On $\ell$-torsion in degree $\ell$ superelliptic Jacobians over $\mathbf{F}_{q}$

For all the results that follow, we assume $\ell\geq 3$ is prime, $q$ is a prime power coprime to $\ell$, and $f(x)\in\mathbf{F}_{q}[x]$ is an irreducible polynomial with $d:=\deg f$ coprime to $\ell$. Let $C$ be the smooth projective curve with affine equation given by $y^{\ell}=f(x)$; such a curve is an example of a “superelliptic curve.” Let $J$ be the Jacobian of $C$, so the degree $0$ subgroup of the divisor class group of $C$ is isomorphic to $J(\mathbf{F}_{q})$. The $\ell$-torsion of $J(\mathbf{F}_{q})$ can be equipped with the structure of a vector space over $\mathbf{F}_{\ell}$, and we define the $\ell$-rank of $C$ to be the dimension of this $\mathbf{F}_{\ell}$ vector space,

The function field of $C$ is isomorphic to $\mathbf{F}_{q}(\sqrt[\ell]{f},x)$, and up to isomorphism $C$ is the only smooth projective curve with this function field. We define the divisor class group of $\mathbf{F}_{q}(\sqrt[\ell]{f},x)$ to be the divisor class group of $C$. Then an equivalent definition for $r_{\ell}(C)$ is that it is the largest power of $\mathbf{Z}/\ell\mathbf{Z}$ that occurs as a subgroup of the divisor class group of the function field $\mathbf{F}_{q}(\sqrt[\ell]{f},x)$.

Let $\gamma=\textnormal{ord}_{\ell}(q)$ be the multiplicative order of $q\bmod\ell$ in $(\mathbf{Z}/\ell\mathbf{Z})^{\times}$, that is, the smallest positive integer such that $q^{\gamma}\equiv 1\bmod\ell$. This is an important invariant for this problem because $\mathbf{F}_{q^{\gamma}}$ is the smallest extension of $\mathbf{F}_{q}$ containing $\ell$-th roots of unity, and hence the Galois closure of $\mathbf{F}_{q}(\sqrt[\ell]{f},x)/\mathbf{F}_{q}(x)$ is a degree $\gamma$ extension field, namely $\mathbf{F}_{q^{\gamma}}(\sqrt[\ell]{f},x)$.

The parity constraint $r_{\ell}(C)\equiv B\bmod 2$ is proved using the Weil pairing on $J[\ell]$. This phenomenon does not appear to occur in the analogous situation in number fields, namely ideal class groups of cyclic extensions $\mathbf{Q}(\sqrt[p]{N})$ for $N$ prime discussed in [12]; see Section 1.4 for a discussion of the number field case.

In one special case, we can prove a lower bound that equals the upper bound in Theorem 1.3, allowing us to construct families of curves with large $\ell$-torsion subgroups in their divisor class groups.

Theorem 1.1 follows immediately from Theorem 1.3 and Proposition 1.5. Both Theorem 1.3 and Proposition 1.5 can be proven with linear algebra, using linear maps on $J[\ell]$ defined using endomorphisms of $J$. The parity constraint $r_{\ell}(C)\equiv B\bmod 2$ is proved using the Weil pairing on $J[\ell]$. These topics are summarized in Section 1.2 and discussed in depth in Section 2 and Section 3. The proofs of Theorem 1.3 and Proposition 1.5 are then completed in Section 7.

