On a variant of the uniform boundedness conjecture for Drinfeld modules. 2nd Kyoto-Hefei Workshop on Arithmetic Geometry Shun Ishii

On a variant of the uniform boundedness conjecture for Drinfeld modules.

2nd Kyoto-Hefei Workshop on Arithmetic Geometry

Shun Ishii

RIMS, Kyoto University

August 21, 2020

This talk is based on the speaker’s master thesis:

On thep-primary uniform boundedness conjecture for Drinfeld modules (2020).

Contents

1 Backgrounds and the main result.

2 Proof of the main result.

3 Future work.

2 / 30

1 Backgrounds and the main result.

2 Proof of the main result.

3 Future work.

The Uniform Boundedness Conjecture for abelian varieties.

Conjecture (The UBC for abelian varieties).

L:a finitely generated field over a prime field.

d >0 :an integer.

Then there exists a constantC:=C(L, d)≥0which depends onLandds.t.

|X(L)tors|< C holds for everyd-dim abelian varietyX overL.

Known results.

Mazur^ab: The UBC ford= 1andL=Q. Merel^c: The UBC ford= 1.

aB. Mazur.“ Modular curves and the Eisenstein ideal”. In:

Inst. Hautes Études Sci. Publ. Math. 47 (1977). With an appendix by Mazur and M.

Rapoport, 33–186 (1978).

bB. Mazur.“ Rational isogenies of prime degree (with an appendix by D. Goldfeld)”.In:

Invent. Math. 44.2 (1978), pp. 129–162.

cLoïc Merel.“ Bornes pour la torsion des courbes elliptiques sur les corps de nombres”.In:

Invent. Math. 124.1-3 (1996), pp. 437–449.

4 / 30

The p-primary Uniform Boundedness Conjecture.

Conjecture (ThepUBC for abelian varieties).

L:a finitely generated field over a prime field.

d >0 :an integer.

p:a prime.

Then there exists a constantC:=C(L, p, d)≥0which depends onL,pandds.t.

|X[p^∞](L)|< C holds for everyd-dim abelian varietyX overL.

Known results.

Manin^a:ThepUBC ford= 1.

Cadoret^b: ThepUBC ford= 2with real multiplication assuming the Bombieri-Lang conj.

Cadoret-Tamagawa^c: ThepUBC for every1-dimensional family of abelian varieties.

aJu. I. Manin.“ Thep-torsion of elliptic curves is uniformly bounded”.In:

Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), pp. 459–465.

bAnna Cadoret.“ Theℓ-primary torsion conjecture for abelian surfaces with real multiplication”.In: Algebraic number theory and related topics 2010. RIMS Kôkyûroku Bessatsu, B32. Res. Inst. Math. Sci. (RIMS), Kyoto, 2012, pp. 195–204.

cAnna Cadoret and Akio Tamagawa.“ Uniform boundedness ofp-primary torsion of abelian schemes”.In: Invent. Math. 188.1 (2012), pp. 83–125.

Cadoret-Tamagawa’s result.

Here is the result of Cadoret-Tamagwa^∗. Theorem [Cadoret-Tamagawa].

L:a finitely generated field over a prime field.

S:a1-dimensional scheme of finite type overL.

A:an abelian scheme overS. p:a prime s.t. p̸= ch(L).

Then there exists aN :=N(L, S, A, p)≥0depends onL,S,Aandps.t.

As[p^∞](L)⊂As[p^N](L)

holds for everys∈S(L), i.e. everyL-rationalp-primary torsion point ofAs is annihilated byp^N.

Motivation.

Find a “Drinfeld-module analogue” of Cadoret-Tamagawa’s result.

∗Anna Cadoret and Akio Tamagawa.“ Torsion of abelian schemes and rational points on moduli spaces”.In: Algebraic number theory and related topics 2007. RIMS Kôkyûroku Bessatsu, B12. Res. Inst.

Math. Sci. (RIMS), Kyoto, 2009, pp. 7–29.

6 / 30

Drinfeld modules.

What are Drinfeld modules？

Drinfeld modules are function-field analogues of abelian varieties introduced by Drinfeld^a under the name of “elliptic module”.

aV. G. Drinfeld.“ Elliptic modules”.In: Mat. Sb. (N.S.) 94(136) (1974), pp. 594–627, 656.

