$2$-adic slopes of Hilbert modular forms over $\mathbb{Q}(\sqrt{5})$

We show that for arithmetic weights with a fixed finite order character, the slopes of $U_p$ (for $p=2$) acting on overconvergent Hilbert modular forms of level $U_0(4)$ are independent of the (algebraic part of the) weight and can be obtained by a simple recipe from the classical slopes in parallel weight $3$.


Introduction
Buzzard and Kilford [BK05] studied slopes a of overconvergent (elliptic) modular forms in the special case that the tame level is 1 and p = 2. In this case they proved that, for weights near the boundary of weight space, slopes have a great deal of structure, in particular they are in arithmetic progression and scale to zero as the weight moves further towards the boundary of weight space. From this one can deduce that, over the boundary annulus, the eigencurve is a disjoint union of annuli. This example motivated further work in the area (see [Roe14,WXZ17]) and most notably Liu-Wan-Xiao in [LWX17] showed that, the slopes of quaternionic modular forms (over Q) have structure analogous to what was observed by Buzzard-Kilford. Specifically, they showed that the slopes (for weights near the boundary) are given as a union of arithmetic progressions with common difference and they scale as the weight approaches the boundary. From this they then deduce that over the boundary annulus, the eigenvarieties are are disjoint union of rigid spaces which are flat over this annulus. Then, via Chenevier's overconvergent Jacquet-Langlands correspondence, one deduces similar results for a large class of spaces of overconvergent modular forms. This in turn can be used to prove parity conjectures of Selmer ranks for modular forms.
In the setting of Hilbert modular forms, much less is known. Computations in [Birb] suggest that, again for weights near the boundary, slopes behave in a highly predictable way but in general are not a union of arithmetic progressions with common difference. In this more general setting one observes that the multiset of slopes appears to be completely determined by a simple recipe starting with slopes of classical Hilbert modular a The p-adic valuation of the Up eigenvalues. 1 forms of some small weight. Similar behaviour is also present for elliptic modular forms, yet due to their simpler nature, in this case, the above recipe coincidentally gives unions of arithmetic progressions with common difference. This combined with the scaling behaviour of the slopes gives the above statements about the geometry of the associated eigenvarieties.
In the Hilbert setting, one can also consider the effect the splitting behaviour of p. If p = i p i then U p = i U pi and by making certain restrictions on the weight space b and study the slopes of U pi . If p is totally split, work of Newton-Johansson [JN18], shows that the methods of [LWX17] can be used to describe the slopes of U pi , which for general p is not the case. c Moreover, they construct partial eigenvarieties and prove that over a boundary annulus these partial eigenvarieties decompose as a union of components which are finite over weight space. From this they prove the parity part of the Bloch-Kato conjecture for Galois representations associated to Hilbert modular forms (still with p totally split).
In general, the structure of the slopes of U p in the Hilbert case is unknown. In this note, we show that for an inert prime (p = 2) and for arithmetic weights with a fixed finite order character, the slopes of U 2 acting on spaces of overconvergent Hilbert modular forms of level U 0 (4) (over Q( √ 5)) are completely determined by the slopes of classical Hilbert modular forms of parallel weight 3 with a suitable finite order character. We also give computational evidence that as weights approach the boundary of weight space, the slopes scale and tend to zero (similar to what is seen for elliptic modular forms). Specifically, we prove the following result: Theorem. The slopes of U 2 acting in weight [n 1 , n 2 ]χ on overconvergent Hilbert modular forms of level U 0 (4) is independent of n i . Moreover, they are completely determined by the slopes of U 2 acting on the classical spaces of Hilbert modular forms of level U 0 (4), parallel weight 3 with nebentypus χτ n for a τ a certain character of order 6.
This theorem is in line with [Birb,Conjecture 4.7.9], appropriately generalized to this situation where the level is not sufficiently small. The proof is essentially an extended exercise on p-adic matrix analysis which we do not expect to generalize to many other situations. Specifically, it relies on the fact that, for this example, knowledge of the p-adic valuations of the entries of the matrix associated to U p completely determine the slopes of its eigenvalues, which is not always the case. d Yet we note that the methods used here are of a different nature than those in [LWX17], which rely on studying the associated Hodge polygons.
Acknowledgements. This paper owes it existence to very useful conversations with Kevin Buzzard, for which I am very grateful. The work was done while the author was an EPSRC Doctoral Prize fellow at UCL.

