Newforms mod p in squarefree level, with applications to Monsky's Hecke-stable filtration

We propose an algebraic definition of the space of l-new mod-p modular forms for Gamma0(Nl) in the case that l is prime to N, which naturally generalizes to a notion of newforms modulo p in squarefree level. We use this notion of newforms to interpret the Hecke algebras on the graded pieces of the space of mod-2 level-3 modular forms described by Paul Monsky. Along the way, we describe a renormalized version of the Atkin-Lehner involution: no longer an involution, it is an automorphism of the algebra of modular forms, even in characteristic p.

(1) Develop an algebraic theory of spaces of ℓ-new modular forms modulo p, consistent with the classical characteristic-zero definitions. (2) Introduce a modified Atkin-Lehner "involution" that descends to an finite-order algebra automorphism of the space of modular forms modulo p. The appendix, written by Alex Ghitza, justifies this modification geometrically by viewing modular forms modulo p as regular functions on the Igusa curve with poles only at supersingular points, and interpreting the Atkin-Lehner operator moduli-theoretically. (3) Construct a three-term Hecke-invariant filtration of the space of modular forms modulo p. On an old local component satisfying the level-raising condition at ℓ, the Hecke algebras on the graded pieces of the filtration may be identified with two copies of the ℓ-old Hecke algebra and one copy of the ℓ-new Hecke algebra. We compare this filtration and its Hecke algebras to those found by Monsky in the case ℓ ≡ −1 mod p.
We now discuss each goal in detail. Throughout this section N is an integer level, and ℓ is a prime dividing N exactly once. The ring B is a commutative Z[ 1 ℓ ]-algebra.
1.1. Spaces of ℓ-new forms in characteristic p. The theory of newforms in characteristic zero, developed by Atkin and Lehner [1], traditionally casts new eigenforms as eigenforms that are not old (i.e., do not come from lower level) and the space of newforms as a complement (under the Petersson inner product) to the space of old forms. Alternatively, one can define what it means to be a new eigenform -again, not old -and then the newforms are those expressible as linear combinations of new eigenforms. Viewed from both perspectives, newforms are classically identified by what they are not rather than what they are: in a sense, a quotient space rather than a subspace.
This "anti"-property of newforms creates problems as soon as we move into characteristic p. On one hand, there is no Petersson inner product, so no obvious way to find a complement of the old forms. On the other hand, in fixed level, there are infinitely many forms modulo p, but only finitely many eigenforms, so we cannot rely on eigenforms alone to characterize the newforms.
We propose two different algebraic notions of newness, both based on properties of presence rather than absence. The first is based on the Atkin-Lehner result that an eigenform of level N and weight k that is new at a prime ℓ exactly dividing the level has its U ℓ -eigenvalue equal to ±ℓ k−2 2 [1, Theorem 3]. The second is inspired by an observation of Serre from [23, §3.1(d)]: in the same setup, the ℓ-new forms of level N are exactly those forms f that satisfy both Tr ℓ f = 0 and Tr ℓ w ℓ f = 0. Here Tr ℓ is the trace map from forms of level N to forms of level N/ℓ (see section 4), and w ℓ is the Atkin-Lehner involution at ℓ (see section 3).
More precisely, we define two submodules of S k (N, B), the module of cuspforms of weight k and level N over B: let S k (N, B) U ℓ -new be the kernel of the Hecke operator U 2 ℓ − ℓ k−2 , and let S k (N, B) Tr ℓ -new be the intersection of the kernels of Tr ℓ and Tr ℓ w ℓ . Our first result is that these submodules coincide, and agree with the usual notion of ℓ-newforms for characteristic-zero B: We give similar results for S(N, B), the space of cuspforms of level N and all weights over B, viewed as q-expansions (see subsection 2.1 for definitions), if B is a domain. Theorem A allows us to define a robust notion of the module of ℓ-new forms in characteristic p, and hence a notion of a module of newforms in characteristic p for squarefree levels. In characteristic p the spaces of ℓ-new and ℓ-old forms need not be disjoint; the description of their intersection in section 7 matches the level-raising results of Ribet and Diamond [22,5], supporting our definitions.
1.2. Atkin-Lehner operators as algebra automorphisms on forms mod p. It is well known that Atkin-Lehner operator w ℓ (see section 3) is an involution on M k (N, Z[ 1 ℓ ]), the space of modular forms of level N and weight k over B = Z[ 1 ℓ ], and descends to an involution on M k (N, F p ) as well. Less popular is the (easy) fact that w ℓ is an algebra involution of M (N, Z[ 1 ℓ ]), the algebra of modular forms of level N and all weights at once (here viewed as q-expansions; see subsection 2.1 for definitions). Moreover, because of congruences between forms whose weights differ by an odd multiple of p − 1, the Atkin-Lehner operator w ℓ is not in general well-defined on M (N, F p ), essentially because of the factor of ℓ k 2 that appears in its definition. In section 3 we discuss this difficulty in detail, and propose a renormalization W ℓ of w ℓ that does descend to an algebra automorphism of M (N, F p ), with the property that W 2 ℓ acts on forms of weight k by multiplication by ℓ k .
In Appendix A, Alex Ghitza gives a geometric interpretation of the operator W ℓ on M (N, F p ), constructing it from an automorphism of the Igusa curve covering the modular curve X 0 (N ℓ) Fp .
1.3. Hecke-stable filtrations of generalized eigenspaces modulo p. In the last part of the paper, we focus on using the space of ℓ-new mod-p cuspforms to get information about the structure of the mod-p Hecke algebra of level N . We define a Hecke-stable filtration of K(N, F p ), the subspace of S(N, F p ) annihilated by the U p operator (see (8.2)): Here the t indicates that we've restricted to a generalized Hecke eigencomponent for the eigensystem carried by a pseudorepresentation t landing in a finite extension F of F p (see subsection 7.1 for definitions). If t is ℓ-old but satisfies the level-raising condition, then under certain regularity conditions on the Hecke algebra at level N/ℓ, we show that the Hecke algebra on the graded pieces of this filtration are exactly A(N, F) ℓ-new t , A(N/ℓ, F) t , A(N/ℓ, F) t , the shallow Hecke algebras acting faithfully on K(N, F) ℓ-new t , K(N/ℓ, F) t , and K(N/ℓ, F) t , respectively. See Proposition 8.1.
Finally, we compare this filtration to the filtration given in the case ℓ ≡ −1 mod p by Paul Monsky in [16,17] (see (8.4)): Here again t marks an ℓ-old component satisfying the level-raising condition. It is not difficult to see that the Hecke algebras on the first and third graded pieces are both A(N/ℓ, F) t . Under similar regularity conditions on A(N/ℓ, F) t , we show that the Hecke algebra on the middle graded piece is once again A(N, F) ℓ-new t . See Proposition 8.4.