Notation.

p:a prime.

q:a power ofp.

C:a smooth geometrically irreducible projective curve overFq.

∞:a fixed closed point ofC.

K:the function field ofC.

A:= Γ(C\ {∞},OC).

Drinfeld modules.

Definition (DrinfeldA-modules).

L:anA-field (i.e. a fieldLwith a homomorphismι:A→L).

A DrinfeldA-module overLis a homomorphismϕ:A→End(Ga,L)which satisfies the following two conditions:

1 ϕ(A)̸⊂L

2 A−→^ϕ End(Ga,L)−→^δ Lequalsιwhereδis the differentiation map ofGa,L at0.

Remark.

If we denote theq-th Frobenius byτ, then End_Fq(Ga,L) =L{τ}:={∑

iaiτⁱ(finite sum)|ai∈L}andδ(∑

iaiτⁱ) =a0.

8 / 30

Drinfeld modules.

For every DrinfeldA-moduleϕover anA-fieldL, we can define therankofϕwhich plays a similar role as the dimension of an abelian variety.

Example.

AssumeA=Fq[T].

Then therankofϕequals thedegreeofϕ(T)∈L{τ}as a polynomial inτ. LetIbe an ideal ofA.

TheI-torsion subgroup ofϕis defined by:

ϕ[I] :=∩a∈Iker(ϕa:Ga→Ga).

If the characteristic ofL(:= ker(ι))does not divideI,ϕ[I]is a finite étale group scheme which is étale-locally isomorphic to(A/I)^dwheredis the rank ofϕ.

Letpbe a maximal ideal ofA.

Thep-adic Tate module ofϕis defined by:

Tp(ϕ) := lim←−ϕ[pⁿ](L).

If the characteristic ofLdoes not dividep,Tp(ϕ)is a freeAp-module of rankd.

Drinfeld modules.

Poonen proved the finiteness of torsion submodules of Drinfeld modules^†. Theorem [Poonen].

LetLbe a finitely generatedA-field which containsKandϕa DrinfeldA-module over L. Then the set ofL-rational torsion points ofϕisfinite.

Remark.

L=Ga(L)can be regarded as anA-module throughϕ. ThisA-module is never finitely generated. This shows that an analogue of the Mordell-Weil theorem for Drinfeld module does not hold.

†Bjorn Poonen.“ Local height functions and the Mordell-Weil theorem for Drinfel’d modules”.In:

Compositio Math. 97.3 (1995), pp. 349–368.

10 / 30

The Uniform Boundedness Conjecture for Drinfeld modules.

Conjecture (The UBC for DrinfeldA-modules).

L:a finitely generated field overK.

d >0 :an integer.

Then there exists a constantC:=C(L, d)≥0which depends onLandds.t.

|ϕ(L)tors|< C holds for every DrinfeldA-moduleϕof rankdoverL.

Known results.

Poonen^a: The UBC ford= 1.

Pál^b: IfA=F2[T],Y0(p)(K)is empty for everypwithdeg(p)≥3.

Armana^c: IfA=Fq[T],Y0(p)(K)is empty for everypwithdeg(p) = 3,4.

aBjorn Poonen.“ Torsion in rank1Drinfeld modules and the uniform boundedness conjecture”.In: Math. Ann. 308.4 (1997), pp. 571–586.

bAmbrus Pál.“ On the torsion of Drinfeld modules of rank two”. In: J. Reine Angew. Math.

640 (2010), pp. 1–45.

cCécile Armana.“ Torsion des modules de Drinfeld de rang 2 et formes modulaires de Drinfeld”.In: Algebra & Number Theory 6.6 (2012), pp. 1239–1288.

The p-primary Uniform Boundedness Conjecture.

Conjecture (ThepUBC for Drinfeld A-modules).

L:a finitely generated field overK.

d >0 :an integer.

p:a maximal ideal ofA.

Then there exists aC:=C(L,p, d)≥0which depends onL,pandds.t.

|ϕ[p^∞](L)|< C holds for every DrinfeldA-moduleϕof rankdoverL.