Background
Let us begin with the relevant set-up. We want to study the p-adic slopes of overconvergent Hilbert modular forms over F = Q( √ 5) with p = 2. We note that 2 is inert in F and that F has narrow class number one which will simplify the computations. As is usual, instead of working directly with the space of overconvergent Hilbert modular b Specifically, fixing certain components of the weights. c This is due to the fact that the associated Newton and Hodge polygons do not touch in general (see 3.17). d As one can check in the examples computed in [Birb].
forms, we will instead work with spaces of overconvergent quaternionic modular forms as the geometry is simpler. Specifically, by [Bira, Theorem 1], if we let D be the unique quaternion algebra over F ramifying only at the two infinite places of F , then the eigenvariety associated to Hilbert modular forms (as defined by [AIP16]) is isomorphic to the eigenvariety associated to these quaternionic forms. Therefore, since we are only interested in slopes, there is no loss in working with overconvergent quaternionic forms. We begin by recalling their definition.
Notation 2.1. Throughout D will be the totally definite quaternion algebra, which we note has class number one and we let O D denote a fixed maximal order. Specifically, we let O D be the icosian ring, since any other maximal order is conjugate to this (see [Dem05,Section 3]). Lastly, we fix a splitting of D at p.
Definition 2.2. Let n ∈ Z 2 and v ∈ Z 2 such that n + 2v = (r, r) for some r ∈ Z. Set k = n + 2 and w = v + n + 1 (understood as the obvious component-wise sum). It follows from the above that all the entries of k have the same parity and k = 2w − r. Moreover, note that given k (with all entries paritious and greater than 2) and r we can recover n, v, w. We call the 5-tuple (k, r, n, v, w) a weight tuple and, in order to simplify notation, we denote it simply by [n 1 , n 2 ], where it is understood that k, r, v, w are implicit.
For χ a finite character on O F we consider it as a character on O K via weak approximation. Then for an algebraic weight [n 1 , n 2 ] we define a weight-character [n 1 , n 2 ]χ as the map O K → C p defined by α → α n1 α n2 χ(α), where for α ∈ O K we let α denote the image of α under the action of the non-trivial element in Gal(K/Q p ). Such weight-characters are called arithmetic weights.
The space of overconvergent quaternionic modular forms of weight κ = [n 1 , n 2 ]χ, level U 0 (p s ) (and radius of overconvergence 1), denoted by S D, † κ (U 0 (np s )), is defined as the vector space of functions where (k, r, n, v, w) is the associated weight tuple. To ease notation we are ignoring the radius of overconvergence since this will not affect our result.
There is a bijection between where t ∈ D f is trivial at all finite places different from 2 and at 2 is given by 0 −1 1 0 . Now, by evaluating at t we get an isomorphism Notation 2.5. From now on we let ζ denote a fixed non-trivial cube root of unity in K.
Proposition 2.6. In the above set-up, Γ t /Γ t ∩ O × K is cyclic of order 3 and generated by the matrix Proof. This is just a simple calculation. Now, in level U 0 (p s ), the action of U p is given by (We have not given an explicit description of the a i , d i , b, c since this is cumbersome and we will only need the stated properties.) Proof. This is a simple (computer) calculation.

