Masser-W¨ustholz bound for reducibility of Galois representations for Drinfeld modules of arbitrary rank

In this paper, we give an explicit bound on the irreducibility of mod-l Galois representation for Drinfeld modules of arbitrary rank without complex multiplication. This is a function ﬁeld analogue of Masser-W¨ustholz bound on irreducibility of mod-ℓ Galois representation for elliptic curves over number ﬁeld.


Introduction
In 1993, Masser and Wüstholz [MW93b] proved a famous result on existence of isogeny, with degree bounded by an explicit formula, between two isogenous Elliptic curves.Building upon this achievement, they [MW93a] subsequently employed the isogeny estimation to establish an explicit bound on the irreducibility of mod-ℓ Galois representation associated to elliptic curves over number field without complex multiplication (CM).This bound is then used to deduce a bound on the surjectivity of mod-ℓ Galois representation for elliptic curves over number field without CM.
Analogous to the elliptic curve theory, David and Denis [DD99] introduced an isogeny estimation applicable to Anderson t-modules.In particular, they deduced an isogeny estimation for Drinfeld Fq[T ]-modules over a global function field, see Theorem 2.12 for more details.This naturally prompts the query of whether the strategy employed by Masser-Wüstholz can be adapted to deduce an irreducibility limit for mod-l Galois representations concerning rank-r Drinfeld modules without CM.However, the Masser-Wüstholz strategy can not be applied directly to the context of Drinfeld modules.The main resean is when one computes the degree of isogeny between Drinfeld modules, the degree is always a power of q, which is not a prime number.Thus the computational trick in Lemma 3.1 of [MW93a] does not work for Drinfeld modules.
Nevertheless, the fundamental concept underlying the Masser-Wüstholz approach has inspired us to produce a similar method.By combining this concept with the height estimation on isogenous Drinfeld modules, as established by Breuer, Pazuki, and Razafinjatovo (as detailed in Theorem 2.13), we are equipped to deduce our main result: an explicit bound on the irreducibility of the mod-l Galois representation for Drinfeld modules of any rank, without CM.
Theorem 1.1.Let q = p e be a prime power, A := Fq[T ], and K be a finite extension of F := Fq(T ) of degree d.Let φ be a rank-r Drinfeld A-module over K of generic characteristic and assume that End K (φ) = A. Let l = (ℓ) be a prime ideal of A, consider the mod-l Galois representation ρφ,l : If ρφ,l is reducible, then either Here C is a computable constant depending on q and r, and N d = 10(d + 1) 7 .Furthermore, h(φ) denotes the naive height of Drinfeld module, while hG(φ) is the graded height of Drinfeld module.We refer their explicit definitions to Definition 2.5.
As a corollary of Theorem 1.1, we deduce a sufficient condition for deg T ℓ to have irreducible mod-l Galois representation ρφ,l .See corollary 4.2 for details.
Regarding the specialized scenario of "rank-2 Drinfeld modules over Fq(T )," a more nuanced estimation concerning the irreducibility of the mod-l Galois representation has been advanced by Chen and Lee [CL19].However, their strategy uses the fact that a power of 1dimensional group representation is again a group representation, see proof of Proposition 7.1 in [CL19].In the context of rank-2 scenarios, the reducibility of mod-l Galois representation always contributes a 1-dimensional subrepresentation.But this is not true for higher rank Drinfeld modules.
On the other hand, Chen and Lee [CL19] gave an explicit bound on surjectivity of mod-l Galois representations for rank-2 Drinfeld modules over Fq(T ) without CM.Such an explicit bound is still unknown for higher rank Drinfeld modules.The main difficulty is the classification of maximal subgroups (up to conjugacy classes) in GLr over finite field is much more complicated comparing to the GL2 case, where one only need to take care of the Borel and Cartan cases.

Preliminaries
Let A = Fq[T ] be the polynomial ring over finite field with q = p e an odd prime power, F = Fq(T ) be the fractional field of A, and K be a finite extension over F .Throughout this paper, "log" refers to the logarithm with base q.