Wayfinding: In section 2 we set the notation for the various spaces of modular forms that we consider. In section 3, we discuss problems with the Atkin-Lehner operator in characteristic p (when considering all weights at once) and introduce a modified version. In section 4 we discuss the trace-at-ℓ operator. In section 5 we discuss ℓ-old forms. In section 6 we discuss and propose a space of ℓ-new forms over rings that are not subrings of C. Intersections between spaces of ℓ-old and ℓ-new spaces, especially restricted to local components of the Hecke algebra (defined in subsection 7.1) are discussed in section 7. Finally in section 8, we discuss two Hecke-stable filtrations and compare the Hecke algebras on the corresponding graded pieces.
For any f ∈ B q and n ≥ 0, write a n (f ) for the coefficient of q n : that is, f = n≥0 a n (f )q n . If f ∈ M k (N, B) or M (N, B), then a n (f ) is the n th Fourier coefficient of f . For m ≥ 0, write U m for the formal B-linear operator B q → B q given by a n (U m f ) = a mn (f ).  (N, B) and r is a prime not dividing N (and again either k ≥ 2 or 1 r ∈ B), then the action of T r s is determined by the definition of T r on q-expansions 2) a n (T r f ) = a rn (f ) + r k−1 a n/r (f ), where we interpret a n/r (f ) to be zero if r ∤ n, and the recurrence for all s ≥ 0. On the other hand, if m divides N , then the action of T m on f ∈ M k (N, B) is given by a n (T m f ) = a mn (f ), so that T m coincides with the formal U m operator defined earlier.
Finally, if the characteristic of B is c > 0, and m divides c, then the action of T m on M k (N, Z) coincides with the action of U m so long as k ≥ 2. We always work with and write U m instead of T m for m dividing N or the (positive) characteristic of B.
All of these classical Hecke operators commute with each other. Moreover, if B is a domain, then all of them extend to the algebra of modular forms M (N, B). Indeed, this is immediate if B has characteristic zero (as M (N, B) is the direct sum of the M k (N, B)). If B has characteristic p and r is a prime not dividing N p, then T r is well-defined on M (N, B) from the q-expansion formula (2.2) because M (N, B) is a direct sum of weight-modulo-(p − 1) spaces (2.1) and r k−1 is well-defined in characteristic p for k modulo p − 1. The action of T m on M (N, B) for prime power m relatively prime to N p follows from the recurrence (2.3). The action of U m for m dividing N p is independent of the weight and hence always well defined.
We can streamline these arguments by introducing a weight-separating operator. If B is a domain and m is invertible in B, we define the operator S m : Note that S m extends to an algebra automorphism of M (N, B). If every m prime to N and the (positive) characteristic of B is invertible in B (for example, if B is a Q-algebra or a finite extension of F p ), then the action of all the T m is generated by the action of the T r and S r for primes r not dividing N or the (positive) characteristic of B.

The Atkin-Lehner involution at ℓ
We now fix an additional prime ℓ not dividing N . From now on, we assume that B is a Z[ 1 ℓ ]-domain. Our eventual goal is to meaningfully compare the Hecke action on the algebras M (N ℓ, B) and M (N, B). In this section, we discuss how to extend the Atkin-Lehner involution on M k (N ℓ, B) to an algebra involution on M (N ℓ, B).
3.1. The Atkin-Lehner involution at ℓ in weight k. For k ∈ 2Z ≥0 , we recall the definition and properties of the Atkin-Lehner involution on M k (N ℓ, B) as in [1].
Let H be the complex upper half plane. We extend the weight-k right action of SL 2 (Z) on Here, for γ = a b c d ∈ GL 2 (Q) + , we write γz for az+b cz+d (this is the usual conformal action of GL 2 (Q) + on H + = H ∪ P 1 (Q) leaving P 1 (Q) invariant); and j(γ, z) := cz + d is the usual automorphy factor. The normalization of (det γ) k 2 is chosen so that the scalars GL 2 (Q) + act trivially. We use a different normalization here so that Sm extends to an algebra automorphism on M (N, B). We will eventually work with S ℓ for ℓ is a prime exactly dividing the level.
Let γ ℓ ∈ GL 2 (Q) + be any matrix of the form ℓ a N ℓ ℓb , where a and b are integers such that ℓb − aN = 1, which can be found as we've assumed that ℓ ∤ N . Let w ℓ be the operator on functions f : H → C sending f to f | k γ ℓ . One can check that (1) the matrix γ ℓ normalizes Γ 0 (N ℓ), so that w ℓ maps M k (N ℓ, C) to M k (N ℓ, C); (2) any two choices of γ ℓ differ by an element of Γ 0 (N ℓ), so that the action of w ℓ on M k (N ℓ, C) is defined without ambiguity; , and therefore w ℓ is an involution, called the Atkin- 3.2. Atkin-Lehner as an algebra involution in characteristic zero. If B has characteristic zero, then it is clear from the definitions above and the direct sum property of M (N ℓ, B) that w ℓ extends to an algebra involution on M (N ℓ, B). However, if B has characteristic p and ℓ is not a square modulo p, then we incur a sign ambiguity, essentially because of the factor of ℓ k 2 coming from the determinant term in (3.1).
In the next section, we discuss the extent to which the Atkin-Lehner involutions on M k (N ℓ, B) patch together to an algebra involution on M (N ℓ, B) when B has characteristic p.

3.3.
Atkin-Lehner as an algebra involution in characteristic p: difficulties. In this section we work with B = F p and finite extensions. We also assume the theory of oldforms and newforms in characteristic zero [1], which will be reviewed in section 5 and section 6 below. From item (6) above, we know that if f and f ′ are characteristic-zero modular forms of the same weight and level N ℓ that are congruent modulo p, then w ℓ f and w ℓ f ′ are congruent modulo p as well. Indeed, this is what it means for w ℓ to descend to an involution on M k (N ℓ, B). However, if f and f ′ appear in weights that differ by an odd multiple of p − 1, then w ℓ f will be congruent to w ℓ f ′ up to a factor of ℓ p only. (1) Newform example: Let p be an odd prime. If f ∈ M k (N ℓ, Z p ) is a new eigenform, then f is an eigenform for w ℓ as well, so that So ε(f ′ ) will not be congruent to ε(f ) modulo p unless ℓ k−k ′ 2 ≡ 1 (mod p). In particular, if p is odd and k − k ′ is an odd multiple of p − 1, then ε(f ) ≡ ε(f ′ ) mod p if and only if ℓ is a square modulo p.
For example, write S k (ℓ, Q) new,± for the new subspace on which w ℓ acts by ±1. For ℓ = 3 the spaces S 12 (3, Q) + and S 16 (3, Q) − are one-dimensional, spanned by Then f + 12 and f − 16 are congruent mod 5, but w 3 f + 12 = f + 12 and w 3 f − 16 = −f − 16 are not. (2) Oldform example: Let f ∈ M k (N, Z p ) be any form, not necessarily eigen. Then is congruent to f . Then we similarly see that w ℓ f ≡ w ℓ f ′ mod p if and only if either ℓ is a square modulo p or k − k ′ is a multiple of 2(p − 1). Indeed, for any p ≥ 5, compare f = E p−1 ∈ M p−1 (1, Z p ) and the constant form and w ℓ (1) = 1; these are congruent modulo p exactly when ℓ p = 1.

