on the prime pif the

ﬁeld is not totally split at p. In another direction, one can try to remove

the inﬁnite monodromy condition in the deﬁnition of L(¯

C, Z): the main

obstruction here is that such local systems are not known to come from

abelian varieties, or some other family of varieties with a “uniform” de-

scription: instead, they are known to arise in the cohomology of moduli of

shtukas, whose geometry depends heavily on ¯

Cand q.

1.5. Context. Our work is motivated by Maeda’s conjecture in the function

ﬁeld setting. For example, for each curve ( ¯

C, Z)/Fqof type (g, n) = (0,4),

computations of Kontsevich [Kon09, §0.1] shows that, in almost all cases,

there are four trace ﬁelds, each of degree roughly (q+1)/41. This contributes

to the Maeda philosophy that, generically, trace ﬁelds should be as large as

1the splitting into four ﬁelds comes from the Atkin-Lehner operators

BOUNDEDNESS OF TRACE FIELDS OF RANK TWO LOCAL SYSTEMS 3

possible. We refer the reader to [Lam22] for more on the analogue of Maeda’s

conjecture over function ﬁelds, as well as references for the number ﬁeld case.

We were also heavily inﬂuenced by the results of Faltings [Fal83] on

Arakelov’s theorem for abelian varieties, as well as Deligne’s ﬁniteness theo-

rem [Del87]. More precisely, we were motivated by the possibility of unifor-

mity in Deligne’s ﬁniteness theorem: the latter says that on a ﬁxed complex

curve C, only ﬁnitely many rank NQ-local systems can come from algebraic

geometry, and by uniformity we mean whether this ﬁnite number depends

on the underlying curve in its moduli space. As a ﬁrst step towards this

question, for a ﬁxed N, one may ask about the contribution to rank NQ-

local systems coming from abelian varieties: in this case uniformity is known

and is a corollary of Faltings’ theorem [Fal83, Theorem 1]. Note that this

uniformity from Faltings’ proof seems, at least to the author, to be stronger

than the subsequent re-proofs of the same result due to Deligne [Del87], as

well as Jost and Yau [JY93], although these works are more general in that

they apply to local systems not coming from abelian schemes.

In any case, for ﬁxed (g, n), one sees that there are only ﬁnitely many F’s

of ﬁxed degree such that a curve of type (g, n) carries a non-trivial family

of abelian varieties of GL2(F)-type. Theorem 1.2 is the analogue of this in

positive characteristic. It would be interesting to investigate the analogue

of the full strength version (i.e. beyond abelian varieties of GL2-type) of

Faltings’ theorem in positive characteristic. For example, is it true that,

in characteristic p, any abelian scheme of dimension Don a curve ( ¯

C, Z)

of type g, n is isogenous to one whose Hodge bundle has degree bounded

by g, n, p, D? We should also mention a related result of Litt [Lit21] which

is an analogue of Deligne’s result in positive characteristic, but for -adic

coeﬃcients.

1.6. Sketch of proof of Theorem 1.2. For this sketch, we will focus

on the case of local systems with unipotent monodromy around each cusp,

which is the key part. By work of Drinfeld, we know that the rank two local

Qℓ-local systems in question come from abelian schemes of GL2(F)-type,

with Fbeing the trace ﬁeld of Frobenii; suppose that there are inﬁnitely

many such F’s of degree d. Using Zarhin’s trick, we obtain prinipally polar-

ized abelian schemes of dimension N:= 8dover C, and therefore inﬁnitely

many maps C→AN, which extend to maps ¯

C→A∗

N, where the latter

denotes the minimal compactiﬁcation. The ﬁrst key idea is that, using par-

tial Hasse invariants and a Frobenius untwisting result, we can pass to an

isogenous abelian scheme and bound the degree (with respect to the Hodge

bundle of A∗

N) of such maps in terms of just (g, n, p, d). This step may

be seen as an enhancement of the recent beautiful work [KYZ22] of Krish-

namoorthy, Yang, and Zuo by the use of partial Hasse invariants; it is also

where the unramiﬁed at pcondition is used.

