Proof. Recall that \(\mathcal{O}_X(nD)|_{nD}\) is naturally isomorphic to \(\mathcal{O}_{nD}(nD)\), and the bottom row can also be obtained by twisting the closed subscheme exact sequence. There is a similar interpretation for the row of differentials. Thus the rows are exact. Now the degrees of \(\Omega^1_X(-nD)\) and \(\mathcal{O}_X(-(n+1)D)\) are negative as \(\deg(nD) > \max(2g_X-2,0)\). Hence by Serre duality \(\mathcal{O}_X(nD)\) and \(\Omega^1_X((n+1)D)\) are acyclic. The third term in each row is acyclic as it is supported in dimension zero. ◻

Proof. The algebraic de Rham cohomology of \(X\) is the hypercohomology of the de Rham complex, which we can compute as the cohomology of the total complex of the global sections of any acyclic resolution. Using the resolution from Proposition 10, the total complex is \[\label{eq:totalycomplex} 0 \to \Gamma(\mathcal{O}_X(nD)) \overset{\delta_0} \to \Gamma(\Omega^1_X((n+1)D)) \oplus \Gamma(\mathcal{O}_X(nD)|_{nD}) \overset{\delta_1} \to \Gamma(\Omega^1_X((n+1)D)|_{(n+1)D}) \to 0\tag{3}\] Now \(\mathcal{O}_X(nD)|_{nD}\) is a skyscraper sheaf supported on \(D\), with the stalk at \(Q\) consisting of functions with a pole of order at most \(n\) at \(Q\) modulo functions regular at \(Q\). Similarly \(\Gamma(\Omega^1_X((n+1)D)|_{(n+1)D})\) consists of “tails” of differentials at \(S\). Recalling the definition of the maps \(\delta_i\) in the total complex, we see \(\ker \delta_1\) consists of pairs \((\omega, (f_Q)_{Q \in S})\) where \[\delta_1(\omega, (f_Q)_{Q \in S}) = \pm (\omega - d(f_Q))_{Q \in S} =0 \in \Omega^1_X((n+1)D).\] In other words, \(\omega - d(f_Q)\) is regular at \(Q\) for each \(Q \in S\). Thus there is an isomorphism \(\ker \delta_1 \to E_{X,nD}\). For \(f \in \Gamma(\mathcal{O}_X(nD))\), note that \(\delta_0(f)= (df, (f_Q)_{Q \in S})\). Thus we obtain an isomorphism \(\ker \delta_1 / \textrm{Im} \delta_0 \simeq E_{X,nD} / d_X(\Gamma(\mathcal{O}_X(nD))\). ◻

Proof of Theorem 9. When \(n \leq m\), there are natural maps \(\mathcal{D}(n) \to \mathcal{D}(m)\) and \(E_{X,nD} \to E_{X,mD}\). These induce maps \(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(X) \to \mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(X)\) and \(E_{X,nD} / d_X(\Gamma(\mathcal{O}_X(nD))) \to E_{X,mD} / d_X(\Gamma(\mathcal{O}_X(mD)))\) which are compatible with the identification of Theorem 11. Taking the limit, we obtain an isomorphism between \(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(X)\) and \(E_{X,S}/d_{X,S}(A_S)\). A similar argument taking the limit over nonempty finite sets \(S\) of \(X\) gives the isomorphism between \(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(X)\) and \(E/d_X(k(X))\).

It remains to establish the formula for the pairing. We will obtain it indirectly in Proposition 14, by comparing the description of \(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(X)\) in terms of differentials of the second kind to the Čech description and using the known form of the pairing in that case. ◻

Proof. Note that \(\imath\) factors through the quotient as if \((\omega_1,\omega_2,f)\) is a Čech 1-coboundary, then \(\left( \omega_1, (f \mod \widehat{\mathcal{O}}_{X,Q})_{Q \in S} \right)\) is equal to \(d_{X,S}(f_1)\). To see that this map is an isomorphism, we describe the inverse. Let \((\omega, (f_Q)_{Q \in S})\) be an \(S\)-enhanced differential. Let \(f\) be a function in \(\mathcal{O}_X(U_1\cap U_2)\) with \(f \equiv f_Q \mod \widehat{\mathcal{O}}_{X,Q}\) for each \(Q \in S\). Such an \(f\) exists by Riemann-Roch. Then \((\omega, \omega-df, f)\) is a Čech 1-cocycle. A short calculation shows that a different choice of \(f\) gives a \(1\)-cocycle that differs by a \(1\)-coboundary. If \((\omega, (f_Q)_{Q \in S})\) is exact, so equal to \(d_{X,S}(f_1)\), then we can take \(f=f_1\), so the Čech 1-cocycle obtained is \((\omega,0,f_1)\), which is a Čech 1-coboundary. It is straightforward to verify this induces an inverse to the map in ?? . ◻

Proof. First, we show that \([\cdot,\cdot]\) is independent of the lift \(\tilde{f}_Q\) and \(\tilde{g}_Q\). Since the right side of ?? is clearly anti-symmetric, it is enough to show that the expression is independent of lift of \(g_Q\). Any other lift is of the form \(\tilde{g}_Q + h_Q\) where \(h_Q \in \widehat{\mathcal{O}}_{X,Q}\). Write \(\omega=d\tilde{f}_Q + \omega_0\), where \(\omega_0\) is holomorphic. Then \(\mathrm{Res}_Q(h_Q\omega_0)=0\). In particular, we see that \(\mathrm{Res}_Q(h_Q\omega)= \mathrm{Res}_Q(h_Qd\tilde{f}_Q)\). A short calculation shows \([\cdot,\cdot]\) does not depend on the lift.