3.3.2.
Sometimes we get an algebra involution compatible with reduction. In light of these examples, it is not true in general that w ℓ descends to an algebra involution of M (N ℓ, F). However it does work in certain cases: (1) If ℓ is a square modulo p, then there is no sign ambiguity, and w ℓ is an algebra involution of M (N ℓ, F p ). This is easy to show by moving around different weights by multiplying by E p−1 and using the fact that w ℓ (E p−1 ) = ℓ p E p−1 (q ℓ ). (Use E 4 and E 6 in place of E p−1 if p = 2 or 3.) In particular, p = 2 never poses a problem.
(2) Restricting to M (N ℓ, F p ) 0 and p ≥ 3, we can define w ℓ as an algebra involution compatible with reduction of some lift. Namely, f ∈ M (N ℓ, F p ) 0 is the reduction of somef ∈ M k (N ℓ, Z p ) with k divisible by 2(p − 1); define w ℓ f as the reduction of w ℓf . Since any two suchf s differ (multiplicatively) by a power of E 2 p−1 , this construction is independent of the choice off .
For p ≥ 5, this construction is equivalent to the following geometric definition (see [24,Corollaire 2] for level one). By dividing f ∈ M (p−1)k (N ℓ, F p ) by E k p−1 , we can identify M (N ℓ, F p ) 0 with the algebra of regular functions on the affine curve obtained by removing the supersingular points from X 0 (N ℓ) Fp . The geometric Atkin-Lehner involution on X 0 (N ℓ) Fp preserves the supersingular locus and hence induces an algebra involution on M (N ℓ, F p ) 0 .

3.3.3.
Sometimes no algebra involution compatible with reduction is possible. However, it is not always possible to see w ℓ as an algebra involution on M (N ℓ, F p ) compatible with reduction.
On one hand, we're assuming that W is an algebra involution, so that . On the other hand, g ∈ M (ℓ, F p ) 0 and so that W g = w ℓ g = g(q ℓ ) by the recipe in subsubsection 3.3.2 (2) above. Therefore ε Question 1. For p ≡ 3 mod 4 (if ℓ is not a square modulo p) the argument above fundamentally fails: one can indeed "extend" the definition of w ℓ on M (N ℓ, F p ) 0 as in subsubsection 3.3.2 (2) to an algebra involution W on the ℓ-old forms in M (N ℓ, F p ) in a reduction-compatible manner by setting W f = (−ℓ) k/2 f (q ℓ ) for f ∈ M k (N, F p ), well-defined as −ℓ is now a square modulo p. But can this be extended in an algebra-involution way to all of M (N ℓ, F p ) compatible with reductions? And can one show that any algebra involution on M (N ℓ, F p ) compatible with some reduction of w ℓ restricts to the construction from subsubsection 3.3.2 (2) on M (N ℓ, F p ) 0 ?
3.4. Modified Atkin-Lehner as an algebra automorphism in characteristic p. To fix this difficulty, we will renormalize w ℓ to be compatible with algebra structures.
For any m ∈ Z, possibly depending on k, the weight-k right action of SL 2 (Z) on functions f : H → C can be extended to GL 2 (Q) + via the formula, for z ∈ H, Scalar matrices ( a 0 0 a ) then act via multiplication by a 2m−k . The usual choice in the definition of the Atkin-Lehner operator is m = k 2 (scalars act trivially; see, for example, [1, p.135]); another possibility that appears in the literature is m = k − 1 (used to define Hecke operators; see, for example, [7, Exercise 1.2.11]). For our renormalized Atkin-Lehner operator, we adopt m = k, so that scalars act through their k th power.
We define a new map Here γ ℓ is again a matrix of the form ℓ a N ℓ ℓb , where a and b are integers such that ℓb − aN = 1, as in subsection 3.1. Since W ℓ = ℓ k 2 w ℓ , it is clear that this map is well-defined independent of the choice of γ ℓ . Moreover, W ℓ satisfies the following properties.
(2) W ℓ extends to an algebra automorphism of M (N ℓ, B) for any characteristic-zero Z[ 1 ℓ ]domain B. This algebra automorphism preserves the ideal S(N ℓ, B).
(3) W ℓ descends to an algebra automorphism for any characteristic-p domain B. This algebra automorphism restricts to the involution on M (N ℓ, F p ) 0 defined in subsubsection 3.3.2 (2). For p ≥ 3, the order of W ℓ divides p − 1; for p = 2, W ℓ coincides with w ℓ and hence has order 2.
Only the last item requires justification. It relies on the following: Proof. It suffices to consider f, g appearing in single weights, so let these be k(f ), k(g), respectively. Since w ℓ already has this property for k(f ) = k(g), so does W ℓ . It therefore suffices prove the case k(f ) < k(g). By a theorem of Serre (see equation (2.1)) k(g) − k(f ) = n(p − 1) for some n ∈ Z + . But then E n p−1 f and g are congruent in the same weight, so Appendix A shows that the renormalized Atkin-Lehner operator W ℓ in characteristic p is induced geometrically on modular forms by an automorphism of the Igusa curve.

For any characteristic-zero
given, for B = C, by (1) For f ∈ S(N ℓ, B), we have Tr ℓ f ∈ S(N, B). ( The shape of these equations suggest that it might be more natural to renormalize T ℓ and U ℓ by scaling them by ℓ, so that the Hecke operators are true "trace" rather than a scaled trace and stay integral even in weight 0. In fact, this renormalization would amount to using the | k,k -action discussed in subsection 3.4 to define the Hecke operators, which we are already using to define W ℓ . But we will not do so here. But this means that f has to be a constant! Indeed, suppose n > 0 is the least integer such that a n (f ) = 0. Since the right-hand side is in B q ℓ 2 , we must have n = mℓ 2 for some m < n.
But the q n -coefficient on the right-hand side is ℓ k a m (f ), which must be zero as n was the least index of a nonzero coefficient of f .
Alternatively, we can deduce Proposition 5.1 in characteristic zero from [1, Theorem 1] and in characteristic p from the following more recent theorem of Ono-Ramsey. . Let p be a prime, and f a form in M k (N, Z) withf = a n q n ∈ M k (N, F p ) its mod-p image. Suppose that there exists an m prime to N p and a power series g ∈ F p q so thatf = g(q m ). Thenf = a 0 .