If $\gamma\leq 2$ then $r_{\ell}(C)$ is completely determined by Theorem 1.1, so for the remainder of this section we assume $\gamma\geq 3$. We can compute more refined bounds on $r_{\ell}(C)$ if we additionally assume $\gamma\mid d$. This constraint ensures that $f$ totally splits in the extension $\mathbf{F}_{q^{\gamma}}(x)/\mathbf{F}_{q}(x)$; this is analogous to the constraint $N\equiv 1\bmod p$ in [12] which guarantees that $N$ totally splits in $\mathbf{Q}(\operatorname{\zeta}_{p})/\mathbf{Q}$. Over $\mathbf{F}_{q^{\gamma}}[x]$, $f(x)$ splits into $\gamma$ irreducible factors, which we label $f_{1}(x),f_{2}(x),\ldots,f_{\gamma}(x)$ in such a way that the Frobenius automorphism on $\mathbf{F}_{q^{\gamma}}$ sends $f_{i}(x)$ to $f_{i+1}(x)$ for all $i$ (and $f_{\gamma}(x)$ to $f_{1}(x)$). Set

The polynomials $h_{n}(x)$ are associated via Kummer theory to certain cyclic degree $\ell$ extensions of $\mathbf{F}_{q^{\gamma}}(x)$; see Section 6.3 for more on how these polynomials arise.

Since $|\mathcal{T}|\leq\gamma-1=\gcd(d,\gamma)-1$ we have $B^{\prime}\leq B$, and from $\min\{B,1\}\leq r_{\ell}(C)\leq B^{\prime}$ we can conclude $\min\{B^{\prime},2\}\geq\min\{B,1\}$. Thus Theorem 1.6 gives both upper and lower bounds that are at least as strong as those in Theorem 1.3.

In addition to the linear algebra on $J[\ell]$ discussed in Section 2 and Section 3, the proof of Theorem 1.6 requires techniques from Kummer theory and Galois cohomology. These techniques are introduced in Section 1.3 and discussed in depth in Sections 4, 5, and 6. The proof of Theorem 1.6 is then completed in Section 7.

In some cases, Theorem 1.3 and Theorem 1.6 are sufficient to determine $r_{\ell}(C)$ precisely.

For larger values of $\ell$, Theorem 1.3 and Theorem 1.6 are not sufficient to determine $r_{\ell}(C)$. For example, we have the following options when $\ell=7$:

For the first two rows, we can exhibit curves attaining both possible values of $r_{7}(C)$, demonstrating that the parameters $\ell,q,d,|\mathcal{T}|$ are not sufficient to fully determine the value of $r_{\ell}(C)$ in general. For instance, consider the case $\ell=7$, $q=3$ (so $\gamma=6$), and $d=12$ from the introduction, summarized in Table 1. We may categorize these function fields further by the sets $\mathcal{T}$ associated to each. See Table 2, and note in particular the last two columns, consisting of curves with the same $\mathcal{T}$ but different values of $r_{\ell}(C)$. ¹¹1The astute reader may notice in Table 2 that $\mathcal{T}\setminus\{1\}$ is closed under $n\mapsto 1-n\bmod\gamma$. This symmetry does hold in general, following from Theorem 1.11 and Theorem 1.13 below, and can be used to cut down the number of computations needed in order to find the set $\mathcal{T}$.

The most important feature of working with function fields is that we can represent elements of the ideal class group of $\mathbf{F}_{q}(\sqrt[\ell]{f},x)$ using geometric objects, as described in Section 1.1. This allows us to use morphisms from $C$ to itself to study $r_{\ell}(C)$.

The $q$-power Frobenius map $(x,y)\to(x^{q},y^{q})$ on $C(\overline{\mathbf{F}}_{q})$ induces a linear operator $\operatorname{Frob}$ on the $\mathbf{F}_{\ell}$-vector space $J[\ell]$. The eigenspace of eigenvalue $1$ for this action is $J[\ell](\mathbf{F}_{q})$, so $r_{\ell}(C)$ can be recovered as the dimension of this eigenspace. The primary difficulty we will encounter is that the action of $\operatorname{Frob}$ on $J[\ell]$ is not semi-simple in general. The action of Frobenius on the $\ell$-adic Tate module $J[\ell^{\infty}]$ is semi-simple, but this property does not descend to the mod $\ell$ reduction. So even though we can determine the full characteristic polynomial of $\operatorname{Frob}$ acting on $J[\ell]$ with relatively little work (Remark 2.10), this is not enough to determine the dimension of any particular eigenspace.