Known results.

Poonen : ThepUBC ford= 2 andA=Fq[T].

Cornelissen-Kato-Kool^a: A strong version of thepUBC ford= 2.

aGunther Cornelissen, Fumiharu Kato, and Janne Kool.“ A combinatorial Li-Yau inequality and rational points on curves”.In: Math. Ann. 361.1-2 (2015), pp. 211–258.

Main result.

ThepUBC for every 1-dimensional family of Drinfeld modules of arbitrary rank.

12 / 30

Main Result.

Theorem [I].

L:a finitely generated field overK.

S:a1-dimensional scheme of finite type overL.

ϕ:a DrinfeldA-module of rankdoverS.

p:a maximal ideal ofA.

Then there exists an integerN :=N(L, S, ϕ,p)≥0which depends onL,S,ϕandps.t.

ϕs[p^∞](L)⊂ϕs[p^N](L)

holds for everys∈S(L), i.e. everyL-rationalp-primary torsion point ofϕsis annihilated byp^N.

Corollary.

The theorem implies thepUBC ford= 2.

(∵)Apply this theorem for Drinfeld modular curves.

1 Backgrounds and the main result.

2 Proof of the main result.

3 Future work.

14 / 30

The strategy of the proof.

A model case: S =Y1(p)andϕis the universal DrinfeldA-module.

In this case, the claim of the theorem⇔Y1(pⁿ)(L) =∅forn≫0.

1 First, consider the tower of modular curves

· · · →X1(pⁿ⁺¹)→X1(pⁿ)→X1(pⁿ⁻¹)→ · · · and prove that the genusg(X1(pⁿ))goes to∞whenn→ ∞.

2 SinceX1(pⁿ)is notFp-isotrivial,X1(pⁿ)(L)is finite ifg(X1(pⁿ))≥2by a positive characteristic analogue of the Mordell conjecture proved by Samuel.

3 IfY1(pⁿ)(L)̸=∅, thenlim←−Y1(p)(L)̸=∅for everyn, which shows a Drinfeld module overLhas infinitely manyL-torsion points.

HenceY1(pⁿ)(L)is empty forn≫0.

1 For a generalS, we will define an analogue of the modular curve, and prove that the genus goes to infinity.

2 However, since we do not assumeS is non-isotrivial, so we cannnot conclude the theorem as above.

The strategy of the proof.

We have divided the proof into three parts:

The strategy of the proof.

Assume that the assersion of the theorem does not hold.

Step 1: Show that every component of a tower of finite connected étale covers ofS (= an analogue of the modular tower) has anL-rational point.

Step 2: Prove the genus of that tower goes to infinity.

Step 3: Take nice models ofSpec(L)andS overFpand extends the Drinfeld module ϕto such a model. Specializing thep-primary torsion subgroup ofϕat some closed point of the model leads to a contradiction.

16 / 30

Step 1 (1/2)

Notation.

η: Spec(L(S))→S:the generic point ofS.

η: Spec(L(S))→S:a geometric generic point ofS.

π1(S, η) :the étale fundamental group ofS.

Π :=π1(S×LL, η) :the geometric fundamental group ofS.

Sinceϕ[pⁿ]is a finite étale group scheme, we have an actionπ1(S, η)↷ϕη[pⁿ](L(S)).

Definition (Sv_n→S).

n >0 :an integer.

Forvn∈ϕη[pⁿ]^∗(L(S)) :=ϕη[pⁿ](L(S))\pϕη[pⁿ](L(S)), writeSv_n for the connected étale cover ofS which correponds to the stabilizer ofvnw.r.t. π1(S)↷ϕη[pⁿ](L(S)).

Svn plays a role analogous toY1(pⁿ).

Step 1 (2/2)

Lemma.

Assume that the main result does not hold.

Then∃v:= (vn)∈Tp(ϕ)\pTp(ϕ)s.t.

Svn(L)̸=∅ holds for everyn≥0.

Hence we have a “modular tower” of geometrically connected finite étale covers ofS:

· · · →Sv_n+1→Svn→Sv_n−1→ · · ·.

Observation.