Proof. Let f, g ∈ M (N, B) be forms so that f = W ℓ (g) ∈ B q ℓ . In light of Proposition 5.1, it suffices to show that we may assume that both f and g appear in a fixed weight k. As a Z[ 1 ℓ ]-domain, B is flat over either Z[ 1 ℓ ] or over F p for some p prime to ℓN . In either case, from subsection 2.1, we know that we can express both f and g as finite sums of forms f = f i and 5.2. ℓ-Old forms. Following Atkin-Lehner [1] and others, define the ℓ-old forms in M k (N ℓ, Q) as the span of M k (N, Q) and W ℓ M k (N, Q): Note that M k (N ℓ, B) ℓ-old may a priori be bigger than M k (N, B) + W ℓ M k (N, B). For example, if E k is the normalized (i.e., with a 1 = 1) weight-k level-one Eisenstein series and B = Z p , then , since E p−1 is has p in the denominator of its constant term. (ii) For our purposes, the following will suffice: Proof. Since we are in a single weight, it suffices to consider  6. The space of ℓ-new forms 6.1. ℓ-New forms in characteristic zero.

Analytic notion.
For B = C one can follow Atkin-Lehner's characterization of newforms to define the space S k (N ℓ, C) ℓ-new of cuspidal ℓ-new forms of level N ℓ and weight k as the orthogonal complement to the space of ℓ-old forms under the Petersson inner product [1, p. 145]. Alternatively, the space of ℓ-new cuspforms is the C-span of the ℓ-new eigenforms: those eigenforms that are not in S k (N ℓ, C) ℓ-old [1, Lemma 18]. This latter definition can be extended to Eisenstein forms as well, to obtain well-defined spaces M k (N ℓ, C) ℓ-new and S k (N ℓ, C) ℓ-new , which we here identify with their q-expansions.
and, for any characteristic-zero domain B, as usual. In characteristic zero, of course, this sum is direct.
Proof of Proposition 6.1. It suffices to prove that the kernel of D ℓ | M k (N ℓ,B) is M k (N ℓ, B) ℓ-new in a single weight k. Moreover, since B is flat over Z it suffices to prove the statement for B = Z; and since M k (N ℓ, C) ℓ-new has a basis over Z, it suffices to take B = C.
The module M k (N ℓ, C) is a direct sum of C-spans of eigenforms ℓ-new and ℓ-old. Since D ℓ preserves away-from-ℓ Hecke eigenspaces, it suffices to see that D ℓ annihilates all ℓ-new eigenforms and never annihilates ℓ-old eigenforms. If f ∈ M k (N ℓ, C) is Eisenstein, then it must be old at ℓ, the ℓ-stabilization of a form g ∈ M k (N, C) with a ℓ (g) = χ(ℓ)ℓ k−1 + χ(ℓ) −1 for some Dirichlet character χ of modulus M with M 2 | N (see, for example, [7,Theorem 4.5.2]). Hence the absolute value of the U ℓ -eigenvalue of f is either ℓ k−1 or 1. If f ∈ M k (N ℓ, C) is a cuspidal ℓ-new form, then by [1, Theorem 5], its U ℓ -eigenvalue is ±ℓ (N ℓ, C) is a cuspidal ℓ-old eigenform, then f is the ℓ-stabilization of some normalized eigenform g ∈ M k (N, C), and the U ℓ -eigenvalue of f is a root of the polynomial P ℓ,g (X) = X 2 − a ℓ (g)X + ℓ k−1 . If one root of P ℓ,g is ±ℓ k 2 −1 , then the other root must be ±ℓ k 2 , so that a ℓ (g) = ±(ℓ + 1)ℓ k−2 2 , which impossible by Lemma 6.2.  Proof. Since B is flat over Z[ 1 ℓ ], we may replace Z by Z[ 1 ℓ ] in the beginning of the proof of Proposition 6.1 to see that it suffices to establish this in a single weight k for B = C. Since both Tr ℓ and W ℓ commute with Hecke operators prime to ℓ, it suffices to consider separately the one-dimensional eigenspaces spanned by ℓ-new eigenforms and the two-dimensional ℓ-old eigenspaces coming from eigenforms of level N . If f ∈ M k (N ℓ, C) is ℓ-new eigen, then both Tr ℓ f and Tr ℓ W ℓ f are forms of level N with the same eigenvalues away from ℓ as f , which is impossible by [1, Lemma 23]. Therefore both Tr ℓ f = 0 and Tr ℓ W ℓ f = 0, so that ker Tr ℓ ∩ ker Tr ℓ W ℓ does indeed contain M (N ℓ, B) ℓ-new . For the reverse containment, if f is in M k (N ℓ, C) ℓ-old , then it suffices to consider to f contained in the two-dimensional span of g and W ℓ (g) for some eigenform g ∈ M k (N, C). From the identities in section 4, the operators Tr ℓ and Tr ℓ W ℓ , on the ordered basis {g, W ℓ (g)} of the ℓ-old subspace of M k (N ℓ, C) associated to g, have matrix form The kernels of matrices of the form a b 0 0 and c d 0 0 have a nontrivial intersection if and only if ad = bc. In our case that would mean that a ℓ (g) 2 = (ℓ + 1) 2 ℓ k−2 , which is again impossible by the Weil bounds (Lemma 6.2).
6.2. Newforms over any domain: a proposal. Inspired by the algebraic characterisations of Proposition 6.1 and Proposition 6.3 of newforms in characteristic zero, we make the following two definitions.  To prove Theorem 1, we first establish S(N ℓ, B) ℓ-old U ℓ -new = S(N ℓ, B) ℓ-old Tr ℓ -new : (N, B) for some weight k. Then the following are equivalent. ( (1) Proposition 6.4 may be rewritten more symmetrically in terms of w ℓ , the involutionnormalized Atkin-Lehner operator on S k (N ℓ, B). Namely, let λ k = −(ℓ + 1)ℓ k−2 2 . Then the claim of the proposition is that The constant λ k appears in connection with level-raising theorems of Ribet [22] and Diamond [5]. See also subsection 7.2 for more details. Proof of Proposition 6.4. We use the identities from section 4 repeatedly, including the fact that for f ∈ M (N, B), (N, B). On one hand we have From Proposition 5.1, the intersection of S k (N, B) and holds if and only if The second equation reduces to Inserting this into the first equation, combining like terms, and eliminating S ℓ reveals (6.2) ℓ T ℓ g = −(ℓ + 1)f, as required.
For (2) ⇐⇒ (3), we recall that for f ∈ M k (N, B), Proof of Theorem 1. If B has characteristic zero, then this statement is already known (Proposition 6.1 & Proposition 6.3), but we prove it again without using the Weil bound. As in the proof of Proposition 6.1, we may assume that we are in a single weight k and that B = C, and note that each one-dimensional ℓ-new eigenspace is annihilated by all three operators D ℓ , Tr ℓ , and Tr ℓ W ℓ . Now Proposition 6.4 establishes the desired statement for each two-dimensional ℓ-old away-from-ℓ Hecke eigenspace and completes the proof.
