$\wp$-adic continuous families of Drinfeld eigenforms of finite slope

Let $p$ be a rational prime, $v_p$ the normalized $p$-adic valuation on $\mathbb{Z}$, $q>1$ a $p$-power and $A=\mathbb{F}_q[t]$. Let $\wp\in A$ be an irreducible polynomial and $\mathfrak{n}\in A$ a non-zero element which is divisible by $\wp$. Let $k\geq 2$ be an integer. We denote by $S_k(\Gamma_1(\mathfrak{n}))$ the space of Drinfeld cuspforms of level $\Gamma_1(\mathfrak{n})$ and weight $k$ for $A$. Let $n\geq 1$ be an integer and $a\geq 0$ a rational number. Suppose that $\mathfrak{n}$ has a prime factor of degree one and the generalized eigenspace in $S_k(\Gamma_1(\mathfrak{n}))$ of slope $a$ is one dimensional. In this paper, under an assumption that $a$ is sufficiently small, we construct a family $\{F_{k'}\mid v_p(k'-k)\geq \log_p(p^n+a)\}$ of Hecke eigenforms $F_{k'}\in S_{k'}(\Gamma_1(\mathfrak{n}))$ of slope $a$ such that, for any $Q\in A$, the Hecke eigenvalues of $F_k$ and $F_{k'}$ at $Q$ are congruent modulo $\wp^\kappa$ with some $\kappa>p^{v_p(k'-k)}-p^n-a$.


Introduction
Let p be a rational prime, q ą 1 a p-power and F q the field of q elements. Put A " F q rts and K " F q ptq. Let ℘ P A be an irreducible polynomial of positive degree and n a non-zero element of A which is divisible by ℘. We denote by K ℘ the ℘-adic completion of K, by C ℘ the ℘-adic completion of an algebraic closure of K ℘ and by v ℘ : C ℘ Ñ QYt`8u the ℘-adic additive valuation on C ℘ normalized as v ℘ p℘q " 1. Similarly, we denote by K 8 the p1{tq-adic completion of K and by C 8 the p1{tq-adic completion of an algebraic closure of K 8 . LetK be the algebraic closure of K inside C 8 and we fix an embedding of Kalgebras ι ℘ :K Ñ C ℘ . For any x PK, we define its normalized ℘-adic valuation by v ℘ pι ℘ pxqq. Let Ω " P 1 pC 8 qzP 1 pK 8 q be the Drinfeld upper half plane, which has a natural structure of a rigid analytic variety over K 8 .
Let Γ be a subgroup of SL 2 pAq and k an integer. A Drinfeld modular form of level Γ and weight k is a rigid analytic function on Ω satisfying fˆa z`b cz`d˙" pcz`dq k f pzq for any γ "ˆa b c d˙P Γ, z P Ω and a holomorphy condition at cusps. It is considered as a function field analogue of the notion of elliptic modular form.
Recently, ℘-adic properties of Drinfeld modular forms have attracted attention and have been studied actively (for example, [BV1,BV2,BV3,Gos,Hat1,Hat2,PZ,Vin]). However, though we have a highly developed theory of p-adic analytic families of elliptic eigenforms of finite slope, ℘-adic properties of Drinfeld modular forms are much less well-understood compared to the elliptic case. One of the difficulties in the Drinfeld case is that, since the group OK ℘ is topologically of infinitely generated, analogues of the completed group ring Z p rrZp ss are not Noetherian, and it seems that we have no good definition of characteristic power series applicable to non-Noetherian base rings, as mentioned in [Buz2,paragraph before Lemma 2.3].
In this paper, we will construct families of Drinfeld eigenforms in which Hecke eigenvalues vary in a ℘-adically continuous way. For the precise statement, we fix some notation. For any m P A, we put Γ 1 pmq " " γ P SL 2 pAqˇˇˇˇγ "ˆ10 1˙m od m * .
Let k ě 2 be an integer. For any non-zero element Q P A, the Hecke operator T Q acts on the C 8 -vector space S k pΓ 1 pnqq of Drinfeld cuspforms of level Γ 1 pnq and weight k. The operator T ℘ is also denoted by U. Since they stabilize an A-lattice V k pAq (Proposition 2.2), every eigenvalue of T Q is integral over A. The normalized ℘-adic valuation of an eigenvalue of U is called slope, and we denote by dpk, aq the dimension of the generalized U-eigenspace for the eigenvalues of slope a. For any Hecke eigenform F , its T Q -eigenvalue is denoted by λ Q pF q. We denote by v p the p-adic valuation on Z satisfying v p ppq " 1. Then the main theorem of this paper (Theorem 4.1) gives the following, which we will prove in §4.1.