Now consider the moduli space Mof curves Cof type (g, n), along with

maps ¯

C→A∗

Nof some ﬁxed degree. Using the previous step, we have

4 YEUK HAY JOSHUA LAM

inﬁnitely many points si∈ M, corresponding to inﬁnitely many ﬁelds Fi’s.

Since Mis of ﬁnite type, taking the Zariski closure of the si’s give a positive

dimensional family of curves C → S, and an abelian scheme A→C, such that

the ﬁber of Aat si∈Sis a (power of an) abelian scheme of GL2(Fi)-type. If

we were in characteristic zero, this would already give a contradiction since

by isomonodromy all the monodromy reprsentations of π1(Csi) must be the

same. In our situation, the Q-structure on Betti cohomology is of course not

available, and we crucially make use of Chin’s theorem on -independence of

monodromy groups and some tricks, such as the ﬁniteness of number ﬁelds

of ﬁxed degree and bounded ramiﬁcation, to conclude.

Acknowledgements. I am grateful to Maxim Kontsevich for sharing his

computations and insights with me, as well as to Mark Kisin, Bruno Klingler

and Sasha Petrov for several enlightening discussions and comments on a

previous draft. As is hopefully clear throughout the text, we owe a great

intellectual debt to the authors of [KYZ22], as well as the previous work

[ST18]; we thank both sets of authors for their beautiful work.

1.7. Notation. Throughout, p > 0 will denote a prime, qa power of p. For

a scheme Z,π1(Z) will always denote the ´etale fundamental group.

2. Drinfeld’s work on function field Langlands

We ﬁrst recall the results of Drinfeld from his work on function ﬁeld

Langlands for GL2. Throughout this section, we use notation as follows. As

in the introduction, let ¯

C/Fqbe a smooth, projective curve of genus g, and

Zan eﬀective Cartier divisor on ¯

C/Fqof degree n; we write C:=¯

C−Zfor

the open curve. We will refer to ( ¯

C, Z) as a pointed curve of type (g, n).

Let be a prime distinct from p, and Lbe a rank two Qℓ-local system

on C/Fq, such that det L∼

=Qℓ(−1). Let F⊂Qℓbe the ﬁeld generated

by Frobenius traces of L; we write [F:Q] = d., and sometimes refer to F

simply as the Frobenius trace ﬁeld, or simply trace ﬁeld, of L.

Theorem 2.1. Suppose Lhas inﬁnite local monodromy around some z∈

Z. Then there exists an abelian scheme πDrinf :BDrinf →Cof relative

dimension d, such that EndC(BDrinf )⊗Q=F, and Lappears as a direct

summand of R1πDrinf,∗Qℓ. Moreover, if we write D:=R1πDrinf,∗Qp∈

F-Isoc(C)Qp, then we have a decomposition

D=M

Dτ,

where the above sum is over embeddings τ:F −→ Qp, each Dτis two dimen-

sional, and the induced action of Fon Dτis through τ.

We refer the reader to [KYZ22, Theorem 2.2] as well as to the remark in

loc. cit. that follows for how to deduce the above theorem from the works

of Drinfeld. Our next goal is to reﬁne the abelian scheme BDrinf as follows,

under further assumptions.

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BOUNDEDNESSOFTRACEFIELDSOFRANKTWOLOCALSYSTEMSYEUKHAYJOSHUALAMAbstract.Letpbeafixedprimenumber,andqapowerofp.ForanycurveoverFqandanylocalsystemonit,wehaveanumberfieldgeneratedbythetracesofFrobeniiatclosedpoints,knownasthetracefield.Weshowthataswerangeoverallpointedcurvesoftype(g,n)incharacteristicpan...

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BOUNDEDNESS OF TRACE FIELDS OF RANK TWO LOCAL SYSTEMS YEUK HAY JOSHUA LAM

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