If B has characteristic p, then we may assume that B = F p and again as in the proof of Corollary 5.3 work in a single weight k. We will have to distinguish between coefficients in Z p and quotients, so for any ring B, write X B for the operator X acting on S k (N ℓ, B).
Take f ∈ S k (N ℓ, F p ). Then there exist integral formsf ℓ-new andf ℓ-old in S k (N ℓ, Z p ) ℓ-new and S k (N ℓ, Z p ) ℓ-old , respectively, and a b ∈ Z ≥0 so that f is the mod-p reduction of In other words, the form f old is in ker D ℓ Z/p b+1 Z , where f old ∈ S k (N ℓ, Z/p b+1 Z) is the image off old under the reduction-mod-p b+1 map. By Proposition 6.4, f old is in ker(Tr ℓ ) Z/p b+1 Z ∩ ker(Tr ℓ W ℓ ) Z/p b+1 Z . By lifting back up to characteristic zero, we see that both Tr ℓ Zp (f old ) and (Tr ℓ W ℓ ) Zp (f old ) are in p b+1 Z p q .
In light of Theorem 1, we introduce the following definition: We will also use the notation M (N ℓ, B) ℓ-new := M (N ℓ, B) Tr ℓ -new . Observe that the space of ℓ-new forms is stable under W ℓ .

Interactions between ℓ-old and ℓ-new spaces mod p
In characteristic zero, spaces of ℓ-new and ℓ-old forms are disjoint. This fails in characteristic p because of congruences between ℓ-new and ℓ-old forms. A related phenomenon: over a field of characteristic zero, ℓ-new and ℓ-old forms together span the space of forms of level N ℓ. This already fails over a ring like Z p , again because of congruences between ℓ-new and ℓ-old forms. A guiding scenario: if f ∈ S k (N ℓ, Z p ) ℓ-new is nonzero modulo p but congruent to g ∈ S k (N ℓ, Z p ) ℓ-old modulo p but not modulo p 2 , then 1 Example 1. Take N = 5, ℓ = 3, p = 7, k = 4. There is only one cuspform at level N , namely, f = q − 4q 2 + 2q 3 + 8q 4 − 5q 5 − 8q 6 + 6q 7 − 23q 9 + O(q 10 ) ∈ S 4 (5, Z 7 ). In level N ℓ, there are two newforms, forming a basis of S 4 (15, Z 7 ) (but not over Z, as they are congruent modulo 2): In this section, we describe the intersection of the ℓ-old and the ℓ-new subspaces modulo p and comment on the failure of these to span the whole level-N ℓ space. We will fix a prime p and work with B = F p or a finite extension, suppressing B from notation. We start with the following corollary to Proposition 6.4 and the first remark following: (2) If p = 2, then in fixed weight k with λ k = −(ℓ + 1)ℓ To offer a more detailed analysis, we will pass to generalized Hecke eigenspaces. In subsection 7.1 we recall definitions and notations for mod-p big Hecke algebras. And in subsection 7.2 we state our conclusions on the intersection of ℓ-old and ℓ-new subspaces in characteristic p.  N )) for this contruction. This is the big (iii) Indeed, the level-raising condition for f at 3 modulo 7 is satisfied, so that the existence of such a congruence is guaranteed by Diamond [5]. See also subsection 7.2.
shallow Hecke algebra acting on the space of modular forms of level N modulo p, the only kind of Hecke algebra we study here. (iv) One can show that A(N ) is a complete noetherian semilocal ring that factors into a product of its localizations at its maximal ideals, which by Deligne and Serre reciprocity (formerly Serre's conjecture) correspond to Galois orbits of odd dimension-2 Chenevier pseudorepresentations (t, d) : Here ω p is the mod-p cyclotomic character, and G Q,N p is the Galois group Gal(Q N p /Q), where Q N p is the maximal extension of Q unramified outside the support of N p∞. Since the d in each pseudorepresentation is entirely determined by t in this Γ 0 (N ) setting (indeed, if p > 2 we have d(g) = t(g) 2 −t(g 2 ) 2 for any g ∈ G Q,N p ; and if p = 2 then d = 1), we will frequently suppress it from notation. For more on Chenevier pseudorepresentations see [3] or [2, 1.4]. If we assume that F is large enough to contain all the finitely many Hecke eigenvalue systems appearing in M (N ), then the Galois orbits become trivial; from now on we assume that this is done.
Let K(N ) ⊂ M (N ) be the kernel of the U p operator. Since U p in characteristic p is a left inverse of the raising to the p th power operator V p , given any form f ∈ M (N ℓ) the form g = (1−V p U p )f has the property that a n (g) = a n (f ) unless p | n, in which case a n (g) = 0. Therefore K(N ) is a nontrivial subspace of M (N ). Further, since U p preserves the grading from (2.1), we can set K(N ) k := K(N ) ∩ M (N ) k for k ∈ Z/(p − 1)Z and then K(N ) = k K(N ) k . One can show that A(N ) acts faithfully on K(N ), so that A(N ) is also Hecke(K(N )). Studying this smaller space eliminates minor complications caused by the behavior of our Hecke eigensystems at p. Since the operators Tr ℓ , W ℓ , D ℓ used to define the ℓ-old and ℓ-new subspaces of M (N ℓ), commute with Hecke operators away from ℓ, the spaces M (N ℓ) ℓ-new and M (N ℓ) ℓ-old also decompose into (iv) One can also consider the big partially full Hecke algebra A(N ) pf , topologically generated inside End F M (N ) by the action of Tn for all (n, N p) = 1 as well as U ℓ for ℓ | N , and the big full Hecke algebra A(N ) full , which also includes the action of Up. Many authors also consider the "smaller" algebras A k (N ), A k (N ) pf , A k (N ) full acting on forms in a single weight. generalized eigenspaces for the various t ∈ PS(N ℓ). For a Hecke module C ⊂ M (N ℓ), write C t := C ∩ M (N ℓ) t , so that we define S(N ℓ) t , S(N ℓ) ℓ-old t and S(N ℓ) ℓ-new t . Theorem 2. Fix κ ∈ 2Z/(p − 1)Z (or κ = 0 if p = 2) and t ∈ PS κ (N ℓ). For k even with k ≡ κ mod (p − 1), let λ k be the image of −(ℓ + 1)ℓ k− 2 2 in F p . Note that the set {±λ k } depends only on κ.
In part (2(b)ii), note that ε k w ℓ depends only on κ, not on k (in other words, ε k w ℓ is well defined on S(N ℓ) t ). It also straightforward to see that ε k w ℓ = (ε k λ k )ℓ(ℓ + 1) −1 S −1 ℓ W ℓ . The statements of Theorem 2 dovetail nicely with the level-raising results [22,5]: if f is an integral eigenform of level N and weight k whose mod-p representation is absolutely irreducible, then there is another eigenform of level N ℓ congruent modulo p to f (away from N ℓp) if and only if a ℓ (f ) 2 ≡ λ 2 k modulo p. For a level-N pseudorepresentation t mod p, we will say that the level-raising condition is satisfied for (t, ℓ) if t(Frob ℓ ) = ±λ k .