Let a be any non-negative rational number satisfying a ă mintDpn, d, εq, k´1u.
Suppose dpk, aq " 1. Then, for any integer k 1 ě k satisfying v p pk 1´k q ě log p pp n`a q, there exists a Hecke eigenform F k 1 P S k 1 pΓ 1 pnqq of slope a such that for For example, in the case of n " ℘ " t, we have d " ε " 1 and Dpn, 1, 1q " ? 2p n´1 2 . In this case, Theorem 1.1 implies that, for any Hecke eigenform F k of slope zero in S k pΓ 1 ptqq, the T Q -eigenvalue λ Q pF k q is t-adically arbitrarily close to those coming from Hecke eigenforms with A-expansion [Pet], which shows λ Q pF k q " 1 for any Q (Proposition 4.2). We also note that families constructed in Theorem 1.1 contain Hecke eigenforms whose Hecke eigenvalue at Q is not a power of Q ( §4.2), and thus they capture a more subtle ℘-adic structure of Hecke eigenvalues than the theory of A-expansions.
Let us explain the idea of the proof of Theorem 1.1. Note that a usual method to construct p-adic families of eigenforms of finite slope in the number field case is the use of the Riesz theory [Col,Buz2], which is not available for our case at present, due to the lack of a notion of characteristic power series over non-Noetherian Banach algebras. Instead, we follow an idea of Buzzard [Buz1] by which he constructed p-adically continuous families of quaternionic eigenforms over Q.
First we will prove a variant of the Gouvêa-Mazur conjecture (Proposition 3.8), which implies dpk, aq " dpk 1 , aq if k and k 1 are highly congruent p-adically and a is sufficiently small. With the assumption dpk, aq " 1, it produces Hecke eigenforms F k and F k 1 of slope a in weights k and k 1 , respectively. For this part, we employ the same idea as in [Hat2]: a lower bound of elementary divisors of the representing matrix of U with some basis and a perturbation lemma [Ked,Theorem 4.4.2] yield the equality. To obtain such a bound (Corollary 3.6), we need to define Hecke operators acting on the Steinberg complex (2.2) with respect to Γ 1 pnq, which is done in §2.3. Note that similar Hecke operators on a Steinberg complex in an adelic setting are given in [Böc,§6.4]. Then, a weight reduction map ( §3.1) yields a Drinfeld cuspform G of weight k such that, for m " v p pk 1´k q, the element G mod ℘ p m is a Hecke eigenform with the same eigenvalues as those of F k 1 mod ℘ p m . Now the point is that, if two lines generated by F k and G are highly congruent in some sense, then we can show that the eigenvalues of F k and G mod ℘ p m are also highly congruent, which gives Theorem 1.1; otherwise the two lines are so far apart that, again by the Gouvêa-Mazur variant mentioned above, they produce U-eigenvalues of slope a with multiplicity more than one, which contradicts dpk, aq " 1 (Theorem 4.1).
Acknowledgements. The author would like to thank Gebhard Böckle for suggesting him to look for ℘-adically continuous families of Drinfeld eigenforms instead of ℘-adically analytic ones, and David Goss for a helpful discussion. This work was supported by JSPS KAKENHI Grant Number JP17K05177.

Drinfeld cuspforms via the Steinberg module
In this section, we first recall an interpretation of the space of Drinfeld cuspforms using the Steinberg module due to Teitelbaum [Tei,p. 506], following the normalization of [Böc,§5]. We also introduce Hecke operators acting on the Steinberg complex. Using them, we define an A-lattice of the space of Drinfeld cuspforms which is stable under Hecke actions.
2.1. Steinberg module. For any A-algebra B, we consider B 2 as the set of row vectors, and define a left action˝of GL 2 pBq on it by γ˝x " xγ´1. Let T be the Bruhat-Tits tree for SL 2 pK 8 q. We denote by T 0 the set of vertices of T , which is the set of K8-equivalence classes of O K8 -lattices in K 2 8 , and by T 1 the set of its edges. The oriented graph associated with T and the set of oriented edges are denoted by T o and T o 1 , respectively. For any oriented edge e, we denote its origin by opeq, its terminus by tpeq and the opposite edge by´e. The group t˘1u acts on T o 1 by p´1qe "´e. Let Γ be an arithmetic subgroup of SL 2 pAq [Böc, §3.4], and we assume Γ to be p 1 -torsion free (namely, every element of Γ of finite order has p-power order). The group Γ acts on T and T o via the natural inclusion Γ Ñ GL 2 pK 8 q. We say a vertex or an oriented edge of T is Γ-stable if its stabilizer subgroup in Γ is trivial, and Γ-unstable otherwise. We denote by T st 0 and T o,st 1 the subsets of Γ-stable elements. For any Γ-unstable vertex v, its stabilizer subgroup in Γ is a non-trivial finite p-group and thus fixes a unique rational end which we denote by bpvq [Ser,Ch. II,§2.9].