Let \(f\) (resp. \(g\)) be a function in \(\mathcal{O}_X(U_1\cap U_2)\) with \(f \equiv f_Q \mod \widehat{\mathcal{O}}_{X,Q}\) (resp. \(g \equiv g_Q \mod \widehat{\mathcal{O}}_{X,Q}\)) for each \(Q \in S\). Then using \(f\) (resp. \(g\)) for the lift of \(f_Q\) (resp. \(g_Q\)) we obtain \[\left[ (\omega, (f_Q)_{Q \in S}), (\tau, (g_Q)_{Q \in S}) \right] = \sum_{Q \in S} \left( \mathrm{Res}_Q(g\omega) - \mathrm{Res}_Q(f\tau) - \mathrm{Res}_Q(gdf). \right)\] We then see that \([\cdot,\cdot]\) agrees with the pairing \(\langle \cdot, \cdot\rangle\) by looking explicitly at the isomorphism \(\imath\) described in Section §2.3 and the formula 4 for the pairing of Čech \(1\)-cocycles. ◻

Proof. On Čech \(1\)-cocycles we have \(V(\omega_1,\omega_2,f)=(V(\omega_1),V(\omega_2),0)\) and \(F(\omega_1,\omega_2,f)=(0,0,f^p)\) (see [7]). The result then follows using the comparison in Section 2.3. ◻

Proof. First, viewing \(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(Y)\) as a \(k[V]\)-module, (a semilinear version of) Fitting’s lemma yields a decomposition \(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(Y) = U \oplus R\), where \(V\) acts bijectively on \(U\) and nilpotently on \(R\). Since \(F \circ V = 0\), it follows that \(F\) acts trivially on \(U\). Then, viewing \(R\) as a \(k[F]\)-module, Fitting’s lemma gives a decomposition \(R=Z \oplus L\), where \(F\) acts bijectively on \(Z\) and nilpotently on \(L\). Again, it follows from \(V \circ F = 0\) that \(V\) must act trivially on \(Z\).

It remains to verify the claim about the pairing. For this, we use the fact \(\langle Fm_1 , m_2\rangle = \langle m_1 , Vm_2 \rangle ^p\) (see [24]), which implies that elements of \(U\) can only pair non-trivially with elements of \(Z\) and vice versa. Finally, the statement follows from the fact that \(U\) and \(Z\) have the same dimension and that the pairing on \(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(Y)\) is perfect. ◻

Proof. Let \(t\) be a uniformizer of \(K_{P_i}\), and consider an Artin-Schreier equation defining \(K_{Q_i}\) over \(K_{P_i}\) \[\label{eq:32AS32explicit} y^2 + y= f(t).\tag{7}\] We may take \(f(t)\) to have a pole of order \(d\). First, we claim that there exists a uniformizer \(t_i\) of \(K_{P_i}\) with \(t_i^{-d_i}=f(t)\). To see this, note that \(t^{d_i}f(t)=c\cdot u(t)\), where \(c \in k^\times\) and \(u\) is a \(1\)-unit. Then \(u\) has a \(d_i\)-th root since \(d_i\) is coprime to the characteristic. Also, \(c\) has a \(d_i\)-th root, since \(k\) is algebraically closed. In particular, we have \(t^{d_i}f(t)=v^{d_i}\). Then we can take \(t_i\) to be \(tv^{-1}\), so that 7 becomes \[\label{eq:32AS32explicit2} y^2+y=t_i^{-d_i}.\tag{8}\] Let \(u_i=yt_i^{\frac{d_i+1}{2}}\). Since \(y\) has a pole of order \(d_i\) at \(Q_i\), we see that \(u_i\) is a uniformizer of \(K_{Q_i}\). Multiplying 8 by \(t_i^{d_i+1}\) gives \[\label{eq:32AS32explicit3} t_i = u_i^2 + t_i^{\frac{d_i+1}{2}}u_i.\tag{9}\] Recursively expanding 9 gives \[t_i = u_i^2+\left( u_i^2 + t_i^{\frac{d_i+1}{2}}u_i\right)^{\frac{d_i+1}{2}}u_i=u_i^2 + u_i^{d_i+2} + O(u_i^{2d+2}).\qedhere\] ◻