Proof of Theorem 2. If t does not factor through G Q,N p , then there are no ℓ-old eigenforms and every form is ℓ-new: this will be true mod p because it is true overZ p . So assume t ∈ PS κ (N ), carried by some eigenform f ∈ S(N ). If M (N ℓ) ℓ-new t = ker D ℓ | M (N ℓ)t is nonzero, then it contains an eigenform g, cuspidal after twisting by θ p−1 if necessary, which by assumption is also an eigenform for U ℓ with eigenvalue ±ℓ k−2 2 . Since g is ℓ-old (more precisely, since g can be lifted to an ℓ-old eigenform in characteristic zero by the Deligne-Serre lifting lemma), it is the ℓ-refinement of some eigenform f ∈ M k (N ) for some weight k, and its U ℓ -eigenvalue is a root of For (2b): if λ k = 0, then remark (1) after Proposition 6.4 restricted to S(N ℓ) t gives us if and only if f and g are in S(N ) t and killed by T ℓ . If λ k is nonzero (so p = 2), then only one of ±λ k , namely ε k λ k , appears as a T ℓ -eigenvalue in S(N ) t . In particular, from the formulation in Corollary 7.1, we see that if and only if f is in the kernel of T ℓ − ε k λ k and g = ε k f . But any f and g in S(N ) t appear together in some weight k. For B = F p and extensions, we no longer expect a direct sum in general, but we may still ask whether ℓ-old and ℓ-new forms together span all cuspforms. To illuminate the behavior most effectively, we restrict to a generalized eigenspace for some t ∈ PS(N ℓ).
To this end, fix t, let F be an extension of F p containing its values, and let O := W (F), the unique unramified extension of Z p with residue field F. We have defined S(N ℓ, F) t as the set of generalized eigenforms in S(N ℓ, F) for the (shallow) Hecke eigensystem carried by t. We define S(N ℓ, O) t as the subspace of S (N ℓ, O) consisting of linear combinations of eigenforms whose corresponding shallow Hecke eigensystem is a lift of t. Unlike in characteristic p, it will no longer be true that every eigensystem is defined over O, but if F is large enough to contain the values of all the elements of PS(N ℓ), then it is still true that S (N ℓ, O) (4) Either t is new at ℓ, or (t, ℓ) does not satisfy the level-raising condition.
We demonstrate (1) ⇐⇒ (2): Since S(N ℓ, O) ℓ-old t breaks up into a graded sum of its fixedweight pieces, and since D ℓ is weight-preserving, surjectivity on S(N ℓ, O) ℓ-old t is equivalent to surjectivity on S k (N ℓ, O) ℓ-old t . By right-exactness of tensoring or Nakayama's lemma (depending on the direction) this last is equivalent to surjectivity on S k (N ℓ, F) ℓ-old t . This space is a finitedimensional vector space, so D ℓ acts surjectively if and only if it has trivial kernel, which is equivalent by definition to S k (N ℓ, F) ℓ-old Finally, if t is absolutely irreducible, then the level-raising theorems [22,5] hold. Therefore if t ∈ PS(N ) and (t, ℓ) satisfies the level-raising condition, then there exists an ℓ-new form congruent to an ℓ-old form (over some extension of O), which implies that (v) That is, t is not the sum of two characters G Q,Nℓp →Fp. Question 2. Is it always true that S(N ℓ, F p ) ℓ-new t + S(N ℓ, F p ) ℓ-old t = S(N ℓ, F p ) t ? A positive answer would furnish additional support for the present definition of ℓ-new forms.

Hecke-stable filtrations mod p
In this section we describe a filtration for the space of modular forms of level N ℓ modulo p, and compare it to the filtration described by Monsky in [16,17], which appears if ℓ ≡ −1 modulo p. We assume that B = F, a finite extension of F p big enough to contain all mod-p eigensystems, throughout, and suppress B from notation.
8.1. The standard filtration (after Paul Monsky). For simplicity, we will restrict to the kernel of the U p operator K(N ℓ) ⊂ M (N ℓ), where formulas are simpler but no Hecke eigensystem information is lost. See also subsection 7.1 and subsection 7.2 for additional notation. Then K(N ℓ) contains two subspaces Here the action of all operators is restricted to K(N ℓ), so that ker D ℓ = ker D ℓ | K(N ℓ) , etc.
The Hecke algebra A(N ℓ) = Hecke K(N ℓ) has quotients A(N ℓ) ℓ-new := Hecke K(N ℓ) ℓ-new and To study the Hecke structure on K(N ℓ) more closely, we consider the following filtration by Hecke-invariant submodules, which we'll call the standard filtration: For any t ∈ PS(N ℓ), we can pass to the sequence on the t-eigenspace: We also consider the following two conditions relative to a pseudorepresentation t ∈ PS(N ) and a Hecke operator T ∈ A(N ) t .
Condition Surj (t, T ): Operator T ∈ A(N ) t acts surjectively on K(N ) t . Condition NZDiv(t, T ): Element 0 = T ∈ A(N ) t is not a zero divisor on K(N ) t .
Note that Surj (t, T ) implies NZDiv (t, T ): suppose T K(N ) t = K(N ) t , and suppose there exists T ′ ∈ A(N ) t with T ′ T = 0. Then T ′ annihilates K(N ) t ; since the action of A(N ) t is faithful, we must have T ′ = 0. Both conditions are satisfied if A(N ) t is a regular local F-algebra of dimension 2. (vi) See section subsection 8.3 below for more details.
We are now ready to analyze the standard filtration (8.2).
(vi) It's not unreasonable to expect that this is always the case for N = 1. No counterexamples are known; for reducible t ∈ PS(1), Vandiver's conjecture implies that A(1)t is a regular local ring of dimension 2: see [2, §10].
(1) If EITHER ℓ ≡ −1 modulo p, OR ℓ ≡ −1 modulo p and Surj (t, T ℓ ) holds , then (2) If EITHER ℓ ≡ −1 mod p and Surj (t, T 2 ℓ − λ 2 k ) holds OR ℓ ≡ −1 mod p and Surj (t, T ℓ ) holds , then In other words, under regularity conditions on A(N ) t , the Hecke algebras acting on the graded pieces of the standard filtration are one copy of A(N ℓ) ℓ-new t and two copies of A(N ℓ) ℓ-old t . Note that K(N ℓ) ℓ-new t and A(N ℓ) ℓ-new t will be zero if the level-raising condition for (t, ℓ) is not satisfied.