For any ring R and any set S, we write RrSs for the free R-module with basis trss | s P Su. When S admits a left action of Γ, the Rmodule RrSs also admits a natural left action of the group ring RrΓs which we denote by˝. In this case, we also define a right action of Γ on RrSs by rss| γ " γ´1˝rss, which makes it a right RrΓs-module.
We consider it as a left ZrΓs-module via the natural inclusion Γ Ñ GL 2 pKq. Then the Steinberg module St is a finitely generated projective ZrΓs-module which sits in the split exact sequence We consider these three left ZrΓs-modules as right ZrΓs-modules via the action rss Þ Ñ rss| γ .
2.2. Drinfeld cuspforms and harmonic cocycles. For any integer k ě 2 and any A-algebra B, we denote by H k´2 pBq the B-submodule of the polynomial ring BrX, Y s consisting of homogeneous polynomials of degree k´2. We consider the left action of the multiplicative monoid M 2 pBq on H k´2 pBq defined by pγ˝X, γ˝Y q " pX, Y qγ. On GL 2 pBq, it agrees with the natural left action on Sym k pHom B pB 2 , Bqq induced by the action˝on B 2 after identifying pX, Y q with the dual basis for the basis pp1, 0q, p0, 1qq of B 2 . Put V k pBq " Hom B pH k´2 pBq, Bq.
We denote the dual basis of the free B-module V k pBq with respect to the basis We also denote by˝the natural left action of GL 2 pBq on V k pBq induced by that on H k´2 pBq. For γ "ˆa b c d˙P GL 2 pBq, P pX, Y q P H k´2 pBq and ω P V k pBq, this action is given by pγ˝ωqpP pX, Y qq " ωpγ´1˝P pX, Y qq " detpγq 2´k ωpP pdX´cY,´bX`aY qq as in [Böc,p. 51]. The group Γ acts on H k´2 pBq and V k pBq via the natural map Γ Ñ GL 2 pBq. Moreover, the monoid rpeq " ε e γ e e pε e P t˘1u, γ e P Γ, rpeq P Λ 1 q.
Note that rpeq, ε e and γ e depend on the choice of Λ 1 . The right ZrΓsmodule ZrT o,st 1 s is free with basis tres | e P Λ 1 u and thus, for any A-algebra B, any element x of L 1,k pBq can be written uniquely as Definition 2.1. Let M be a module. A map c : T o 1 Ñ M is said to be a harmonic cocycle if the following conditions are satisfied: (1) For any v P T 0 , we have (2) For any e P T o 1 , we have cp´eq "´cpeq. Any harmonic cocycle c is determined by its values at Γ-stable edges, as follows. For any e P T o 1 , an edge e 1 P T o,st 1 is said to be a source of e if the following conditions hold: ‚ When e is Γ-stable, we require e 1 " e.
‚ When e is Γ-unstable, we require that a vertex v of e 1 is Γunstable, e lies on the unique half line from v to bpvq and e has the same orientation as e 1 with respect to this half line.
For any A-algebra B, we denote by C har k pΓ, Bq the set of harmonic cocycles c : T o 1 Ñ V k pBq which is Γ-equivariant (namely, cpγpeqq " γ˝cpeq for any γ P Γ and e P T o 1 ). For any rigid analytic function f on Ω and e P T o 1 , we can define an element Respf qpeq P V k pC 8 q, which gives an isomorphism of C 8 -vector spaces Tei,Theorem 16], see also [Böc, Theorem 5.10]). By [Böc,(17)], the slash operator defined by On the other hand, the argument in [Tei,p. 506] shows that for any A-algebra B, we have a B-linear isomorphism which is independent of the choice of a complete set of representatives Λ 1 . Thus we obtain an isomorphism In particular, for any A-subalgebra B of C 8 , we have an injection

Hecke operators.
