Dimension variation of Gouv\^{e}a-Mazur type for Drinfeld cuspforms of level $\Gamma_1(t)$

Let $p$ be a rational prime and $q>1$ a $p$-power. Let $S_k(\Gamma_1(t))$ be the space of Drinfeld cuspforms of level $\Gamma_1(t)$ and weight $k$ for $\mathbb{F}_q[t]$. For any non-negative rational number $\alpha$, we denote by $d(k,\alpha)$ the dimension of the slope $\alpha$ generalized eigenspace for the $U$-operator acting on $S_k(\Gamma_1(t))$. In this paper, we prove a function field analogue of the Gouv\^{e}a-Mazur conjecture for this setting. Namely, we show that for any $\alpha\leq m$ and $k_1,k_2>\alpha+1$, if $k_1\equiv k_2 \bmod p^m$, then $d(k_1,\alpha)=d(k_2,\alpha)$.


Introduction
Let p be a rational prime, q ą 1 a p-power, A " F q rts and ℘ P A a monic irreducible polynomial. For K 8 " F q pp1{tqq, we denote by C 8 the p1{tq-adic completion of an algebraic closure of K 8 . Then the Drinfeld upper half plane Ω " C 8 zK 8 has a natural structure of a rigid analytic variety over K 8 .
Let k be an integer and Γ a subgroup of SL 2 pAq. Then a Drinfeld modular form of level Γ and weight k is a rigid analytic function f : Ω Ñ C 8 satisfying fˆa z`b cz`d˙" pcz`dq k f pzq for any z P Ω,ˆa b c d˙P Γ and a holomorphy condition at cusps. The notion of Drinfeld modular form can be considered as a function field analogue of that of elliptic modular form and the former often has properties which are parallel to the latter. However, despite that the theory of p-adic families of elliptic modular forms is highly developed and has been yielding many applications, ℘-adic properties of Drinfeld modular forms are not wellunderstood yet. A typical difficulty in the Drinfeld case seems that a naïve analogue of the universal character Zp Ñ Z p rrZp ssˆis not locally analytic by [Jeo,Lemma 2.5] and thus similar constructions to those in the classical case including [AIP] will not immediately produce Date: June 25, 2018. an analytic family of invertible sheaves interpolating automorphic line bundles. Still, there seem to exist interesting structures in ℘-adic properties of Drinfeld modular forms. In [BV1,BV2], Bandini-Valentino studied an analogue of the classical Atkin U-operator, which we also denote by U, acting on the space S k pΓ 1 ptqq of Drinfeld cuspforms of level Γ 1 ptq and weight k. The operator U is defined by The normalized t-adic valuation of an eigenvalue of U is called slope. Note that here we adopt the different normalization from that of Bandini-Valentino, and as a result our notion of slope is smaller than theirs by one. For a non-negative rational number α, we denote by dpk, αq the dimension of the generalized eigenspace of U acting on S k pΓ 1 ptqq for the eigenvalues of slope α. Then they proposed a conjecture on a p-adic variation of dpk, αq with respect to k [BV2, Conjecture 6.1] which can be regarded as a function field analogue of the Gouvêa-Mazur conjecture [GM1,Conjecture 1]. In this paper, we will prove it.
First note that, as is mentioned in [Wan,§4,Remarks], the arguments of [GM2] and [Wan] can be generalized over suitable Drinfeld modular curves (including X ∆ 1 pnq of [Hat]). In particular, the characteristic power series of U acting on the spaces of ℘-adic overconvergent Drinfeld modular forms of weight k 1 and k 2 are congruent modulo ℘ p m . Also in our setting, we can show the congruence P pk 1 q pXq " P pk 2 q pXq mod t p m up to some factor. However, though with this we can prove Theorem 1.1 for p ě 3, it is not enough to settle the case of p " 2 on which Bandini-Valentino stated their conjecture.
Instead, we investigate the formula of the representing matrix of U given by (3.1)] more closely. Luckily, the representing matrix is of very special form: each entry on the j-th column (with the normalization that the leftmost column is the zeroth) is an element of F q t j . Thanks to this fact, we can give a lower bound of elementary divisors of the representing matrix (Lemma 2.2). Then a perturbation argument shows that the n-th coefficients of P pkq pXq and P pk`p m q pXq are much more congruent than modulo t p m up to some factor of slope ě k´1 (Corollary 2.7), which is enough to yield Theorem 1.1 for any p.