To study $r_{\ell}(C)$, we use a filtration coming from the automorphism $(x,y)\mapsto(x,\operatorname{\zeta}_{\ell}y)$ on the space $J[\ell]$ denoted as

that is preserved by $\operatorname{Frob}$, and we consider the intersection of generalized eigenspaces for $\operatorname{Frob}$ with this filtration.

Note that for $n\in\mathbf{Z}/\gamma\mathbf{Z}$, multiplication by $q^{n-k+1}$ is a well-defined scalar operator on $J[\ell]$ because $q^{\gamma}=1$ in $\mathbf{F}_{\ell}$. We will show that $J[\ell](\mathbf{F}_{q})$ has a basis formed by taking one true eigenvector of $\operatorname{Frob}$ from each $F_{k-1}^{k}$, whenever such an eigenvector exists. Thus $r_{\ell}(C)$ is directed related to $F_{n}^{k}$ in the following result.

For a visual interpretation, see LABEL:fig:lifting_charts. Each of the three figures represents a different possible curve. If the cell with coordinates $(n,k)$ is shaded dark gray, this means that there exists a $\operatorname{Frob}$ eigenvector with eigenvalue $q^{n-k+1}$ in $V^{k}\setminus V^{k-1}$. The circles correspond to cells $(n,k)$ with $q^{n-k+1}=1$ in $\mathbf{F}_{\ell}$, so any dark gray circle corresponds to a $\operatorname{Frob}$ eigenvector of eigenvalue $1$, that is, an element of $J[\ell](\mathbf{F}_{q})$. The number of dark gray circles equals to $r_{\ell}(C)$. See Remark 2.7 for a more thorough guide to reading these diagrams.

We prove several constraints that determine when $F_{n}^{k}$ contains an eigenvector of $\operatorname{Frob}$. A number of relations follow fairly directly from linear algebra on $J[\ell]$. We say that $F_{n}^{k}$ is a “rooftop” if $F_{n}^{k}$ has a $\operatorname{Frob}$ eigenvector but there is no $k<k^{\prime}\leq\ell-1$ such that $F_{n}^{k^{\prime}}$ has an eigenvector. This notion is justified by statement (b) of the following Proposition. We say a rooftop $F_{n}^{k}$ is “non-maximal” if $k\neq\ell-1$.

These relations are proven in Section 2.5. Part (a) is obtained by constructing an explicit basis of $\operatorname{Frob}$ eigenvectors for $V^{1}$, and part (b) follows from a relation between $\operatorname{Frob}$ and the map used to define the filtration. If $F_{n}^{k}$ is a non-maximal rooftop, we will show that applying $(\operatorname{Frob}-q^{n-k})$ to a vector in $F_{n}^{k+1}$ yields an eigenvector in $F_{n-k}^{1}$ (a “Jordan chain” of length $2$), giving part (c). Part (d) comes from that if two such Jordan chains exist, a linear combination of the two generalized eigenvectors will produce an eigenvector.

The remaining three constraints Theorem 1.11, Theorem 1.12, and Theorem 1.13 are less straightforward, and form the main technical contributions of this paper. The first two can be proven using the Weil pairing on $J[\ell]$, by proving a numerical relation between pairs of Jordan chains (Lemma 3.3).

When $k\equiv 2n\bmod\gamma$, the above result is vacuous; however in this case we will see that the numerical relation gives us the following result.

Both Theorem 1.11 and Theorem 1.12 are proved in Section 3. Theorem 1.11 says that $F_{n}^{k}\mapsto F_{k-n}^{k}$ defines an involution on the set of non-maximal rooftops, and Theorem 1.12 imposes a parity constraint on the fixed points of this involution; these two observations will be used together in Section 7.3 to prove the parity constraint $r_{\ell}(C)\equiv B\bmod 2$.