For any non-zero element Q P A, we have a Hecke operator T Q acting on S k pΓq defined as follows. Write where tξ i | i P IpΓ, Qqu is a complete set of representatives of the right coset space ΓzΓˆ1 0 0 Q˙Γ . For any f P S k pΓq, we define For any A-algebra B, we define a Hecke operator T har Q on C har k pΓ, Bq as follows. Note that ξ´1 i is an element of the monoid M´1. For any c P C har k pΓ, Bq and e P T o 1 , we put Since c is Γ-equivariant, we see that T har Q pcq is a harmonic cocycle which is independent of the choice of a complete set of representatives tξ i | i P IpΓ, Qqu. For any δ P Γ, the set tξ i δ | i P IpΓ, Qqu is also a complete set of representatives of the same right coset space. This yields T har Q pcq P C har k pΓ, Bq. By [Böc,(17)], for any A-subalgebra B of C 8 , the endomorphism T har Q is identified with the restriction on C har k pΓ, Bq Ď C har k pΓ, C 8 q of the Hecke operator T Q on S k pΓq via the isomorphism Res Γ : S k pΓq Ñ C har k pΓ, C 8 q. We also introduce a Hecke operator T 1,Q on L 1,k pBq as follows. We denote by C1 ,k pΓ, Bq the set of Γ-equivariant maps c : is independent of the choice of Λ 1 . By the uniqueness of the expression (2.3), we see that it is an isomorphism. For any c P C1 ,k pΓ, Bq and e P T o,st 1 , we put By (2.5), it is independent of the choice of tξ i u, and the same argument as in the case of T har Q shows that it defines an endomorphism T1 ,Q on C1 ,k pΓ, Bq. Now we put From the construction, we see that T 1,Q is independent of the choices of Λ 1 and tξ i u.
For an explicit description of T 1,Q , fix a complete set of representatives Λ 1 and take any element x " ř ePΛ 1 res b ω e of L 1,k pBq. For any where ε e 1 , γ e 1 and rpe 1 q are defined as (2.3) using Λ 1 . Hence we obtain Proof. Take any c P C har k pΓ, Bq. Since cprpe 1 qq " ε e 1 γ e 1 cpe 1 q, (2.4) yields which agrees with Φ Γ pT har Q pcqq.

Variation of Gouvêa-Mazur type
Let n P A be a non-zero polynomial which is divisible by ℘. In the rest of the paper, we only consider the case Γ " Γ 1 pnq. For any Let dpk, aq be the dimension of the generalized U-eigenspace in S k pΓ 1 pnqq of slope a. In this section, we prove p-adic local constancy results for dpk, aq with respect to k, which generalize the Gouvêa-Mazur conjecture [Hat2, Theorem 1.1] for the case of level Γ 1 ptq.
3.1. Weight reduction. Let Q P A be any non-zero element. Write For any γ P Γ 1 pnq and i P IpQq, we have Consider the Hecke operator T Q acting on the C 8 -vector space S k pΓ 1 pnqq, which preserves the A-lattice V k pAq by Proposition 2.2. To describe it explicitly for the case where Q is irreducible, we fix a complete set of representatives R Q of A{pQq. When Q divides n, we have IpQq " R Q and When Q does not divide n, we can find R, S P A satisfying RQ´nS " 1. Put γ˛"ˆR S n Q˙, ξ˛"ˆR Q S nQ Q˙" γ˛ˆQ 0 0 1˙.
Then we have IpQq " t˛u \ R Q and where xQy n is the diamond operator acting on S k pΓ 1 pnqq defined by f Þ Ñ f | k γ˛. Let n ě 0 be any non-negative integer. For any A-algebra B, the B-linear map Lemma 3.1. Let B be any A p n -algebra. Let ξ P M 2 pAq be any element satisfying ξ "ˆa b c d˙, a " 1, c " 0 mod ℘.
In particular, the map ρ k,n : V k`p n pBq Ñ V k pBq is Γ 1 pnq-equivariant.
Proof. For any integer i P r0, k´2s, the assumption ℘ p n B " 0 implies ξ˝µ k,n pX i Y k´2´i q " paX`cY q p n`i pbX`dY q k´2´i " pa p n X p n`c p n Y p n qpaX`cY q i pbX`dY q k´2´i " X p n paX`cY q i pbX`dY q k´2´i " µ k,n pξ˝pX i Y k´2´i qq.
Taking the dual yields the lemma.
By Lemma 3.1, for any A p n -algebra B we obtain the map 1 b ρ k,n : V k`p n pBq Ñ V k pBq and similarly for L 1,k pBq.
Lemma 3.2. For any A p n -algebra B, the maps 1 b ρ k,n : V k`p n pBq Ñ V k pBq, L 1,k`p n pBq Ñ L 1,k pBq commute with Hecke operators.