Acknowledgments. The author would like to thank Gebhard Böckle for informing him of Valentino's table computing characteristic polynomials of U, and Maria Valentino for pointing out an error in the author's previous computer calculation. This work was supported by JSPS KAKENHI Grant Number JP17K05177.

Dimension variation
Let k ě 2 be an integer. Put On the space S k pΓ 1 ptqq of Drinfeld cuspforms of level Γ 1 ptq and weight k, we consider the U-operator for t defined by (1.1). Note that we follow the usual normalization of the U-operator which differs from that of [BV1,§2.4] by 1{t. Then Bandini-Valentino [BV1, (3.1)] explicitly describe the action of U with respect to some basis c pkq 0 , . . . , c pkq k´2 , which reads as follows with our normalization: (2.1) Here it is understood that the binomial coefficient`c d˘i s zero if any of c, d, c´d is negative and the terms involving c pkq j`hpq´1q are zero if j`hpq1 q R r0, k´2s. We denote by U pkq " pU pkq i,j q 0ďi,jďk´2 the representing matrix of U for this basis. Then we have U pkq P M k´1 pAq. We identify the t-adic completion of A with F q rrtss naturally and consider U pkq as an element of M k´1 pF q rrtssq.
Definition 2.1. Let B " pB i,j q 0ďiďm´1,0ďjďn´1 be an element of M m,n pF q rrtssq. We say B is glissando if B i,j P F q t j for any i, j.
Lemma 2.2. Let B " pB i,j q 0ďiďm´1,0ďjďn´1 be a glissando matrix in M m,n pF q rrtssq. Let s 1 ď s 2 ď¨¨¨ď s r be the elementary divisors of B (namely, they are integers or`8 such that t s i is the pi´1, i´1q-entry of the Smith normal form of B). Then we have s l ě l´1 for any l.
Proof. We prove the lemma by induction on n. For n " 1, we have s 1 " 0 if B ‰ O and s 1 "`8 otherwise. For n ą 1, we may assume B ‰ O and let c be the integer with 0 ď c ď n´1 such that the leftmost non-zero column of B is the c-th one. Since B is glissando, the first elementary divisor of B is c ě 0 and the rest are equal to the elementary divisors of a matrix t c`1 B 1 , where B 1 is also glissando with n´1 columns. Let s 1 1 ď¨¨¨ď s 1 r 1 be the elementary divisors of B 1 . By the induction hypothesis, we have s 1 l ě l´1 and thus s l " c`1`s 1 l´1 ě l´1 for l ě 2. This concludes the proof. Let v t be the t-adic additive valuation normalized as v t ptq " 1. For any element P pXq " ř 8 n"0 p n X n P F q rrtssrrXss, the Newton polygon of P pXq is by definition the lower convex hull of the set tpn, v t pp n qq | n ě 0u.
Lemma 2.3. Let B P M m pF q rrtssq be a glissando matrix. For any non-negative integer l, put P pXq " detpI´t l XBq " m ÿ n"0 p n X n P F q rrtssrXs.
(2) Any slope of the Newton polygon of P pXq is no less than l.
Proof. First note that, for the characteristic polynomial QpXq " detpXIt l Bq, we have P pXq " X m QpX´1q and thus p n is, up to a sign, equal to the sum of the principal nˆn minors of t l B. Since B is glissando, this shows (1). Since p 0 " 1, the resulting inequality v t pp n q ě ln implies (2). Now we put P pkq pXq " detpI´XU pkq q " k´1 ÿ n"0 a pkq n X n and a pkq n " 0 for any n ě k. Let y " N pkq pxq be the Newton polygon of P pkq pXq. For any non-negative rational number α, we denote by dpk, αq the dimension of the generalized eigenspace for the eigenvalues of normalized t-adic valuation α. Then dpk, αq is equal to the width of the segment of slope α in the Newton polygon N pkq .