Another method we use to study the $\ell$-rank of $J[\ell](\mathbf{F}_{q})$ comes from Galois cohomology. This approach is more closely related to the approach used to study the number field version of this problem, where there is no direct analogue of the geometric $\ell$-torsion subgroup $J[\ell](\overline{\mathbf{F}}_{q})$ (see Section 1.4). In Section 4 and Section 5 we use Kummer theory to relate the two perspectives. The culmination of these sections is Proposition 5.7, which relates the existence of an eigenvector of $\operatorname{Frob}$ in $F_{n}^{k}$ to the existence of a certain cohomology class in a Selmer group associated to a $(k+1)$-dimensional representation of $\textnormal{Gal}(\mathbf{F}_{q}(x)^{\textnormal{sep}}/\mathbf{F}_{q}(x))$.

In Section 6 we show that $F_{n}^{k}$ is a non-maximal rooftop if and only if a certain cup product of cohomology classes does not vanish. The vanishing of this cup product can be determined by a residue field calculation, leading us to the last constraint.

We note that Proposition 1.10 and Theorem 1.11 can also be proven entirely using this Galois cohomology framework: for Theorem 1.11, instead of the Weil pairing we one can use Poitou-Tate duality. On the other hand, we have not yet found a way to prove Theorem 1.12 using this framework. The key difficulty comes from determining whether a Selmer class associated to a self-dual representation lifts to a Selmer class associated to a higher-dimensional representation. These self-dual representations are quite difficult to work with compared to their non-self-dual counterparts, so our geometric proof of Theorem 1.12 using the Weil pairing illustrates a method that can be used to work with them in the function field setting. On the other hand, we were only able to prove Theorem 1.13 using cohomological techniques. Thus, using both the geometric approach (Sections 2–3) and the cohomological approach (Sections 4–6) allows us to prove stronger results than any one approach individually.

Together with Proposition 1.10 and Theorem 1.9, constraints on $r_{\ell}(C)$ can be obtained by counting arguments, analyzing the various restrictions on pairs $(n,k)$. Section 7 contains proofs of some such constraints, including all the results stated in Section 1.1.

The study of $\ell$-torsion in divisor class groups of superelliptic extensions $K(\sqrt[\ell]{f},x)/K(x)$ (for some field $K$) has been explored in many other contexts; for some examples [14, 7]. Most of these explorations are largely independent from the content of this paper; for instance, some take $K=\mathbf{Q}$ instead of $K=\mathbf{F}_{q}$, and they impose different conditions on $\ell$ and $f(x)$. Further, these works typically focus on a particular subgroup of the $\ell$-torsion generated by divisors supported at the ramification locus of $f(x)$, which is the first stage $V^{1}$ in the filtration of $J[\ell]$ discussed in Section 1.2.

In the case of hyperelliptic function fields $\mathbf{F}_{q}(\sqrt{f},x)/\mathbf{F}_{q}(x)$, this first stage $V^{1}$ contains the entirety of the $2$-torsion; Cornelissen uses this to compute the $2$-rank of $J[2](\mathbf{F}_{q})$ for arbitrary hyperelliptic curves over $\mathbf{F}_{q}$ (allowing $f(x)$ to be reducible) [4]. However, for $\ell>2$, there is more to the filtration than this first stage, and these deeper filtration stages are one of the primary focuses of this paper. The primary difficulty we face is that unlike the action of Frobenius on $V^{1}$, the action of $\operatorname{Frob}$ on $J[\ell]$ as a whole is not semi-simple. See Remark 2.10 for a discussion.

The aforementioned filtration can be defined using an endomorphism $(x,y)\mapsto(x,\operatorname{\zeta}_{\ell}y)$ on $C$, where $\operatorname{\zeta}_{\ell}$ is some $\ell^{\text{th}}$ root of unity in $\overline{\mathbf{F}}_{q}$. This endomorphism and the filtration it defines were used by Poonen–Schaefer [11] and were further explored by Arul [1].