Proof. From Lemma 2 we know \(\frac{t_i}{u_i^2} = 1 + O(u_i^{d_i})\). Thus, \(\frac{t_i^j}{u_i^{2j}} = 1 + O(u_i^{d_i})\) (e.g. use the generalized binomial theorem). The corollary follows by multiplying by \(u_i^{2j}\). ◻

Proof. By Riemann-Roch there exists a differential \(w'_{i,j}\) on \(X\) that is regular away from \(P_i\) and whose local expansion at \(P_i\) is \[w'_{i,j} = (t_i^{-j-1} + O(1))dt_i.\] The difference of two such differentials is regular everywhere, so the choice of \(w'_{i,j}\) is unique up to adding an element of \(H^0(X,\Omega_X^1)\). By Corollary 2 we know that \(t_i^{-j-1} + O(1) = u_i^{-2j-2} + O\left(u_i^{\min(0,-2j-2+d_i)}\right)\). Then by Lemma 2 we have \[\begin{align} \omega_{i,j} &= \left(u_i^{-2j-2} + O\left(u_i^{\min(0,-2j-2+d_i)}\right)\right)\left( c_i u_i^{d_i + 1} + O(u_i^{2d_i + 2})\right) du_i \\ &= \left(c_iu_i^{-2j+d_i-1} + O\left(u_i^{\min(d_i+1,-2j+2d_i-1)}\right) \right)du_i. \end{align}\] When \(j=1,\dots,\frac{d_i-1}{2}\) the minimum in the exponent is \(d_i+1\) and when \(j=\frac{d_i+1}{2}, \dots, d_i-1\) the minimum in the exponent is \(-2j+2d_i - 1\). Finally rescale so the leading term in \(\omega_{i,j}\) is monic. ◻

Proof. Since \(w_{i,j}\) is regular at \(P_{i'}\), we have \(w_{i,j} = O(1)dt_{i'}\). The corollary follows immediately from Lemma 2. ◻

Proof. For \(j=1,\dots,\frac{d_i-1}{2}\), we know from Lemma 3 and Corollary 3 that \(\omega_{i,j}\) is a global holomorphic differential on \(Y\), which gives the desired result. For \(j=\frac{d_i+1}{2},\dots,d_i-1\), recalling Definition 4 we need to prove that \[\begin{align} \omega_{i,j} &= [u_i^{-2j + d_i - 1} + O(1)]du_i. \end{align}\] But Lemma 3 shows that if \(-2j + 2d_i - 1 \geq 1\) then \[\begin{align} \omega_{i,j} &= [u_i^{-2j + d_i - 1} + O(u_i^{})]du_i \end{align}\] which suffices to give the result.

Since the differentials \(w_{i,j}\) on \(X\) are unique up to adding an element of \(\mathop{\mathrm{\mathrm{H}}}^0(X,\Omega_X^1)\), the \(S\)-enhanced differentials \(\widehat{\omega}_{i,j}\) are unique up to adding \(S\)-enhanced differentials of the form \((\pi^*\eta , (0, \ldots ,0))\), for \(\eta \in \mathop{\mathrm{\mathrm{H}}}^0(X,\Omega_X^1)\). ◻

Proof. Note first that \(V\) kills \(\widehat{\omega}_{i,j}^{\mathrm{bij}}\) since it is in the image of \(F\). Thus it suffices to show the lemma for the \(\widehat{\omega}_{i,j}\).

Since \(\omega_{i,j}\) is regular away from \(P_i\), so is \(V(\omega_{i,j})\). Locally at \(P_i\), we have \[V(w_{i,j}) = V((t_i^{-j-1} +O(1))dt_i) = \begin{cases} \left(t_i^{-\frac{j}{2}+1}+O(1)\right)dt_i & if j is even \\ O(1)dt_i &if j is odd. \end{cases}\] Pulling back along \(\pi\) and using Proposition 16, the local expansions show that \(V(\widehat{\omega}_{i,j}) - \omega_{i,j/2}\) is the pullback of a differential which is regular on \(X\). Noting that \(\widehat{\omega}_{i,j/2}= \widehat{\omega}_{i,j/2}^{\mathrm{nil}}\) since a regular differential is killed by \(F\), we obtain the formula in the lemma. It remains to be shown that we may assume that \(V\) acts nilpotently on \(\eta_{i,j}\) (and hence on \(\widehat{\omega}_{i,j}\)). For this, recall the decomposition of \(k[V]\)-modules from Fitting’s lemma \[\mathop{\mathrm{\mathrm{H}}}^0(X,\Omega_X^1) = \mathop{\mathrm{\mathrm{H}}}^0(X,\Omega_X^1)^{\mathrm{bij}} \oplus \mathop{\mathrm{\mathrm{H}}}^0(X,\Omega_X^1)^{\mathrm{nil}}\] and write \(\eta_{i,j} = \eta_{i,j}^{\mathrm{bij}} + \eta_{i,j}^{\mathrm{nil}}.\) There exists a differential \(\gamma_{i,j}\in \mathop{\mathrm{\mathrm{H}}}^0(X,\Omega_X^1)\) such that \(V(\gamma_{i,j})=\eta_{i,j}^{\mathrm{bij}}\). Recall that the differential \(w_{i,j}\) on \(X\) is chosen up to an element of \(\mathop{\mathrm{\mathrm{H}}}^0(X,\Omega_X^1)\), so we may replace \(w_{i,j}\) by \(w_{i,j}-\gamma_{i,j}\). Then we have \[V(w_{i,j}-\gamma_{i,j})=V(w_{i,j}) - V(\gamma_{i,j}) = \begin{cases} w_{i,j/2} + \eta_{i,j}^{\mathrm{nil}} &if j is odd \\ \eta_{i,j}^{\mathrm{nil}} &if j is even. \end{cases}\] Thus we may choose \(w_{i,j}\) such that \(V\) is nilpotent on \(w_{i,j}\) and therefore \(V\) is nilpotent on \(\widehat{\omega}_{i,j}^{\mathrm{nil}}\). ◻