Slopes
In order for the space of modular forms of weight [n 1 , n 2 ]χ to be non-trivial one requires is the associated weight tuple. With this in mind we note that, since p = 2, there is are no non-trivial characters of O F with conductor 4 such that χ(x) = N F/Q (x) 2 for all x ∈ O × F and there is a unique character with χ(x) = N F/Q (x). We now set-up some notation that will be used throughout.
• Our arithmetic weights will be of the form [n 1 , n 2 ]χ where n 1 , n 2 are both odd and χ is the unique non-trivial of O F with conductor 4 on O F which sends the fundamental unit to −1 and sends −1 to 1.

2-ADIC SLOPES OF HILBERT MODULAR FORMS OVER Q(
√ 5) 5 • Throughout we will have the notational issue that the matrix representing the U p operator will be in terms of a basis of monomials X i Y j as this is the natural basis of K X, Y . Moreover, our main technical results will be about understanding precisely the p-adic valuations of each entry of the U p operator matrix, which will depend on the monomials to which the entry corresponds. For this reason we will speak of the (i, j)-th entry of a matrix, with i = (i 1 , i 2 ), where i corresponds to the basis element X i1 Y j1 and similarly for j.
Corollary 3.3. In weight [n 1 , n 2 ]χ, a basis of K X, Y Γ t is given by X i1 Y i2 with n 2 − n 1 ≡ i 1 − i 2 mod 3.
Proof. Throughout the proof we are using the notation as in Proposition 2.7. The first part is immediate from Corollary 3.2 and the fact that U p := 4 s=1 | gs . We now need to prove the claimed simplification.
We begin by noting that since g 1 is diagonal, it will only contribute to U 0 p (i, i, n 1 , n 2 ) and it is easy to see from 2.7 and 3.2 that Ω n (g 1 , i, i) Finally, one checks that χ(d 1 ) = −1 and (2ζ +1) 2 = −3, which combines to give We now move on to the other three terms. Again, using Proposition 2.7, one checks easily that 4 s=2 Ω n (g s , i, j) reduces to which is the same as The result now follows by conjugating by the diagonal matrix whose (i, i)-entry is (bd 2 ) i1 bd 2 i2 .
We now choose a specific ordering for our basis of U 0 p acting in weight κ := [n 1 , n 2 ]χ. Obviously the choice will not alter the result, but some choices will make it easier to prove what the slopes are. Specifically, we want to order the basis in such a way that the block diagonal submatrix of U 0 p with 2 × 2-blocks has the same Newton polygon as U 0 p (as we will prove below). We begin by first ordering the basis elements using the graded lexicographic order. Let X a1 Y b1 be the first basis element under this ordering. We then set B 1 (κ) := {X a1 Y b1 , X a1+1 Y b1+1 } and for n ≥ 1, define where X an+1 Y bn+1 the next basis element not already contained in B n . Lastly, we set B ∞ (κ) = n B n (κ).
Notation 3.6. Let U be an infinite matrix with respect to the basis B ∞ (κ). Let D(U ) denote the block diagonal submatrix of U with blocks of size 2. Note that each matrix along the diagonal will correspond to a pair of basis elements, say, {X a Y b , X a+1 Y b+1 }. So we will denote these matrices by D a,b (U ).

Proposition 3.7.
(1) If a ≡ 0 mod 2 or b ≡ 0 mod 2 then the slopes of D a,b (U 0 p ) are a + b + 1 and a + b + 3.