Other authors have studied the $\ell$-torsion subgroups of divisor class groups of different kinds of degree $\ell$ extensions of $\mathbf{F}_{q}(x)$. For instance, Wittmann considered degree $\ell$ Galois extensions $K/\mathbf{F}_{q}(x)$, and studied the Galois module structure of the $\ell$-torsion in the divisor class group of $K$ [15].

The question this paper is exploring has a direct analogue in number fields: namely, to study the $p$-rank of the ideal class group of $\mathbf{Q}(N^{1/p})$ for distinct primes $N$ and $p$. Several authors have studied this question under the assumption $N\equiv 1\bmod p$, which is analogous to the assumption $\gamma\mid d$ we make in Theorem 1.6. Using deformations of Galois representations, Calegari–Emerton [2] determined conditions under which the $p$-part is cyclic (i.e. $p$-rank $1$). For instance, one of their results is that if $\prod_{i=1}^{(N-1)/2}i^{i}$ is a $p$-th power modulo $N$, then the $p$-rank of $\textnormal{Cl}(\mathbf{Q}(N^{1/p}))$ is at least $2$ [2, Theorem 1.3(ii)]. These results were generalized by Wake–Wang-Erickson [13, Proposition 11.1.1]; in particular, they interpreted the congruence condition as a cup product on Galois cohomology. The techniques of Wake–Wang-Erickson were used by Karl Schaefer and the third author to prove a full converse of Calegari–Emerton’s result by imposing additional congruence conditions.

The cohomological methods used in Sections 4–6 of this paper closely follow the work of Schaefer and the third author, using the Galois cohomology framework developed by Wake–Wang-Erickson. In particular, the upper bound in Theorem 1.6 is directly analogous to [12, Theorem 1.1.1]. On the other hand, [12, Table 3] shows that there is no parity constraint on the $p$-rank in the number field setting; the parity constraint on $r_{\ell}(C)$ appears to be a phenomenon unique to function fields.

The bulk of this research was conducted while all three authors held a CRM-ISM postdoctoral fellowship. The first author was partially supported by NSF grant DMS-2302511, and the second author was partially supported by ERC Starting Grant 101076941 (‘Gagarin’).

The authors thank Patrick Allen, Jordan Ellenberg, Jaclyn Lang, Bjorn Poonen, Karl Schaefer, Jacob Stix, Yunqing Tang, Carl Wang-Erickson for conversations that pointed them in helpful directions. The first two authors want to thank the third author for suggesting this project and for introducing them to the technical details of Galois cohomology needed for this paper, and the third author wishes to thank his collaborators for seeing this project through after he left academia.

We use the following notation throughout the paper.

• •

  $\ell\geq 3$ is a prime.

• •

  $q$ is a prime power coprime to $\ell$.

• •

  $\gamma$ is the multiplicative order of $q$ in $(\mathbf{Z}/\ell\mathbf{Z})^{\times}$.

• •

  $\operatorname{\zeta}\in\mathbf{F}_{q^{\gamma}}$ is a fixed nontrivial $\ell^{\text{th}}$ root of unity.

• •

  $f(x)\in\mathbf{F}_{q}[x]$ is an irreducible polynomial with $d:=\deg f$ coprime to $\ell$.

• •

  $C/\mathbf{F}_{q}$ is the smooth projective curve with affine equation $y^{\ell}=f(x)$, and $C_{\overline{\mathbf{F}}_{q}}$ its base change to $\overline{\mathbf{F}}_{q}$.

• •

  $J/\mathbf{F}_{q}$ is the Jacobian of $C$, and $J_{\overline{\mathbf{F}}_{q}}$ its base change to $\overline{\mathbf{F}}_{q}$.