Proof. Recall that we had defined \(\widetilde{\omega}_{i',j'} = \widehat{\omega}_{i',j'} - \widehat{\omega}_{i',j'}^{\mathrm{bij}}\). Since \(j \leq (d_i-1)/2\), we have \(\widetilde{\omega}_{i,j} = \widehat{\omega}_{i,j}\). By Proposition 18, \(V\) is nilpotent on \(\widehat{\omega}_{i,j}\), so let \(r\) be such that \(V^r(\widehat{\omega}_{i,j})=0\). Note there exists \(\delta_{i',j'}\) such that \(F^r(\delta_{i',j'})=\widehat{\omega}_{i',j'}^{\mathrm{bij}}\). Then observe \[\begin{align} \langle \widetilde{\omega}_{i,j} , \widetilde{\omega}_{i',j'} \rangle &= \langle \widehat{\omega}_{i,j} , \widehat{\omega}_{i',j'} - \widehat{\omega}_{i',j'}^{\mathrm{bij}} \rangle \\ &= \langle \widehat{\omega}_{i,j}, \widehat{\omega}_{i',j'} \rangle - \langle \widehat{\omega}_{i,j} , F^r(\delta_{i',j'}) \rangle \\ &= \langle \widehat{\omega}_{i,j} ,\widehat{\omega}_{i',j'} \rangle - \langle V^r(\widehat{\omega}_{i,j}), \delta_{i', j'} \rangle^{p^r} \\ &= \langle \widehat{\omega}_{i,j}, \widehat{\omega}_{i',j'} \rangle. \end{align}\] The rest of the proof is devoted to computing \(\langle \widehat{\omega}_{i,j}, \widehat{\omega}_{i',j'} \rangle\) making use of Theorem 11.

First assume \(i'\neq i\). Since \(j \leq \frac{d_i-1}{2}\), the class \(\widehat{\omega}_{i,j}\) only consists of a regular differential. If \(j' \leq \frac{d_{i'}}{2}\), then we immediately obtain \(\langle \omega_{i,j}, \omega_{i',j'} \rangle =0\). Otherwise, write \(\widehat{\omega}_{i',j'} = (\omega_{i',j'}, (0, \ldots , u_{i'}^{-2j'+d_{i'}}, \ldots ,0).\) Then the pairing is given by \[\langle \widehat{\omega}_{i,j}, \widehat{\omega}_{i',j'} \rangle = \mathrm{Res}_{Q_{i'}} (u^{-2j'+d_{i'}} \omega_{i,j}).\] Now, since \(w_{i,j}\) is regular at \(P_{i'}\), it follows from Lemma 2 that \(\mathrm{ord}_{Q_{i'}}(\omega_{i,j}) \geq d_{i'}+1,\) so that \[\mathrm{ord}_{Q_{i'}}(u^{-2j'+d_{i'}} \omega_{i,j}) \geq -2j'+d_{i'} + (d_{i'}+1) \geq 3,\] since \(j' \leq d_{i'}-1\). Therefore the residue vanishes.

Next we compute the shape of the pairing restricted to \(W_i\). For \(j' \leq \frac{d_i-1}{2}\), we know \(\omega_{i,j}\) and \(\omega_{i,j'}\) are regular differentials, and thus \(\langle \widehat{\omega}_{i,j}, \widehat{\omega}_{i,j'}\rangle=0\). When \(\omega_{i,j'}\) is not regular, we obtain \[\begin{align} \langle \widehat{\omega}_{i,j}, \widehat{\omega}_{i,j'}\rangle&= \mathrm{Res}_{Q_i}(u_i^{-2j'+d_i} \omega_{i,j}). \end{align}\] By Lemma 3 we have \[u_i^{-2j'+d_i} \omega_{i,j} = [u_i^{-2(j+j')+2d_i-1} + O(u_i^{-2j'+2d_i-1})]du_i\] Since \(j'\leq d_i-1\) we know that \(O(u_i^{-2j'+2d_i-1})du_i\) has zero residue. Thus, we see that \(u_i^{-2j'+d_i} \omega_{i,j}\) has residue one if \(j+j'=d_i\) and zero otherwise. ◻

Proof. The claims about the vector space decompositions and dimensions are elementary. Note that \(M\) and \(M_0\), being pullbacks of enhanced differentials on the base, are certainly preserved by \(V\). The other two spaces are stable under \(V\) by Proposition 18.