Proof. Write
Then it follows from Proposition 3.5 that with u a,b a unit and n i = 2m i + 1. Similarly, we have Now a simple calculation shows that where e ≥ 0 is an error term. From this, the result follows at once.
Remark 3.8. Note that if a, b are not both even, then the slopes of D a,b are determined by the diagonal entries and for a ≡ b ≡ 0 mod 2 the slopes are determined by the anti-diagonal entries.
Let us now give an example which will motivate the proof of the main theorem.
Example 3.9. We begin by describing the basis in the case κ = [1, 1]χ (recall that this is our notation for the weight which would usually de called parallel weight 3 with nebentypus χ). From Corollary 3.3 we see that, for this weight, the basis for the space of overconvergent forms is given by B := {X i Y j : i ≡ j mod 3}. Now, with the grlex ordering the first basis element is 1, then B 1 (κ) = {1, XY }. From this we get where the first entry denotes the slope and the second its multiplicity. To explain this, we consider the matrix V p whose (i, j)-th entry is the p-adic valuation of the (i, j)-th entry of U 0 p . Using Proposition 3.5, we see that V p (10) is * 0 0 0 0 0 0 0 0 0 4 * 3 3 3 3 6 2 7 2 * * 4 6 * * 4 3 4 3 * * 9 6 * * 9 8 6 5 * * * * 4 6 4 3 * * * * * * 9 6 9 8 * * * * * * * * 8 4 * * * * * * * * 8 11 * * * * 10 9 * * 10 10 12 6 * * 13 12 * * 15 12 10 11 where here * denote an entry with infinite p-adic valuation, i.e., a zero entry of U 0 p . The above computation suggest that it is enough to understand the slopes of the matrices lying on the block diagonal. Looking at the above matrix one sees that the slopes are completely determined by the valuations of the entries. In fact, we will show that for U 0 p the valuations of the entries completely determine the slopes, which in general is not true.
This example motivates the following definition: Definition 3.10. Let M κ denote the set of infinite matrices with respect to the basis B ∞ (κ), such that if A ∈ M κ then the entries of D a,b (A) have valuations as given in the proof of Proposition 3.7 and if a i,j does not lie on D(A) then val p (a i,j ) ≥ i 1 + i 2 + g(i 1 , j 1 , n 1 ) + g(i 2 , j 2 , n 2 ) (as in Corollary 3.4).

Remark 3.11. Note that if
The following is the key result from which our main theorem will follow: Lemma 3.12. All matrices in M κ have the same Newton polygon.
We will defer the proof of this lemma to the next section and instead show how one can use it to deduce the main result.  Remark 3.14. We note that, in order for the space of Hilbert modular forms to be nontrivial we require χ(e)τ n (e) = −1 where e is the fundamental unit in O F . From this one checks that the only valid exponent of τ is n ≡ 0 mod 3 as τ (e) = ζ 2 (recall ζ is our fixed cube root on unity).
Theorem 3.16. Let n i ∈ Z with n i odd and let κ = [n 1 , n 2 ]χ. Then Proof. This follows from Corollary 3.13 combined with Proposition 3.7 and Table 1.
Note that twisting our weight by τ has the effect of changing the component of weight space that we are in. The above theorem then says that once we know a small set of classical slopes in each component of weight space, then we can obtain all slopes in any component of weight space.