DIMENSION VARIATION FOR DRINFELD CUSPFORMS OF LEVEL Γ1ptq 5
Proof. By (2.1), we have U pkq 0,0 "`k´2 0˘" 1. On the other hand, since U pkq is glissando, we have v t pU pkq i,j q ě j and a pkq 1 "´k´2 Moreover, from Lemma 2.3 (1) we obtain v t pa pkq n q ą 0 for any n ě 2. This yields the lemma.
Lemma 2.5. Let a and b be non-negative integers. Let m ě 1 be an integer. Then we havè a`p m b˘"`a b˘``a b´p m˘m od p.
Here it is understood that`c d˘" 0 if any of c, d, c´d is negative.
Proof. This follows from pX`1q a`p m " pX`1q a pX p m`1 q mod p.
Proposition 2.6. Let m ě 1 be an integer. Then there exist glissando matrices C P M p m ,k´1 pAq and D P M p m ,p m´k`1pAq satisfying Here it is understood that the middle blocks are empty if p m ď k´1.
Proof. Let j be an integer satisfying 0 ď j ď k`p m´2 . By (2.1), the element Upc pk`p m q j q is equal to Note that both of U pk`p m q i,j and U pkq i,j are divisible by t p m for j ě p m . Since U pk`p m q is glissando, what we need to show is (1) For any j ď mintk´2, p m´1 u and i P r0, k´2s, we have U pk`p m q i,j " U pkq i,j , and (2) If k ď p m , then for any j P rk´1, p m´1 s and i P r0, k´2s, we have U pk`p m q i,j " 0.
First we suppose j ď mintk´2, p m´1 u. By Lemma 2.5, the element Upc pk`p m q j q equals p´tq j´`k´2´j j˘``k´2´j j´p m˘¯c pk`p m q j t j ÿ hPZ,h‰0 j`hpq´1qPr0,k´2s Since j ă p m , we have`k´2´j j´p m˘" 0. For the case of j`hpq´1q P r0, k´2s, we also have´hpq´1q´p m ď j´p m ă 0 and`k´2´j´h pq´1q hpq´1q´p m˘" k´2´j´hpq´1q j´p m˘" 0. This proves (1). Next we suppose k ď p m and j P rk´1, p m´1 s. For any i P r0, k´2s, the element U pk`p m q i,j is equal tó if we can write i " j`hpq´1q with some h ‰ 0, and zero otherwise. Since i ď k´2, we have k´2´j´hpq´1q ě 0 and Lemma 2.5 implies Since i " j`hpq´1q P r0, k´2s and j ă p m , we have`k´2´j´h pq´1q hpq´1q´p m˘" k´2´j´hpq´1q j´p m˘" 0 as is seen above. Since j ě k´1, we also havè k´2´j´hpq´1q hpq´1q˘"`k´2´j´h pq´1q j˘" 0. This proves (2) and the proposition follows.
Let V P M k`p m´1pAq be the matrix of the right-hand side of Proposition 2.6. Let D 1 be the upper pp m´k`1 qˆpp m´k`1 q block of D if k ď p m and D 1 " O otherwise. Then D 1 is also glissando. Put P pXq " detpI´XV q " P pkq pXq detpI´t k´1 XD 1 q and writeP pXq " ř k`p m´1 n"0ã n X n . We denote byÑ the Newton polygon ofP pXq.
Corollary 2.7. Let m and n be integers satisfying m ě 1 and 0 ď n ď k`p m´1 . Then we have v t pa pk`p m q n´ã n q ě p m`n´1 ÿ l"1 mintl´1, p m u.
Proof. Write U pk`p m q " V`t p m W with some W P M k`p m´1pAq. Let s 1 ď¨¨¨ď s k`p m´1 be the elementary divisors of V . Since V is glissando, by Lemma 2.2 we obtain s l ě l´1 for any l. Then [Ked,Theorem 4.4.2] shows v t pa pk`p m q n´ã n q ě p m`n´1 ÿ l"1 mints l , p m u ě p m`n´1 ÿ l"1 mintl´1, p m u.
Lemma 2.8. Let m and n be integers satisfying m ě 1 and n ě 2.