For the final statement about \(T_0\), note that the trace map \(\pi_* :\mathop{\mathrm{\mathrm{H}}}^0(Y,\Omega_Y^1) \to \mathop{\mathrm{\mathrm{H}}}^0(X, \Omega_X^1)\) is surjective. Hence, given any \(\omega_1 \in \mathop{\mathrm{\mathrm{H}}}^0(X,\Omega_X^1)\), let \(\eta\) be an element of \(\mathop{\mathrm{\mathrm{H}}}^0(Y,\Omega_Y^1)\) whose trace is \(\omega_1\). Note that the trace map commutes with \(V\). To see this, let \(y^2+y=\psi\) be the equation defining our \(\mathbf{Z}/2\mathbf{Z}\)-cover and note that \(\eta=\omega_0 + \omega_1 y\) where \(\omega_0\) is a meromorphic differential on \(X\). Then \(V(\pi_*(\eta))=V(\omega_1)\). On the other hand, we have \(V(\omega_1 y) = V(\omega_1)y + V(\omega_1\psi)\), so \(\pi_*(V(\eta))=V(\omega_1)\). It follows that \(\pi_{T_0}(V(\eta))=V(\omega_1)\) and thus the projection of \(V^n(T_0)\) to \(T_0\) has dimension \(l_X - a^n_X\). ◻

Proof. Given \(\eta \in \mathop{\mathrm{\mathrm{H}}}^0(X,\Omega_X^1)\) and \(\widetilde{\omega}_{i,j} \in W_i\), note \[\langle \pi^*\eta , \widetilde{\omega}_{i,j} \rangle = \text{Res}_{Q_i} (u_i^{-2j+d_i} \pi^*(\eta)).\] However, since \(\eta\) is regular at \(P_i\), Lemma 2 implies \(\mathrm{ord}_{Q_i} (\pi^* (\eta)) \geq d_i+1\). Furthermore, since \(j \leq d_i-1\), we obtain \(\mathrm{ord}_{Q_i}(u_i^{-2j+d_i} \pi^*(\eta)) \geq 3\) and the residue vanishes, as desired. ◻

Proof. Recall \(M = \pi^* \mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(X)\), so any class in \(M\) can be represented as \[\xi = (\pi^* (\omega), (\pi^*(f_1), \ldots ,\pi^*(f_m))),\] where \(f_i\) is the tail of a local function at \(P_i\). If \(\sum_{i=1}^m (d_i+1)/2 > 2g_X -2\), then for any \(n > (d_i+1)/2\) the Riemann-Roch theorem gives a function \(h\) on \(X\) which is regular away from \(\{P_1,\ldots, P_m\}\) with a pole of order exactly \(n\) at \(P_i\) and a pole of order at most \((d_j+1)/2\) at \(P_j\) when \(j \neq i\). By modifying the de Rham class representative using these functions (i.e. replacing \(\xi\) with \(\xi + dh\)), we may assume that \[\mathrm{ord}_{Q_i}(\pi^*(f_i)) = 2\mathrm{ord}_{P_i}(f_i) \geq - (d_i+1)\] for every \(i\). By Corollary 2, we may write \[f_i = \sum_{l=\frac{d_i+1}{2}}^0 c_l t_i^{-l} + O(1)\quad \text{and} \quad \pi^*(f_i) = \sum_{l=-\frac{d_i+1}{2}}^0 \left(c_lu_i^{2l} + O\left(u_i^{2l+d_i}\right) \right). \label{eq:32pi944240f95i41}\tag{13}\] In particular, \(\mathrm{ord}_{Q_i}(f_i) \geq - (d_i+1)\).

If \(\eta\in \mathop{\mathrm{\mathrm{H}}}^0(X,\Omega_X^1)\) then Lemma 2 and the bound on the order show that \(\pi^*(f_i \eta)\) is regular at \(Q_i\) and hence \[\langle \pi^*(\eta), \xi \rangle = \sum_{i=1}^m \mathrm{Res}_{Q_i}(\pi^*(f_i) \pi^*(\eta)) = 0.\] Hence \(M\) pairs trivially with \(M_0\).