• •

  For a field extension $\mathbf{F}/\mathbf{F}_{q}$, $J(\mathbf{F})$ denotes the $\mathbf{F}$-points of $J$. Elements of $J(\overline{\mathbf{F}}_{q})$ can be interpreted as divisors on $C_{\overline{\mathbf{F}}_{q}}$ modulo linear equivalence, and if $\mathbf{F}/\mathbf{F}_{q}$ is a finite extension, elements of $J(\mathbf{F})$ correspond to divisor classes in $J(\overline{\mathbf{F}}_{q})$ that are invariant under $\textnormal{Gal}(\overline{\mathbf{F}}_{q}/\mathbf{F})$.

• •

  $J[\ell](\mathbf{F})$ denotes the $\ell$-torsion subgroup of $J(\mathbf{F})$, and $J[\ell]:=J[\ell](\overline{\mathbf{F}}_{q})$ the geometric $\ell$-torsion group.

• •

  The $\ell$-torsion rank of $C$ is defined to be

  $$r_{\ell}(C):=\dim_{\mathbf{F}_{\ell}}(J[\ell](\mathbf{F}_{q})).$$

We also define two morphisms of $C_{\mathbf{F}_{q^{\gamma}}}$ by giving their actions on geometric points $(x,y)\in C(\overline{\mathbf{F}}_{q})$. Using the $\ell^{\text{th}}$ root of unity $\operatorname{\zeta}\in\mathbf{F}_{q^{\gamma}}$ chosen above, by abuse of notation we also let $\operatorname{\zeta}:C_{\mathbf{F}_{q^{\gamma}}}\to C_{\mathbf{F}_{q^{\gamma}}}$ denote the morphism defined by

We let $\operatorname{Frob}:C_{\mathbf{F}_{q^{\gamma}}}\to C_{\mathbf{F}_{q^{\gamma}}}$ denote the relative Frobenius map,

Note that $\operatorname{\zeta}$ is an automorphism of $C_{\mathbf{F}_{q^{\gamma}}}$, while $\operatorname{Frob}$ is a degree $q$ endomorphism; both act invertibly on $C(\overline{\mathbf{F}}_{q})$. On $C(\overline{\mathbf{F}}_{q})$, we also have the relation

By further abuse of notation, we also let $\operatorname{\zeta}$ and $\operatorname{Frob}$ denote the respective endomorphisms of $J_{\overline{\mathbf{F}}_{q}}$ induced by their namesakes, as well as the induced linear maps on $J[\ell]$ considered as a vector space over $\mathbf{F}_{\ell}$.

The automorphism $\operatorname{\zeta}$ and endomorphism $1-\operatorname{\zeta}$ on $J_{\overline{\mathbf{F}}_{q}}$ were discussed in detail in [1, Section 2.3]. Here we use them to construct a filtration on $J[\ell]$.

Noting that the endomorphism $\operatorname{\zeta}$ is annihilated by the $\ell^{\text{th}}$ cyclotomic polynomial, we can derive the relation

in the endomorphism ring $\textnormal{End}(J_{\overline{\mathbf{F}}_{q}})$. For each $1\leq i\leq\ell-1$, $1-\operatorname{\zeta}^{i}$ is equal to $1-\operatorname{\zeta}$ times a unit, and so $(1-\operatorname{\zeta})^{\ell-1}$ is $\ell$ times a unit. We can conclude that the kernel of $(1-\operatorname{\zeta})^{\ell-1}$ on $J_{\overline{\mathbf{F}}_{q}}$ is exactly $J[\ell]$.

Define $V^{k}$ to be the $\overline{\mathbf{F}}_{q}$-points of $\ker\,(1-\operatorname{\zeta})^{k}$; these $V^{k}$ then give a filtration

Note that each subgroup $V^{k}$ has the structure of a $\mathbf{F}_{\ell}$-vector space.