Then we have Proof. First we assume n´2 ě p m . Then the left-hand side of the lemma is equal to For m ě 1, we have 1 2 p m ě m and thus 1 2 p m pp m`3 q ě mpp m`2 q. Hence the right-hand side of (2.2) is greater than mpn´1q.
Next we assume n´2 ă p m . In this case, the left-hand side of the lemma equals p m`1 2 pn´1qpn´2q. It is greater than mpn´1q if and only ifˆn´ˆm`3 2˙˙2`2 p m´m pm`1q´1 4 ą 0.
Since m and n are integers, the first term is no less than 1 4 . Since we can show 2p m ą mpm`1q for any p and m ě 1, the lemma follows also for this case.
Lemma 2.9. The part of the Newton polygonÑ ofP pXq of slope less than k´1 agrees with that of N pkq .
Proof. For any QpXq P F q rrtssrXs and any non-negative rational number α, the Newton polygon of QpXq has a segment of slope α and width l if and only if it has exactly l roots of normalized t-adic valuation´α. By Lemma 2.3 (2), every root of the polynomial detpI´t k´1 XD 1 q has normalized t-adic valuation no more than´pk´1q. Thus, forP pXq and P pkq pXq, the sets of roots of normalized t-adic valuation more thań pk´1q agree including multiplicities. This shows the lemma.
Theorem 2.10. Let k and m be integers satisfying k ě 2 and m ě 0. Let α be a non-negative rational number satisfying α ď m and α ă k´1. Then we have dpk`p m , αq " dpk, αq.
Proof. Let tα 1 , . . . , α N u be the set of slopes of the Newton polygons N pk`p m q and N pkq which is no more than m and less than k´1, and renumber them so that α i ă α i`1 for any i. We proceed by induction, following the proof of [Wan,Lemma 4.1]. By Lemma 2.4, we have α 1 " 0 and dpk`p m , 0q " dpk, 0q " 1. Thus we may assume m ě 1 and N ě 2. Suppose that for some r ď N´1, the equality dpk`p m , α i q " dpk, α i q holds for any i satisfying 1 ď i ď r. By Lemma 2.9, this means that the Newton polygons N pkq , N pk`p m q andÑ agree with each other on the part of slope no more than α r . Put α " α r`1 ą α 1 " 0 and let us prove dpk`p m , αq " dpk, αq. We choose k 1 P tk, k`p m u such that the slope α occurs in N pk 1 q and let k 2 be the other. Let β be the slope of N pk 2 q on the right of α r . Then β ě α.
Let pn, v t pa pk 1 q n qq be the endpoint of the segment of N pk 1 q of slope α. Since the Newton polygon N pk 1 q has a segment of slope zero, we have n ě 2 and N pk 1 q pnq " v t pa pk 1 q n q ď αpn´1q ď mpn´1q.
Then Corollary 2.7 and Lemma 2.8 imply (2.3) v t pa pk 1 q n q ă v t pa pk`p m q n´ã n q. If k 1 " k, then Lemma 2.9 shows v t pa pk 1 q n q " v t pa pkq n q " v t pã n q and from (2.3) we obtain v t pa pk`p m q n q " v t pã n q " v t pa pkq n q. This equality implies α " β and dpk, αq ď dpk`p m , αq. In particular, the slope α also occurs in N pk`p m q .
If k 1 " k`p m , then (2.3) gives v t pã n q " v t pa pk`p m q n q. Let γ be the slope of the Newton polygonÑ on the right of α r . Then this equality implies γ ď α ă k´1. By Lemma 2.9, we have β " γ ď α. Therefore, we have α " β " γ and the width of the segment of slope α inÑ is no less than that in N pk`p m q . Thus Lemma 2.9 again shows dpk, αq ě dpk`p m , αq. In particular, the slope α also occurs in N pkq . Combining these two cases, we obtain dpk, αq " dpk`p m , αq. This concludes the proof of Theorem 2.10.

Remarks
Computations using (2.1) with Pari/GP indicate that the slopes appearing in S k pΓ 1 ptqq have some patterns (see also [BV2,§6]). The below is a table of the case p " q " 2, where the bold numbers denote multiplicities.
This could be thought of as a function field analogue of Emerton's theorem [Eme] which asserts that the minimal slopes of S k pΓ 0 p2qq are periodic of period 8.