We now show that any \(\widetilde{\omega}_{i,j} \in W \cap L_0\) pairs trivially with \(\xi\). (Necessarily \(j \leq (d_i-1)/2\).) Note \[\langle \widetilde{\omega}_{i,j}, \xi \rangle = \sum_{i'=1}^m \mathrm{Res}_{Q_{i'}} (\pi^*(f_{i'}) \omega_{i,j}).\] The terms with \(i' \neq i\) are zero, as \(\pi^*(f_{i'}) \omega_{i,j}\) is regular at \(Q_{i'}\) by Equation 13 and Lemma 2. Combining the local expansion \(\omega_{i,j} = \left[u_i^{-2j+d_i-1} + O\left(u_i^{d_i+1}\right)\right ]du_i\) at \(Q_i\) from Lemma 3(ii) with Equation 13 gives \[\pi^*(f_i) \omega_{i,j} = \sum_{l=\frac{d_i+1}{2}}^0 \left(c_lu_i^{-2l} + O\left(u_i^{-2l+d_i}\right) \right)\left(u_i^{-2j+d_i-1} + O\left(u_i^{d_i+1}\right)\right)du_i,\] which does not have a \(u_i^{-1}\) term since \(d_i\) is odd. Therefore the residue vanishes, so \(M\) pairs trivially with a basis for \(W \cap L_0\). ◻

Proof. This follows immediately from Proposition 18 and the observation that, since \(X\) is ordinary, there are no nonzero elements of \(\mathop{\mathrm{\mathrm{H}}}^0(X,\Omega_X^1)\) on which \(V\) acts nilpotently. ◻

Proof. Appealing to Equation @{eq:eq: k[F,V] decomposition ordinary rough} , we need to show that \(F\) and \(V\) preserve each \(W_i\). Lemma 7 shows that \(V\) preserves \(W_i\), so it remains to be shown that \(F\) preserves \(W_i\). Equation @{eq:eq: k[F,V] decomposition ordinary rough} gives that \(F(W_i) \subseteq W\), so it suffices to check that the projection of \(F(\omega_{i,j})\) to \(W_{i'}\) is zero when \(i'\neq i\). For this, let \(j'\) be arbitrary. Then using Proposition 20 we have \[\langle \widetilde{\omega}_{i',j'} , F(\widetilde{\omega}_{i,j}) \rangle = \langle V(\widetilde{\omega}_{i',j'} ), \widetilde{\omega}_{i,j}\rangle^p =0,\] since \(V(\widetilde{\omega}_{i',j'})\) is a regular differential in \(W_{i'}\) by Lemma 7. Since \(F(\widetilde{\omega}_{i,j})\) pairs trivially with each \(\widetilde{\omega}_{i',j'}\) when \(i \neq i'\) and the pairing is non-degenerate when restricted to \(W_{i'}\), it follows that the projection of \(F(\widetilde{\omega}_{i,j})\) to \(W_{i'}\) is zero. ◻

Proof. The first case is immediate since \(F\) kills regular differentials. By Lemma 8, \(F(\widetilde{\omega}_{i,j})\) lies in \(W_i\). Moreover, for any \(1 \leq j \leq d_i-1\), Lemma 7 and Proposition 20 yield \[\langle \widetilde{\omega}_{i,j'}, F(\widetilde{\omega}_{i,j}) \rangle = \langle V(\widetilde{\omega}_{i,j'}), \widetilde{\omega}_{i,j} \rangle ^p = \begin{cases} 1 & if \frac{j'}{2} + j =d_i \\ 0 & otherwise. \end{cases}\] Rearranging the condition gives \(j'=2d_i-2j\), so by Proposition 20 we obtain \[F(\widetilde{\omega}_{i,j}) = \widetilde{\omega}_{i,2j-d_i}. \qedhere\] ◻

Proof. Recall the subspaces \(W_{i,l} = \mathrm{span}_k\{ \widetilde{\omega}_{i,j} \; | \; j \leq l\}\) from Definition 9. Note that \(\dim_k W_{i,l} = l\) by construction. Lemma 7 and Lemma 9 establish that the filtration \[0 \subset W_{i,1} \subset \cdots \subset W_{i,d_i-1} = W_i\] is a final filtration. Finally, Lemma 7 yields \(V(N_{i,l})=N_{i,\lfloor l/2 \rfloor}\), giving the desired result. ◻

Proof. Using Lemma 8, we get isomorphisms of Dieudonné modules \[D(\text{Jac}(Y)[p]) = \mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(Y) = U \oplus Z \oplus \bigoplus_{i=1}^m W_i.\] Recalling Lemma 1, we observe that \(D((\underline{\mathbb{Z}/2\mathbb{Z}})^{f_Y}) \cong U\), \(D(\mu_2^{f_Y}) \cong Z\), and \(D(G_{d_i})=W_i\). ◻

Proof. Recall that \(V\) is bijective on \(U\), that \(V(Z)=0\), and that \(U^\perp = U \oplus W\) while \(W^\perp = U \oplus Z\). By Lemma 7, \(V^n(W_{i,l})=W_{i,\left \lfloor l/2^n \right \rfloor}\) and by Proposition 20 \(W_{i,l}^\perp \cap W = W_{i,d_i-1-l} \oplus \bigoplus_{i'\neq i} W_{i'}\). This establishes the first claim. Lemma 8 gives a decomposition compatible with \(V\), so induction gives \[w(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(Y)) = U \oplus \bigoplus_{i=1}^m W_{\phi(d_i,w)}.\] The second result follows as \(\dim_k(W_{i,\phi(d_i,w)}) = \phi(d_i,w)\), by definition \(\phi(w)=\sum_{i=1}^m \phi(d_i,w)\) and finally \(f_Y = \dim_k U\). ◻