Remark 3.17.
(1) The condition that n i is odd is required in order for the relevant space of overconvergent Hilbert modular forms to be non-trivial.
(2) Note that this is a simple generalization of [Birb,Conjecture 4.7.1 and Conjecture 4.7.8] to levels which are not sufficiently small (meaning we have Γ t /Γ t ∩ O × K is non-trivial). Having a level which is not sufficiently small has the effect of making the set over which we index in Theorem 3.16 more complicated.
(3) Computations suggest that in this case, the Newton and Hodge polygons associated to U p never touch (after the first vertex where they touch for trivial reasons) and therefore the methods of [LWX17] cannot be used to describe the slopes. Specifically, if one computes the Newton and Hodge polygons of U p (N ) for N >> 0 then the polygons only touch at the endpoints (which will always be the case by the definition of the Newton and Hodge polygon associated to a finite matrix).
4. Proof of Lemma 3.12 As mentioned above, the proof relies on the coincidence that all matrices in M κ the same slopes. The strategy of proof relies on showing that only the entries on the block diagonal submatrix D(U 0 p ) contribute to the p-adic valuation of a coefficient of the characteristic power series. To show this we will first write down an explicit formula for the coefficients of the characteristic power series in terms of the principal minors and then use our knowledge of valuations of the entries to determine when the valuation of the principal minor is minimized. (1) Let I denote the indexing set for the elements of B ∞ (κ). We will continue to denote these indexes by i.
(3) Let I n denote the subsets of size n of I and I n,R = {J ∈ I n | S(J) = R}.
(4) If A is an infinite matrix with basis given by B ∞ (κ) and J ∈ I n we let [A] J denote the principal minor defined by J. (5) Let r n = min J⊂In {S(J)}. (6) Let J n denote the set of J ∈ I n,rn such that if i ∈ J and (i 1 , i 2 ) ≡ (0, 0) mod 2 then there is j ∈ J such that |i 1 − j 1 | + |i 2 − j 2 | = 2. (7) For J ∈ J n we let σ J ∈ Sym(J) be such that if (i 1 , i 2 ) ≡ (0, 0) mod 2 then σ(i) = i and otherwise σ(i) = j where |i 1 − j 1 | + |i 2 − j 2 | = 2. In particular, σ J is chosen so as to pick out the entries lying on D(U 0 p ) with smallest valuation. Now, recall that we can compute the coefficients of the characteristic polynomial of U 0 p using the following result of Serre. Proof. This is [Ser62,Proposition7]. 2 )] J ) = val p (c ms (D(U 0 2 )). Lastly, for each n the minimal valuation of a n × n principal minor of D(U 0 2 ) lies on or above NP(D(U 0 2 )). Proof. We first note that for any n if we take J ∈ J n then val p ([D(U 0 2 )] J ) will be minimal and lie on or above NP(D(U 0 2 )). To prove that at a vertex we have a unique principal minor of minimal valuation, we note that by taking J to consists of the first m s basis vectors of B ∞ (κ), then J ∈ J ms this will give rise to a minor of minimal valuation. Furthermore, by the definition of m s is it easy to see that such a J is necessarily unique.
With this we can now prove the Lemma. Now, note that: (1) Since our basis is given by elements X i1 Y i2 with n 2 − n 1 ≡ i 1 − i 2 mod 3, it follows that, if a i,j is an entry of A, then |i 1 − j 1 | + |i 2 − j 2 | ≥ 2, with equality if and only if a i,j is an element of D(A) (this is due to how we have ordered our basis). Furthermore, by Corollary 3.4 it is easy to check that if |σ(i) 1 − i i | + |σ(i) 2 − i 2 | > 2 then val p (a ′ σ(i),i a ′ i,σ(i) ) > 2. In other words, if the entries are "far" from the block diagonal then their product has large valuation.
(3) If i ∈ J with (i 1 , i 2 ) ≡ (0, 0) mod 2 then val p (a ′ i,i ) = 1 and if (i 1 , i 2 ) ≡ (0, 0) mod 2 the val p (a ′ i,i ) ≥ 2. From the above it follows that the J ∈ I and σ ∈ Sym(J) for which i∈J val p (a σ(i),i ) is minimal are those with J ∈ J n and σ = σ J . This also shows that the minimal p-adic valuation the n×n principal minors of A and D(A) coincide. Now, note that there could be several such J with val p ([A] J ) being minimal, so we only know that (n, val p (c n (A))) lies on or above NP(D(A))(this follows from 4.3). To get the result we note that for n = m s we know that there is a unique J such that the minor [D(A)] J has minimal valuation and the above shows that the same is true for A, therefore (m s , val p (c ms (A))) is a vertex of NP(A).

Scaling of slopes
We now have the following computational evidence which shows that as our weights approach the boundary of weight space, the slopes scale.
Let χ = χ 2 denote our fixed finite character as in the previous section. Then for n ≥ 3, let χ n be a character of conductor p n , such that χ n (e) = −1 and χ(−1) = 1 for e the fundamental unit in O F . Similarly, for n ≥ 3 we let ψ n denote characters where ψ n (e) = 1 and ψ(−1) = 1.