Proof. For part (1), we show that under the given conditions, the sequence is exact. On the left, exactness is by definition. On the right, if ℓ = −1 modulo p then for any f ∈ K(N ) we have Tr ℓ (f ) = (ℓ + 1)f , which spans f F . Otherwise, Tr ℓ W ℓ f = ℓT ℓ (f ), so condition Surj (t, T ℓ ) suffices.
For part (2), we establish the exactness of Again, left exactness holds since K(N ℓ) ℓ-new = ker Tr ℓ ∩ ker Tr ℓ W ℓ . For right exactness, if ℓ ≡ −1 mod p, then K(N ) t ⊂ ker Tr ℓ , and then Tr ℓ W ℓ f = ℓT ℓ f for any f ∈ K(N ) t . Otherwise use the computations of Proposition 6.4 to see that g = T ℓ f − (ℓ + 1)/ℓW ℓ f is in ker Tr ℓ , and then Tr ℓ W ℓ (ℓ −1 g) = (T 2 ℓ − λ 2 k )f . Proof. From the proof of Proposition 8.1, we see that K(N ℓ) t /(ker Tr ℓ ) t is isomorphic to a Hecke module that sits between T ℓ K(N ) t and K(N ) t . If T ℓ is not a zero divisor on K(N ) t , then A(N ) t acts faithfully on T ℓ K(N ) t : indeed, if any T ∈ A(N ) t annihilates T ℓ K(N ) t , then T T ℓ annihilates K(N ) t . Therefore the Hecke algebra on T ℓ K(N ) t , and hence on K(N ℓ) t /(ker Tr ℓ ) t , is still A(N ) t . The reasoning for the Hecke algebra on (ker Tr ℓ ) t /K(N ℓ) ℓ-new 8.2. Connection to the Monsky filtration. In [16] and [17], Monsky studies K(N ℓ) and related Hecke algebras in the case p = 2, N = 1 and ℓ = 3, 5. For p = 2, there is only one t ∈ PS(1), namely t = 0, the trace of the trivial representation. Monsky describes a different filtration of K(ℓ) = K(ℓ) 0 by Hecke-invariant subspaces, and proves that the Hecke algebras on the graded pieces are two copies of A(1) plus a third "new" Hecke algebra. The goal of this section is to compare the Monsky filtration to the standard filtration from subsection 8.1, and to establish that the "new" Monsky Hecke algebra coincides with A(ℓ) new defined here. The Monsky filtration exists more generally, so long as the level ℓ is congruent to −1 modulo p. As in the previous section, we will assume regularity conditions on t (namely, Surj (t, T ℓ )), guaranteed in Monsky's p = 2 case by work of Nicolas and Serre [19] (via Lemma 8.5).
Fix a t ∈ PS(N ), and let F/F p be an extension containing the image of t. Fix a prime ℓ congruent to −1 modulo p. Then we have the following filtration of K(N ℓ) t by Hecke-invariant subspaces, due to Monsky [16, remark p. 5] (vii) : Indeed, if ℓ + 1 = 0 in F p , then Tr ℓ K(N ) = 0, so that (ker Tr ℓ ) t contains K(N ) t .
As in Proposition 8.1(1), if Surj (t, T ℓ ) holds, then the sequence is exact. Therefore, the Hecke algebra on K(N ℓ) t /(ker Tr ℓ ) t is isomorphic to A(N ) t . (viii) Clearly, the Hecke algebra on K(N ) t is A(N ) t as well. Proof. Denote ker T ℓ | K(N )t by (ker T ℓ ) N,t below. We compare the exact sequences of the middlegraded piece of the Monsky filtration to the same from the standard filtration: (vii) The filtration that appers in Monsky's work is actually conjugated by W ℓ , namely: where the second-to-last term is the kernel of the map W ℓ Tr ℓ W ℓ : K(ℓ) → W ℓ K(1).
Here the Monsky sequence is vertical with solid arrows and the standard sequence (8.3) is horizontal with solid arrows. The inclusion K(N ) t ֒→ (ker Tr ℓ ) t from the Monsky sequence induces the upper horizontal exact sequence; note that the map Tr ℓ W ℓ restricted to K(N ) t coincides with ℓT ℓ . Finally, the snake lemma on the resulting two horizontal short exact sequences gives us a natural isomorphism that we unpack as a short exact sequence below: The first map is natural inclusion; the second map is the composition To see that the induced surjection on Hecke algebras A(N ℓ) ℓ-new Note that A(1) t ∼ = F x, y if t is unobstructed in the sense of deformation theory. See [19] for p = 2, [2] for p ≥ 5, [13] for p = 3, and [12] for more discussion of p = 2, 3, 5, 7, 13.
Proof of Lemma 8.5. In level one, we have a perfect continuous duality between A(1) and K(1) as A(1)-modules under the pairing A(1) t × K(1) t → F given by T, f := a 1 (T f ). Therefore, we may choose a basis {m(a, b)} a≥0,b≥0 of K(1) dual to the "Hilbert basis" {x a y b }: more precisely, one which satisfies x · m(0, b) = y · m(a, 0) = 0 for all a, b, and x · m(a, b) = m(a − 1, b) for a ≥ 1 and y · m(a, b) = m(a, b − 1) if b ≥ 1.
We introduce a total order on pairs of nonnegative integers: we'll say that (a, b) ≺ (c, d) if a + b < c + d, or if a + b = c + d and b < d. (In fact any total order will do.) Suppose T = a+b=k c a,b x a y b + O (x, y) k+1 ∈ F x, y for some k ≥ 0. Let (a 0 , b 0 ) be the ≺-minimal pair among all the pairs (a, b) with c a,b nonzero; by scaling T if necessary, we may assume that c a 0 ,b 0 = 1. For example, if T ℓ = 5x 2 y−y 3 +O (x, y) 4 , then (a 0 , b 0 ) = (2, 1). We induct on ≺ to show that m(a, b) is in the image of T for any pair (a, b). It's clear that T · m(a 0 , b 0 ) = m(0, 0): base case. For the inductive step, suppose that the vector space V a,b = m(c, d) : (c, d) ≺ (a, b) F is in the image of T already. Since T · m(a + a 0 , b + b 0 ) is in m(a, b) + V a,b , in fact m(a, b) is in the image of T as well.
Question 3. Can one prove a similar statement for A(N ) t if it is not a power series ring? At the very least, can one show that condition NZDiv(t, T ) is satisfied?
Appendix A. The Atkin-Lehner automorphism mod p geometrically (Alexandru Ghitza) Our aim is to describe a geometric construction of the modified Atkin-Lehner automorphism W ℓ on the algebra of modular forms M (N ℓ, F p ). This will be an intrinsic characteristic p construction, stemming from an automorphism of the Igusa curve.
A.1. Classical Atkin-Lehner via geometry. Let's start by recalling the geometric construction of the Atkin-Lehner operator w ℓ , following Conrad [21].
Let ℓ be a prime and N a positive integer coprime to ℓ. The noncuspidal points on the modular curve X 0 (N ℓ) have the moduli interpretation (E; C ℓ , C N ) with E an elliptic curve, C j cyclic subgroup of order j.