Iflim←−Sv_n(L)̸=∅, we can conclude the proof.

∵Let(sn)be an arbitary element oflim←−Sv_n(L)and consider the specialization map ϕη[pⁿ](L(S))→ϕs₀[pⁿ](κ(s0))for everyn. Then the image ofvnunder this map is κ(s0)-rational. This contradicts the finiteness of torsion points ofϕs.

In the following, we fixv:= (vn)∈Tp(ϕ)as in the lemma.

We may assume thatSvn(L)is infinite.

18 / 30

Step 2 (1/5)

Next, we prove that the genus of{Svn}goes to infinity.

Proposition.

L:an algebraic closure ofL.

gvn:the genus of the compactification ofSvn×LL.

Then lim

n→∞gv_n=∞holds.

Ingredients for the proof are:

1 The Riemann-Hurwitz formula.

2 Breuer-Pink’s result on the image of the monodromy representation

π1(S×LL)→GLd(Kp)associated top-adic Tate modules of DrinfeldA-modules.

3 Oesterlé’s theorem on the reduction modulo pⁿ of analytic closed subsets inA^dp.

Step 2 (2/5)

Breuer and Pink proved the following thorem^‡. Theorem [Breuer-Pink].

k:finitely generated field overK.

X:variety overkwith the generic pointξ: Spec(k(X))→X.

ψ:DrinfeldA-module overX s.t. ψξis not isotrivial and A= End_k(X)(ψξ).

Then the image of the monodromy representation associated to thep-adic Tate module π1(Xk^sep, ξ)→GL(Tp(ψξ))

is commensurable withSL(Tp(ψξ)).

In the following, for simplicity, we assume that A= End_L(S)(ϕη).

ϕηis notL-isotrivial.

‡Florian Breuer and Richard Pink.“ Monodromy groups associated to non-isotrivial Drinfeld modules in generic characteristic”.In: Number fields and function fields—two parallel worlds. Vol. 239. Progr. Math.

Birkhäuser Boston, Boston, MA, 2005, pp. 61–69.

20 / 30

Step 2 (3/5)

Notation.

λv_n:=_deg(S^2gvn−2

vn→S).

Tp:=Tp(ϕη) :thep-adic Tate module ofϕη. ρ: Π→GLAp(Tp) :the monodromy representation.

G:= Im(ρ).

ρn: Π→GLA/pⁿ(Tp/pⁿTp) :the modpⁿmonodromy representation.

Gn:= Im(ρn).

P1, . . . , Pr:the cusps ofS×LL.

IP_i, . . . , IP_r: the image of the inertia subgroup atPi throughρ.

Property.

{λv_n}nis increasing.

limn→∞gv_n=∞ ⇔limn→∞λv_n>0

Step 2 (4/5)

By the Riemann-Hurwitz formula,λvn satisfies the following inequality:

λvn≥ −2 + ∑

1≤i≤r

(

1− |IP_i\Gnvn|

|Gnvn| )

.

By the affineness of the moduli spaces of Drinfeld modules and the non L-isotriviality ofϕη,we may assume that∃is.t. |IP_i|=∞.

Note that the number of cusps ofSvnabovePi equals|IP_i\Gnvn|.

By Breuer-Pink’s result and the quasi-unipotency of the action of inertia subgroups, one can show thatlim_n→∞|IPi\Gnvn|=∞.

Hence, by replacingSwithSvn, we may assume that|IP_i|=∞for at least threei’s.

22 / 30

Step 2 (5/5)

Hence it suffices to prove that

|IP_i\Gnvn|

|Gnvn| →0.

This follows from the following proposition:

Proposition.

M:a freeAp-module of finite rank.

G⊂GL(M) :an analytic closed subgroup.

v∈M\pM.

I⊂G:a closed subgroup.

Then, under some assumptions,

nlim→∞

|I\Gnvn|

|Gnvn| = 1

|I| holds.

By using Oesterlé’s theorem^§, the assertion of this proposition is reduced to the inequalitydim(Gv)^I <dimGv(the dimension as an analytic space).

§Joseph Oesterlé.“ Réduction modulopⁿdes sous-ensembles analytiques fermés deZ^N_p”.In:

Invent. Math. 66.2 (1982), pp. 325–341.