Drinfeld modules
to be the ring of twisted q-polynomials with usual addition, and the multiplication is defined to be composition of q-polynomials.Definition 2.2.A Drinfeld A-module of rank r over K of generic characteristic is a ring homomorphism For an ideal a =< a > of A, we may define the a-torsion of the Drinfeld module φ over K. Let l be a prime ideal of A, then the l-torsion φ[l] of the Drinfeld module φ is an r-dimensional A/l-vector space.Applying the action of absolute Galois group Gal( K/K) on φ[l], we obtain the so-called mod-l Galois representation ρφ,l : for the Drinfeld module φ over K.
Now we proceed to define various heights associated with Drinfeld modules.Let MK be the set of all places of K including places above ∞.For each place ν ∈ MK , we define nν := [Kν : Fν ] to be the degree of local field extension Kν /Fν.Furthermore, we set | • |ν to be a normalized valuation of Kν .Definition 2.5.Let φ be a rank-r Drinfeld module over K characterized by φT (x) = T x + g1x q + • • • + gr−1x q r−1 + grx q r , where gi ∈ K and gr ∈ K * .
1.The naive height of φ is defined to be where h(gi Corollary 2.6.One can observe from the definition of naive height and graded height that

Isogenies
Definition 2.7.Let φ and ψ be two rank-r Drinfeld A-modules over K.A morphism u : φ → ψ over K is a twisted q-polynomial u ∈ K < x > such that uφa = ψau for all a ∈ A.
A non-zero morphism u : φ → ψ is called an isogeny.A morphism u : φ → ψ is called an isomorphism if its inverse exists.
For L = K, we omit subscripts and write Hom(φ, ψ) := Hom K (φ, ψ) and End(φ) := End K (φ) Definition 2.8.The composition of morphisms makes EndL(φ) into a subring of L < x >, called the endomorphism ring of φ over L. For any rank-r Drinfeld module φ over K with End(φ) = A, we say that φ does not have complex multiplication.Definition 2.9.Let f : φ → ψ be an isogeny of Drinfeld modules over K of rank r, we define the degree of f to be deg f := #ker(f ).
Proposition 2.10.Let f : φ → ψ be an isogeny of Drinfeld modules over K of rank r.There exists a dual isogeny f : ψ → φ such that Here 0 = a ∈ A is an element of minimal T -degree such that ker(f ) ⊂ φ[a].
The following corollary is immediate by counting cardinalities.
Corollary 2.11.As in the setting of Proposition 2.10, we have Now we can state the key tools to derive our main result: Theorem 2.12 ([DD99] Theorem 1.3).Let K be a finite extension over F with [K : F ] := d.Suppose that there are two K-isogenous Drinfeld modules φ and ψ defined over K. Then there is an isogeny f : φ → ψ such that Here c2 = c2(r, q) is a effectively computable constant depends only on r and q.
Theorem 2.13 ([BPR21] Theorem 3.1).Let f : φ → ψ be an isogeny of rank-r Drinfeld modules over K and suppose that ker(f ) ⊂ φ[N ] for some 0 = N ∈ A. Then we have 3 Proof of Theorem 1.1 We are given a rank-r Drinfeld module φ defined over K with End(φ) = A. Suppose the image of mod-l Galois representation Imρ φ,l acting on φ[l] has an invariant A/l-subspace of dimension 1 k r − 1. Denote such an invariant subspace by H. From Proposition 4.7.11 and Remark 4.7.12 of [Gos96], there is an isogeny with ker(f ) = H.Since φ and f both are defined over K, one can see that the Drinfeld module φ/H is a rank-r Drinfeld module defined over K as well.In addition, we have Take a dual isogeny f : φ/H → φ of f .The degree of f can be computed using Corollary 2.11.We get Besides, we can find two isogenies between φ and φ/H with bounded degree from Theorem 2.12: Since End(φ) = A, we can find N1 and N2 in A such that On the other hand, we compute in different order and get Thus we get the equality ℓb = N1N2.As ℓ is prime, we have either case (1): ℓ|N1 or case (2): ℓ|N2.
Then we may write
This complete the proof of Theorem 1.1.

Lower bound on irreducibility of ρφ,l
Under the setting of Theorem 1.1, one may further solve the inequality (1) for deg T ℓ.By setting Theorem 1.1 implies that the mod-l Galois representation is irreducible when (1'): Since when we fix a finite extension K/F and a Drinfeld module φ, the numbers N d and Ω φ are fixed.Elementary Calculus can tell us that the fraction q deg T ℓ deg T ℓ N d tends to infinity as deg T ℓ goes to infinity.Thus we can always find a real number C φ,d such that deg q Ω φ /N d •N d ) ln(q) , and C = c2(r, q) is an effectively computable constant in Theorem 2.12.
Definition 2.3.The a-torsion of a Drinfeld module φ over K is defined to be φ[a] := zeros of φa(x) in K ⊂ K. Now we define the A-module structure on K.For any elements b ∈ A and α ∈ K.We define the A-action of b on α via b • α := φ b (α).This gives K an A-module structure.And the A-module structure inherits to φ[a].As our Drinfeld module φ over K has generic characteristic, we have the following proposition Proposition 2.4.φ[a] is a free A/a-module of rank r.Proof.See [Gos96], Proposition 4.5.3.