We define an involution w ℓ : where φ : E → E/C ℓ is the quotient isogeny.
Conrad explains in what sense this involution can be extended to the cusps of Y 0 (N ℓ), and shows that over C, this construction yields the classical Atkin-Lehner involution on M k (N ℓ). He also proves that, if f (q) ∈ Z[ 1 ℓ ] q , then (w ℓ f )(q) ∈ Z[ 1 ℓ ] q , from which we get the Atkin-Lehner involution w ℓ on modular forms mod p for any prime p = ℓ.
As our setup is simpler (having the extra assumption that p ∤ N ), we think of the classical mod p Atkin-Lehner involution as coming directly from the map w ℓ : Y 0 (N ℓ) Fp → Y 0 (N ℓ) Fp : where E is an elliptic curve in characteristic p and φ : E → E/C ℓ is the quotient isogeny. More explicitly, if f ∈ M k (N ℓ, F p ) and ω is a nonzero invariant differential on E, we have A.2. The Igusa curve I 0 (N ℓ). We summarize the features of Igusa curves that are essential to our construction. We follow mainly Gross's exposition in [9,Section 5], which develops the theory for Γ 1 (N )-structure; this can be adapted to our Γ 0 (N ) situation with minor changes, as summarized in [9,Section 10]. A thorough study of Igusa curves appears in [10,Chapter 12], however without treatment of modular forms. The Γ 0 (1) case is described briefly by Serre in [24, end of p. 416-05]; see also the discussion in MathOverflow question 93059.
Note that when p = 2 we have W ℓ = w ℓ , the classical Atkin-Lehner automorphism. We will henceforth assume that p ≥ 3.
Consider a prime p = ℓ and coprime to N . Given an elliptic curve E in characteristic p, there are morphisms Frobenius F : E → E (p) and Verschiebung V : E (p) → E such that V • F = [p] : E → E and a canonical short exact sequence of group schemes An Igusa structure of level p on E is a choice of generator of (the Cartier divisor) ker V . This is equivalent to choosing a surjective morphism of group schemes E[p] → ker V , or (by Cartier duality) to choosing an embedding of group schemes (ker V ) * ֒→ E[p]. We can be more precise by distinguishing the two cases: • If E is ordinary, then ker V ∼ = Z/pZ and (ker V ) * ∼ = µ p so an Igusa structure is an embedding µ p ֒→ E[p]. • If E is supersingular, then ker V ∼ = α p and (ker V ) * ∼ = α p so an Igusa structure is an embedding α p ֒→ E[p]. In fact, there is a unique such embedding (see [8,Example 3.14]).
If we restrict our attention to ordinary elliptic curves E, the moduli problem defined by the data is representable (as we assume p ≥ 3) by an affine curve I 0 (N ℓ) ord whose coordinate ring we denote S(N ℓ). It has a natural smooth compactification I 0 (N ℓ) with a canonical map π : I 0 (N ℓ) → X 0 (N ℓ) Fp that is totally ramified over the supersingular points. It can be thought of as quotienting by the automorphism group (Z/pZ) × /(±1), which acts freely on I 0 (N ℓ) ord via This defines a grading on the algebra of functions where S α (N ℓ) consists of the functions on I 0 (N ℓ) ord that satisfy d p g = d α g for all d ∈ (Z/pZ) × .
(This is the Γ 0 -analogue of the Γ 1 result in [9, Proposition 5.2], see also [9, Section 10].) We use the section a 2 to trivialize the line bundle π * ω ⊗2 . This allows us to treat sections of ω ⊗k on X 0 (N ℓ) Fp as functions on the ordinary locus I 0 (N ℓ) ord . More precisely, the q-expansion (ix) We abuse notation by writing a 2 even though there is no a itself for Γ0-structures; so whenever we write a k we implicitly assume that k is even and we set a k := (a 2 ) (k/2) . map gives an isomorphism of graded F p -algebras Φ : S(N ℓ) To see that the image of Φ is contained in M (N ℓ, F p ), let g ∈ S α (N ℓ) and let k ≡ α (mod p − 1) be such that a k g is regular on I 0 (N ℓ). Since d p (a k g) = d −k a k d α g = d α−k (a k g) = a k g, we see that a k g descends to a global section f ∈ M k (N ℓ, F p ), and g(q) = f (q) ∈ M (N ℓ, F p ) α .
For the inverse map: given f (q) ∈ M (N ℓ, F p ) α , let f ∈ M k (N ℓ, F p ) be any modular form with q-expansion f (q), and let g = π * f a k Then g is a function on I 0 (N ℓ) ord with d p g = d α g and g(q) = f (q).
A.3. From maps on the Igusa curve to operators on modular forms mod p. A morphism ψ : I 0 (N ℓ) ord → I 0 (N ℓ) ord on the ordinary locus of the Igusa curve determines a homomorphism of graded F p -algebras Ψ : M (N ℓ, F p ) → M (N ℓ, F p ) by setting where, given g ∈ S(N ℓ), ψ * g = g • ψ ∈ S(N ℓ).
In order to recover the modified Atkin-Lehner automorphism W ℓ defined in subsection 3.4, we start with the map w ℓ : I 0 (N ℓ) ord → I 0 (N ℓ) ord given by where φ : E → E/C ℓ is the quotient isogeny. Since we conclude that w 2 ℓ = ℓ p .
We can adapt this into an automorphism w ζ of I 1 (N ℓ) ord by setting We illustrate the various spaces and maps in the following cube diagram whose commutativity is readily checked via calculations similar to that in Lemma A.1, using the moduli interpretation of the covering maps η : I 1 (N ℓ) → I 0 (N ℓ): η(E; β ℓ , α N , i p ) = (E; β ℓ (µ ℓ ), α N (µ N ), i p ) and similarly for η : X 1 (N ℓ) → X 0 (N ℓ).
Remark. The reader is perhaps wondering why we had to involve Γ 1 -structures. It is indeed possible to apply the argument in Lemma A.3 directly to the trivializing section a 2 on I 0 (N ℓ), but that only allows us to conclude that w * ℓ (a 2 ) = ±ℓ −1 a 2 , and we are unable to rule out the possible negative sign when p ≡ 1 (mod 4). The Γ 1 setting provides us with a square root of a 2 1 , which strenghens the argument enough to rule out the unwanted −1. It is possible that working with the moduli stack X 0 (N ℓ) instead of the coarse moduli space X 0 (N ℓ) could also provide the needed flexibility, without the artifice of changing level structures.
Proposition A.4. If f is a modular form of weight k and q-expansion f (q), we have Proof. This is just a matter of combining Lemma A.1 and Lemma A.2: = Φ π * (w ℓ f ) ℓ −k/2 a k = ℓ k/2 w ℓ f (q) = W ℓ f (q).