The strategy of the proof (revisited).

The strategy of the proof.

Assume that the assersion of the theorem does not hold.

Step 1: Show that every component of a tower of finite connected étale covers ofS (= an analogue of the modular tower) has anL-rational point.

Step 2: Prove the genus of that tower goes to infinity.

Step 3: Take nice models ofSpec(L)andS overFpand extends the Drinfeld module ϕto such a model. Specializing thep-primary torsion subgroup ofϕat some closed point of the model leads to a contradiction.

24 / 30

Step 3 (1/3)

Lemma [Cadoret-Tamagawa].

L:a finitely generated extension ofch(L) =p >0.

C:a proper, normal and geometrically integral1-dimensional scheme overLs.t. the genus of the normalization ofC×LLis greater than1.

S:a non-empty open subscheme ofC.

IfS(L)is infinite, we moreover assume thatS isFp-isotrivial.

Then there exists aFp-morphismf:S →T between separated, integral and normal Fp-schemes of finite type which satisfies the following properties:

1 The function fieldFp(T)ofT isFp-isomorphic toL.

2 Under the identificationF(T) =L,S isL-isomorphic to the generic fiberSLoff.

3 Under the identificationS=SL, we haveS(L) =S(T)i.e. everyL-points ofS uniquely extends to aT-point ofS.

Step 3 (2/3)

... ...

Sv_n Sv_n

... ...

Sv1 Sv1

Sv₀(=S) S

Spec(L) T

We show that a non-empty open subscheme ofS is Fp-isotrivial. Then we take a modelf:S →T as in the lemma. We may assume thatϕextends to a Drinfeld A-moduleϕ_S overS.

As the same asSvn, the étale covering corresponding to the stabilizer ofvnw.r.t. π1(S, η)↷ϕη[pⁿ](L(S))is denoted bySv_n→ S.

Then one can show thatSvn(L) =Svn(T)by using S(L) =S(T)and the normality ofSvn. In particular, Sv_n(T)̸=∅for everyn.

26 / 30

Step 3 (3/3)

... ... ...

Svn(L) Svn(T) Svn(κ(t))

... ... ...

Sv₁(L) Sv₁(T) Sv₁(κ(t))

Sv₀(L) S(T) S(κ(t))

∼

∼

∼

∼

Hence, for a closed pointt∈T, Sv_n(κ(t))̸=∅is finite and non-empty.

Take an(xn)∈lim←− S^vn(κ(t))and set x:=x0∈ S(κ(t)).

Then the image ofvn under the specialization map

ϕη[pⁿ](L(S))→(ϕ_S)x[pⁿ](κ(t))is κ(t)-rational for everyn.

Hence(ϕ_S)x has infinitely many κ(t)-rational torsion points. This contradicts Poonen’s theorem.

General case.

In general,End_L(S)(ϕη)may strictly containAandϕηmay beL-isotrivial.

Ifϕη isL-isotrivial, we can prove the following stronger result:

Theorem.

L:a finitely generated field over K.

S:a normal integral scheme of finite type overL.

ϕ:a DrinfeldA-module overS s.t. ϕηisL-isotrivial.

d >0 :an integer.

Then there exists a constantC=C(L, S, d)≥0which depends onL,Sandϕs.t.

|ϕs(L^′)tors|< C

holds for every extensionL^′/Lwith[L^′:L]≤dands∈S(L^′).

Ifϕη is notL-isotrivial andE= End_L(S)(ϕη)strictly containsA, one can roughly regardϕas a DrinfeldE-module. Then almost the same proof as above works well.

28 / 30

1 Backgrounds and the main result.

2 Proof of the main result.

3 Future work.

Future work.

ThepUBC ford= 3.

By the main result, we know that the set of rational points of the moduli space of Drinfeld modules of rank 3with levelpⁿ(which is an affine surface) is empty for n≫0or Zariski dense for everyn >0.

The UBC ford= 2.

This conjecture largely remains open since the formal immersion method (which Mazur and Merel used to prove the UBC for ellptic curves) is difficult to adapt to Drinfeld modular curves.

Thank you for your attention.

30 / 30