Recall from Section 2.1 that the relative Frobenius map $\operatorname{Frob}:J_{\overline{\mathbf{F}}_{q}}\to J_{\overline{\mathbf{F}}_{q}}$ is induced by the action $(x,y)\mapsto(x^{q},y^{q})$ on $C(\overline{\mathbf{F}}_{q})$. The maps $\operatorname{Frob}$ and $1-\operatorname{\zeta}$ on $J_{\overline{\mathbf{F}}_{q}}$ satisfy the relation

Since $1-\operatorname{\zeta}^{q}$ and $1-\operatorname{\zeta}$ are associates in $\textnormal{End}(J_{\overline{\mathbf{F}}_{q}})$, this identity shows that the action of $\operatorname{Frob}$ on $J[\ell](\overline{\mathbf{F}}_{q})$ preserves the filtration stages $V^{k}$. However, the automorphism $\frac{1-\operatorname{\zeta}^{q}}{1-\operatorname{\zeta}}$ of $J_{\overline{\mathbf{F}}_{q}}$ does not preserve the generalized eigenspaces of $\operatorname{Frob}$. To account for this, we introduce a modification of $1-\operatorname{\zeta}$ that interacts in a more predictable way with the Frobenius map.

While the endomorphism $\operatorname{\eta}$ depends on the choice of inverses mod $\ell$, the action of $\operatorname{\eta}$ on $J[\ell]$ is well-defined, independent of the choice of $i^{-1}$ for each $i$. We have the following two important facts about $\operatorname{\eta}$, which both capture the idea that $\operatorname{\eta}$ behaves like a “logarithm” of $\operatorname{\zeta}$. The first statement in the following lemma says that $\operatorname{\eta}$ acts like $\operatorname{\zeta}-1$ up to higher-order terms.

The second statement about $\operatorname{\eta}$ in the following lemma can be thought of as a linearization of the relation $\operatorname{Frob}\circ\operatorname{\zeta}=\operatorname{\zeta}^{q}\circ\operatorname{Frob}$. As a note of caution, the following relation does not hold when $\operatorname{Frob}$ and $\operatorname{\eta}$ are considered as endomorphisms of $J_{\overline{\mathbf{F}}_{q}}$; we obtain the desired equality only when we restrict to the actions on $J[\ell]$.

A consequence of this result is that if $v\in J[\ell]$ is a (generalized) eigenvector of $\operatorname{Frob}$, then so is $\operatorname{\eta}v$.

Any $v\in J[\ell](\mathbf{F}_{q})$ lies in some filtration stage $v\in V^{k}\setminus V^{k-1}$. Then $\operatorname{\eta}^{k-1}v\in V^{1}$, and by Lemma 2.5 we have

so $\operatorname{\eta}^{k-1}v$ is an eigenvector of eigenvalue $q^{k-1}$. So our first goal is to identify which powers of $q$ arise as eigenvalues of $\operatorname{Frob}$ acting on $V^{1}$.

To start, we find a basis of $V^{1}$ which is most suitable for our study of the $\operatorname{Frob}$ action. The dimension of $V^{1}$ is $d-1$ for $d:=\deg f$. The following Lemma, which is well-known in the literature (see e.g. the proof of [5, Theorem 1.7]), shows that the action of $\operatorname{Frob}$ on $V^{1}$ is diagonalizable over $\overline{\mathbf{F}}_{\ell}$.

(Caution: recall that $\operatorname{Frob}$ is induced by the $q$-power Frobenius map on $J_{\overline{\mathbf{F}}_{q}}$, not an $\ell$-power Frobenius map on $\overline{\mathbf{F}}_{\ell}$.)

For $2\leq k\leq\ell-1$, an eigenvector in $V^{1}$ with eigenvalue $q^{n}$ lifts under $\operatorname{\eta}^{k-1}$ to an eigenvector of $\operatorname{Frob}$ in $V^{k}/V^{k-1}$ with eigenvalue $q^{n-k+1}$ (by Lemma 2.4 and Lemma 2.5). This lift is a priori only an eigenvector in the quotient space, but we show in Lemma 2.8 that we can always take the lift to be a generalized eigenvector of $\operatorname{Frob}$ acting on $J[\ell]$ using properties of the operator $\operatorname{\eta}$.