Proof. The Ekedahl-Oort type of \(Y\) is determined by the final type \(\nu_l = \dim_k (V(N_l))\), where \(\{N_l\}\) is any final filtration. Any final filtration is a refinement of the canonical filtration \(\{ w(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(Y))\}\), where \(w\) ranges over all words in \(V\) and \(\perp\). The final type is determined by the values \(\dim_k (w(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(Y)))\), which are given in Lemma 11. ◻

Proof. Proposition 18 almost gives the desired behavior, except the images under \(V\) may include the pullback of a differential regular on \(X\), i.e. a scalar multiple of \(\beta_1 = (dx,0)\). First note that \(\widetilde{\omega}_1\) may be chosen to be the regular differential \((xdx,0)\), which has local expansion \((t^{-2} + O(1) )dt\). Then \(V(\widetilde{\omega}_1) = \beta_1 = (dx,0)\) as desired. Since \(V(\beta_1)=0\) we may freely modify the other \(\widetilde{\omega}_{i,j}\) by adding a multiple of \(dx\) without changing the image under \(V\) to arrange the desired formulas. ◻

Proof. A basis for \(M_n\) is given by \(\{v_1,\ldots,v_n\}\) where \(v_i = x^{-(i+1)} dx\). Thus, it is clear that \(V^r(M_n) = M_{\lfloor n/2^r\rfloor }\), which gives the \(V\)-type. ◻

Proof. From Corollary 5, we know that for \(r=0,\dots,r_n+1\) we have \(V^r(R_n)=R_{\lfloor n/2^r \rfloor}\), so that \(c_r(R_n)=\left \lfloor \frac{n}{2^r} \right \rfloor + 1\). For \(r> r_n+1\) we have \(V^r(R_n)=V(R_0)=0\). ◻

Proof. We know \(H^0(Y,\Omega_Y^1)\) has dimension \(\dim_k(R_\delta)+1 = \delta +2 = (d+3)/2\). Thus, it is enough to show \(\omega_T \not \in R_\delta\). This follows by observing that \(\omega_T\) is not fixed by \(\mathrm{Gal}(Y/X)\), while \(R_\delta\) is fixed by the Galois action. ◻

Proof. This is an easy consequence of Riemann-Roch using that \(X\) has genus one. ◻

Proof Of Theorem 26. We know that \(H^0(Y,\Omega_Y^1) \cong R_\delta \oplus k \cdot \omega_T\), so that \[V^r(H^0(Y,\Omega_Y^1))=V^r(R_\delta) + k \cdot V^r(\omega_T).\] Thus, we see that \(\iota(H^0(Y,\Omega_Y^1))= \iota(R_\delta) + u({\mu_{\psi}})\) where \(\mu_\psi\) is the largest \(r\) such that \(V^r(\omega_T)\not\in V^r(R_\delta)\). We see \[V(\omega_T)=V((z^2 +\psi) dx ) = V(\psi dx),\] so we are looking for the largest value \(r\) such that \(V^r(\psi dx)\not\in V^r(R_\delta)\). By Lemma 15 we know \[\psi dx = c_{d-1}\widetilde{\omega}_{d-1} + \dots + c_1\widetilde{\omega}_1 + c_0 dx,\] where \(c_{d-1}\neq 0\). In particular, we have \[\label{eq:32definition32of32gamma} \psi dx \equiv \underbrace{c_{d-1} \widetilde{\omega}_{d-1} + \dots + c_{\delta+1}\widetilde{\omega}_{\delta+1}}_{\gamma} \mod R_{\delta}.\tag{17}\] From Proposition/Definition 25 it is clear that \(V^r(\psi dx)\in V^r(R_\delta)\) if and only if \(V^r(\gamma)=0\). Thus, \[\label{eq:32max32order32V} \mu_\psi = \max \{\mathrm{ord}_V(c_i\widetilde{\omega}_i)\}_{i=\delta+1,\ldots,d-1} -1.\tag{18}\] Let \(n\) be the largest power of two less or equal to \(d-1\), i.e. \(n = 2^{r_{d-1}}\). We consider two cases:

1. Suppose \(c_{n} \neq 0\). Then \(\mathrm{ord}_V(c_{n}\widetilde{\omega}_{n}) = r_{d-1} + 2\) from Corollary 6. We claim this is the unique maximal term in the right side of 18 . Consider \(i\neq n = 2^{r_{d-1}}\) satisfying \(d-1 \geq i >\delta\). By the definition of \(\delta\) and \(r_{d-1}\), we know that \(i\) is not a power of two. Thus, \(i=2^km\) where \(m >1\) and \(k<r_{d-1}\). Then by Corollary 6 we find \(\mathrm{ord}_V(c_i\widetilde{\omega}_i) \leq k+1< r_{d-1}+1\). It follows that \(\mu_\psi=r_{d-1}+1\).