Proof. It is enough to show the assertion for the latter map. By (2.6), we reduce ourselves to showing that, for any γ P Γ 1 pnq, i P IpQq and ω P V k`p n pBq, we have pγξ i q´1˝ρ k,n pωq " ρ k,n ppγξ i q´1˝ωq.
3.2. U-operators. We denote T ℘ and T 1,℘ also by U and U 1 , respectively. Let E{K ℘ be a finite extension of complete valuation fields. We extend the normalized ℘-adic valuation v ℘ naturally to E. We denote by O E the integer ring of E.
Lemma 3.3. Let m be a non-zero element of A. Suppose that m has a prime factor π of degree one. Then the right ZrΓ 1 pmqs-module St is free of rank rΓ 1 pπq : Γ 1 pmqs, where the rank is independent of the choice of such π.
Proof. We can show that a fundamental domain of Γ 1 pπqzT is the same as the picture of [LM,§7], and that it has no Γ 1 pπq-stable vertex and only one Γ 1 pπq-stable (unoriented) edge. By (2.1), the right ZrΓ 1 pπqsmodule St is free of rank one. Thus the right ZrΓ 1 pmqs-module St is free of rank rΓ 1 pπq : Γ 1 pmqs. Since we have rΓ 1 pπq : Γ 1 pmqs " rSL 2 pAq : Γ 1 pmqs " SL 2 pF q q : the rank is independent of π.
In the sequel, we assume that n has a prime factor π of degree one. Under this assumption, Lemma 3.3 implies that the right ZrΓ 1 pnqsmodule St is free of rank d, where we put d " rΓ 1 pπq : Γ 1 pnqs.
Hence, for any A-algebra B, the B-module V k pBq is free of rank dpk´1q. We fix an ordered basis B k of the free A-module V k pAq, as follows. Take an ordered basis ps 1 , . . . , s d q of the right ZrΓ 1 pnqs-module St. The set forms a basis of the A-module V k pAq, and we order it as v 1,0 , v 2,0 , . . . , v d,0 , v 1,1 , v 2,1 , . . . , v d,1 , v 1,2 , . . . .
For any A-algebra B, the ordered basis of the B-module V k pBq induced by B k is also denoted abusively by B k . We denote by U pkq the representing matrix of U acting on the O E -module V k pO E q with respect to the ordered basis B k .
Proof. For any non-negative integer m, we have the commutative diagram with exact rows Note that ℘ m V k pO E q and ℘ m L 1,k pO E q are the kernels of the left two vertical maps. Thus it suffices to show U 1 pv i,j q P ℘ k´2´j L 1,k pO E q.

Any element of St is a Z-linear combination of elements of ZrT o,st
1 s of the form res| α with e P Λ 1 and α P Γ 1 pnq. Moreover, for any ω P V k pO E q, we have res| α b ω " res b α˝ω. By (2.6), it is enough to show that, for any i P Ip℘q, γ P Γ 1 pnq and integers j, l P r0, k´2s, we have Then the above evaluation is equal to pX j Y k´2´j q _ ppaX`cY q l pbX`dY q k´2´l q.
By (3.1) we have c, d " 0 mod ℘ and the coefficient of X j Y k´2´j in the product paX`cY q l pbX`dY q k´2´l is divisible by ℘ k´2´j . This concludes the proof.
In order to study perturbation of U pkq , we use the following lemma of [Ked]. Note that the assumption B P GL n pF q there is superfluous. Corollary 3.6. Suppose that n has a prime factor π of degree one. Put d " rΓ 1 pπq : Γ 1 pnqs. Let s 1 ď s 2 ď¨¨¨ď s dpk´1q be the elementary divisors of U pkq . Then we have Proof. By Lemma 3.4, the matrix U pkq can be written as U pkq " Bdiagp℘ k´2 , . . . , ℘ k´2 , . . . , ℘, . . . , ℘, 1, . . . , 1q, where B P M dpk´1q pO E q and the diagonal entries of the last matrix are t℘ j | 0 ď j ď k´2u, each with multiplicity d. Then the corollary follows from Lemma 3.5.
Corollary 3.7. Let n ě 0 be any non-negative integer. Then, for some matrices B 1 , B 2 , B 3 , B 4 with entries in O E , we have Proof. By Lemma 3.2, the lower right block is congruent to U pkq and the lower left block is zero modulo ℘ p n . By Lemma 3.4, the entries on the upper left block are divisible by ℘ k´1 . This concludes the proof.
3.3. Perturbation. Let E{K ℘ be a finite extension inside C ℘ . Let V be an E-vector space of finite dimension and T : V Ñ V an Elinear endomorphism. For an eigenvector of T with eigenvalue λ P C ℘ , we refer to v ℘ pλq as its slope. For any rational number a, we denote by dpT, aq the multiplicity of T -eigenvalues of slope a. If A is the representing matrix of T with some basis of V , we also denote it by dpA, aq.
Proposition 3.8. Let d 0 , n and L be positive integers. Let A P M L pO E q be a matrix such that its i-th smallest elementary divisor s i satisfies s i ě t i´1 d 0 u for any i. Put ε 0 " dpA, 0q and Moreover, we put q 1 " r 1 " 0 and for any l ě 2, we write q l " t l´2 d 0 u and r l " l´2´d 0 q l . We define C 2 pn, d 0 , ε 0 q as min " 2p n`d 0 q l pq l´1 q`2q l pr l`1 q 2pl´ε 0 qˇˇˇˇε 0 ă l ď 1`d 0 p n * and put Cpn, d 0 , ε 0 q " mintC 1 pn, d 0 , ε 0 q, C 2 pn, d 0 , ε 0 qu P p0, p n q.
Let B P M L pO E q be any matrix and put A 1 " A`℘ p n B. Let a be any non-negative rational number satisfying a ă Cpn, d 0 , ε 0 q.
Proof. We put Then a l is, up to a sign, the sum of principal lˆl minors of A. Since P A " P A 1 mod ℘, we have dpA 1 , 0q " dpA, 0q " ε 0 . From the assumption on elementary divisors, we see that if i ą d 0 , then any iˆi minor of A is divisible by ℘. This yields ε 0 ď d 0 .
By [Ked,Theorem 4.4.2], for any l ě 0 we have Here we mean that the second term of the right-hand side is zero for l ď 1. Let R be the right-hand side of the inequality. We claim that for any l ą ε 0 , we have a ă Cpn, d 0 , ε 0 q ñ R ą apl´ε 0 q.
Then R ą apl´ε 0 q if and only if (3.2) pp n´a ql´1 2 p n d 0 p1`p n q`aε 0 ą 0.
Since the condition a ă Cpn, d 0 , ε 0 q yields p n ą a, the left-hand side of (3.2) is increasing with respect to l. Thus (3.2) holds for any l ą 1`d 0 p n if and only if it holds for l " 2`d 0 p n , which is equivalent to a ă C 1 pn, d 0 , ε 0 q. On the other hand, when l ď 1`d 0 p n , we have (3.3) R " p n`1 2 d 0 q l pq l´1 q`q l pr l`1 q, from which the claim follows.
Let N A and N A 1 be the Newton polygons of P A and P A 1 , respectively. It suffices to show that the segments of N A and N A 1 with slope less than Cpn, d 0 , ε 0 q agree with each other. Suppose the contrary and take the smallest slope a ă Cpn, d 0 , ε 0 q satisfying dpA, aq ‰ dpA 1 , aq.
Let pl, yq be the right endpoint of the segment of slope a in either of N A or N A 1 . Since dpA, 0q " dpA 1 , 0q, we have a ą 0 and l ą ε 0 . Then the above claim yields Since a is minimal, this implies that slope a appears in both of N A and N A 1 . Applying the same argument to the right endpoint of the segment of slope a in the other Newton polygon, we obtain dpA, aq " dpA 1 , aq. This is the contradiction.
By a similar argument, we can show a slightly different perturbation result as follows.
Proposition 3.9. With the notation in Proposition 3.8, we suppose that the following conditions hold.
Then, for any non-negative rational number a ď n, we have dpA, aq " dpA 1 , aq.
Proof. Let R be as in the proof of Proposition 3.8. We claim R ą npl´ε 0 q for any l ą ε 0 under the assumptions (1) and (2).
Indeed, when l ą 1`d 0 p n , we have R ą npl´ε 0 q for any such l if and only if n ă C 1 pn, d 0 , ε 0 q, namely If p ě 3 or n ě 3, then we have 1 2 p n´n ě 1 2 and the above inequality holds. If p " 2 and n ă 3, it is equivalent to d 0´ε0 ď 1. Thus, under the condition (1), we have R ą npl´ε 0 q in this case.
Let us consider the case of l ď 1`d 0 p n . Note that l " 1 is allowed only if ε 0 " 0, in which case the claim holds by R " p n ą n. For l ě 2, by (3.3) we have R ą npl´ε 0 q if and only if ą 2npr l`2´ε0 q.
Note r l`1 d 0´1 2 P r´1 2 , 1 2 s. Since q l and n are integers, we have Thus the above inequality holds if which is equivalent to the condition (2) and the claim follows. Now the same reasoning as in the proof of Proposition 3.8 shows dpA, aq " dpA 1 , aq.
For the U-operator acting on V k pK ℘ q, we denote dpU, aq also by dpk, aq. Note that it agrees with the previously defined one for S k pΓ 1 pnqq.
Lemma 3.10. Suppose that n has a prime factor of degree one. Then dpk, 0q is independent of k.
Proof. By Corollary 3.7, for any k ě 2 we have U pk`1q "ˆOO U pkq˙m od ℘ and thus detpI´U pk`1q Xq " detpI´U pkq Xq mod ℘, from which the lemma follows. Now the following theorems give generalizations of [Hat2, Theorem 1.1].
Proof. By Lemma 3.10, we may assume k 1 " k`p n . By Corollary 3.7, we can write U pk`p n q`℘p n W " V with W P M dpk`p n´1 q pO K℘ q and V "ˆ℘ k´1 B 1 B 2 O U pkq˙, B 1 P M dp n pO K℘ q, B 2 P M dp n ,dpk´1q pO K℘ q.
Corollary 3.6 and Lemma 3.10 show that U pk`p n q satisfies the assumptions of Proposition 3.8. Hence we obtain dpk`p n , aq " dpV, aq. By [Hat2,Lemma 2.3 (2)], the matrix ℘ k´1 B 1 has no eigenvalue of slope less than k´1. Since a ă k´1, we also have dpV, aq " dpk, aq. This concludes the proof.
Theorem 3.12. Suppose that n has a prime factor π of degree one. Let n ě 1 and k ě 2 be any integers and a ď n any non-negative rational number. Put d " rΓ 1 pπq : Γ 1 pnqs and ε " dpk, 0q. Suppose that the following conditions hold.
Proof. This follows in the same way as Theorem 3.11, using Proposition 3.9 instead of Proposition 3.8.
It will be necessary to use an increasing function no more than Cpn, d, εq instead of itself. Here we give an example.
The right-hand side equals By the inequality of arithmetic and geometric means, it is no less than D 2 pn, d, εq and the lemma follows.

℘-adic continuous family
We say F P V k pC ℘ q is a Hecke eigenform if it is a non-zero eigenvector of T Q for any Q P A. We denote by λ Q pF q the T Q -eigenvalue of F . Since Hecke operators commute with each other, if dpk, aq " 1 then any non-zero U-eigenform in V k pC ℘ q of slope a is a Hecke eigenform. 4.1. Construction of the family. Now we prove the following main theorem of this paper.
Suppose dpk 1 , aq " 1. Let F 1 P V k 1 pC ℘ q be a Hecke eigenform of slope a. Then, for any integer k 2 ě k 1 satisfying k 2 " k 1 mod p n , we have dpk 2 , aq " 1 and thus there exists a Hecke eigenform F 2 P V k 2 pC ℘ q of slope a which is unique up to a scalar multiple. Moreover, for any Q we have v ℘ pλ Q pF 1 q´λ Q pF 2 qq ą p n´pn 1´a .
Proof. By Lemma 3.10, we may assume pk 1 , k 2 q " pk, k`p n q for some integer k ě 2. Theorem 3.11 yields dpk`p n , aq " 1 and any non-zero U-eigenform F 2 P V k 2 pC ℘ q of slope a is a Hecke eigenform. Take a finite extension E{K ℘ inside C ℘ containing λ Q pF i q and λ ℘ pF i q for i " 1, 2. We may assume where each entry of x is the coefficient of v s,t P B k 2 in F 2 with t ă p n . For any integer N and z " t pz 1 , . . . , z N q P O N E , we put v ℘ pzq " mintv ℘ pz i q | i " 1, . . . , Nu.
Replacing F i by its scalar multiple, we may assume v ℘ pF i q " 0.
For any H P V k i pO E q, we denote byH its image by the natural map V k i pO E q Ñ V k i pO E,p n q. Consider the weight reduction map 1 b ρ k,n : V k`p n pO E,p n q Ñ V k pO E,p n q as in §3.1, which we denote by ρ. Then ρpF 2 q " y mod ℘ p n .
Since v ℘ pλ ℘ pF 2 qq " a ă k´1, this forces v ℘ pyq ď a and the claim follows.
Since we have a ă Cpn, d, εq ă p n , the above claim yields v ℘ pG 1 q ď a.
If G 1 P O E F 1 , then G 1 is a Hecke eigenform with the same eigenvalues as those of F 1 . Thus we have Suppose G 1 R O E F 1 , and take H 1 P V k pO E q such that F 1 and H 1 form a basis of a direct summand of V k pO E q containing G 1 . Write Then β ‰ 0. By (4.1), for any R P t℘, Qu we have λ R pF 2 qG 1 " T R pG 1 q " αλ R pF 1 qF 1`β T R pH 1 q mod ℘ p n V k pO E q.
Take an ordered basis pF 1 , H 1 ,ṽ 3 , . . . ,ṽ dpk´1q q of the O E -module V k pO E q, and we denote byŨ pkq the representing matrix of U with respect to it. By (4.7), we can writẽ Note that the elementary divisors ofŨ pkq and U pkq agree with each other. Let V be the element of M dpk´1q pO E q with the same columns as those ofŨ pkq except the second column which we require to bë Then we have dpV, aq ě 2. On the other hand, since p n´b ě p n 1 , the assumption a ă Cpn 1 , d, εq and Proposition 3.8 yield dpV, aq " dpk, aq " 1, which is the contradiction. Thus the case b ď p n´pn 1 never occurs. Now the theorem follows from (4.2) and (4.6).
Proof of Theorem 1.1. Suppose that n, k and a satisfy the assumptions of Theorem 1.1. Take any k 1 ě k satisfying m " v p pk 1´k q ě log p pp n`a q.
Note that, if dpk, aq " 1, then any U-eigenform of slope a in V k pC ℘ q is identified with a scalar multiple of that in V k pKq Ď S k pΓ 1 pnqq via the fixed embedding ι ℘ . Thus Theorem 4.1 produces a Hecke eigenform F k 1 P S k 1 pΓ 1 pnqq such that for any Q we have v ℘ pι ℘ pλ Q pF k 1 q´λ Q pF k qqq ą p m´pn´a .
This concludes the proof of Theorem 1.1.
In the following, we give examples of congruences between Hecke eigenvalues obtained by Theorem 1.1 for this case, using results of [BV2,LM,Pet]. Note that the Hecke operator at Q considered in [BV2,Pet] is QT Q with our normalization.
4.2.1. Slope zero forms. By dpk, 0q " 1, any U-eigenform of slope zero in S k pΓ 1 ptqq is a member of a t-adic continuous family obtained by Theorem 1.1. Some of such eigenforms can be given by the theory of A-expansions [Pet]. For any integer k ě 3 satisfying k " 2 mod q´1, Petrov constructed an element f k,1 P S k pSL 2 pAqq with A-expansion [Pet,Theorem 1.3].
We know that f k,1 is a Hecke eigenform whose Hecke eigenvalue at Q is one for any Q; this follows from a formula for the Hecke action [Pet, p. 2252] and c a " a k´n .
For such k, let f ptq k,1 P S k pΓ 1 ptqq be the t-stabilization of f k,1 of finite slope, namely f ptq k,1 pzq " f k,1 pzq´t k´1 f k,1 ptzq.
It is non-zero by [Pet,Theorem 2.2]. Moreover, we can show that f ptq k,1 is a Hecke eigenform which also satisfies λ Q pf ptq k,1 q " 1 for any Q.
Proposition 4.2. Let k ě 2 be any integer and F k any element of S k pΓ 1 ptqq of slope zero. Then we have λ Q pF k q " 1 for any Q.
Proof. Let r P t0, 1, . . . , q´2u be an integer satisfying k " r mod q´1. For a " 0, the assumptions of Theorem 1.1 are satisfied by n " 1. Then, for any integer s ě 1, we obtain a Hecke eigenform of slope zero F k 1 P S k 1 pΓ 1 ptqq, k 1 " k`pq`1´rqq s such that, with the fixed embedding ι t :K Ñ C t , we have ι t pλ Q pF k 1 qq " ι t pλ Q pF k qq mod t q s´p for any Q.
Thus λ 1`t pG 10 q agrees with the evaluation T 1`t pG 10 qpγ 0 qpX 7 Y q after identifying G 10 with a harmonic cocycle. By [LM,(7.1)], we have λ 1`t pG 10 q " 1´t´t 3 . On the other hand, by computing the characteristic polynomial of T 1`t acting on S 19 pΓ 1 ptqq using [LM,(7.1)] and plugging in X " 1´t´t 3`Z into it, (4.8) implies v t pι t pλ 1`t pG 10 qλ 1`t pG 19 qqq " 9. Note that, since these eigenvalues are not powers of t or 1`t, the Hecke eigenforms G 10 and G 19 are not the t-stabilizations of Hecke eigenforms with A-expansion.