2. Suppose \(c_{n} = 0\). As in the previous case, the non-trivial terms in 18 satisfy \(\mathrm{ord}_V(c_i\widetilde{\omega}_i)<r_{d-1}+1\), . This gives an upper bound of \(\mu_\psi < r_{d-1}\). Furthermore, since \(c_{d-1} \neq 0\) the maximum in 18 must be at least \(v_2(d-1)+1\). Thus, we have \(v_2(d-1) \leq \mu_\psi < r_{d-1}\). To see that each possibility occurs, let \(\mu\) satisfy \(v_2(d-1) \leq \mu < r_{d-1}\). We can find \(i\) satisfying \(d-1 \geq i >\delta\) with \(v_2(i)=\mu\). Then choose \(\psi\) so that \(\psi dx= \widetilde{\omega}_{d-1} + \widetilde{\omega}_i\).

 ◻

Proof. For the first, recall that as \(z^2 = z + \psi\) we have \[\label{eq:32extra32differential32on32bottom32curve} V(\omega_T) = (V(z^2dx + \psi dx) ,0) = (V(\psi dx),0).\tag{19}\] Thus the local expansion of \(\psi\) at \(Q\) controls \(V(\omega_T)\). In particular, if \(\psi = \sum_{i=1}^{d} c_i t^{-i} + O(1)\) (with \(c_d \neq 0\)) then \[V(\psi dx) = \left( \sum_{j=1}^{(d+1)/2} c_{2j-1}^{1/2} t^{-j} + O(1) \right) dt.\] Thus \(V(\omega_T)\) is a \(k\)-linear combination of \(\beta_1, \widetilde{\omega}_1 ,\ldots, \widetilde{\omega}_{(d-1)/2}\), and recalling the local expansions in Lemma 3 we conclude the coefficient of \(\widetilde{\omega}_{(d-1)/2}\) must be nonzero. By similar reasoning, the coefficient of \(\widetilde{\omega}_i\) is nonzero if and only if \(c_{2i+1} \neq 0\).

The second follows from Proposition 20, and the third from Lemma 5 and Lemma 6. The last statement will follow from the second and third. Since the pairing is non-degenerate and \(\beta_1\) pairs trivially with all regular differentials as well as all the \(\widetilde{\omega}_i\) and \(\beta_2\), it must pair nontrivially with \(\omega_T'\). Thus when writing \(\omega\) in the orthogonal complement in terms of the basis, they cannot have a \(\omega_T'\) term. Furthermore, if \(i \leq (d-1)/2\) then \(\widetilde{\omega}_i\) pairs trivially with all basis elements except possibly for \(\omega'_T\) and \(\widetilde{\omega}_{d-i}\) (it is orthogonal to all regular differentials as it is regular). Thus if \(\omega\) is orthogonal to \(\beta_1\) and \(\widetilde{\omega}_i\) then the \(\omega_T'\) and \(\widetilde{\omega}_{d-i}\) terms are zero as desired. ◻

Proof. The claim about \(V(U_i)\) follows from the definition using Proposition 25. Now let us consider \(U_i^\perp\), and let \(j = j(i)\) to simplify notation. By Lemma 16, \(U_i^\perp\) is a codimension one subspace of \[Z(\omega'_T,\widetilde{\omega}_{d-1},\ldots, \widetilde{\omega}_{d-i}) = \mathrm{span}_k\{ \beta_1,\beta_2,\widetilde{\omega}_1,\ldots,\widetilde{\omega}_{d-1-i},\omega_T\}.\] consisting of elements \(\omega\) orthogonal to \(V^{j}(\omega_T)\). Any regular differential is automatically orthogonal to \(\omega_T\), but \(\widetilde{\omega}_{m}\) need not be for \(m > (d-1)/2\). Now \(\widetilde{\omega}_{(d-1)/2}^*(V(\omega_T)) = \widetilde{\omega}_{2^{n-1}}^*(V(\omega_T)) \neq 0\) so \(\widetilde{\omega}^*_{2^{n-j}}( V^j(\omega_T)) \neq 0\), so the orthogonality relation becomes a linear relation of the form \[c_{d-2^{n-j}} \widetilde{\omega}_{d-2^{n-j}}^*(\omega) + \ldots = 0\] where \(c_{d-2^{n-j}}\) is known to be nonzero. This means that \(U_i^\perp\) contains an element of the form \(c' \widetilde{\omega}_{d-2^{n-j}} + \widetilde{\omega}_m\) for each \(m\). Since \(d\) is odd, \[V(c' \widetilde{\omega}_{d-2^{n-j}} + \widetilde{\omega}_m) = V(\widetilde{\omega}_m).\] Then using Proposition 25 we conclude that \[V(U_i^\perp) = U_{i''}\]

For the claim about dimensions, remember that \(\widetilde{\omega}^*_{2^{n-j}}( V^j(\omega)) \neq 0\). Thus \(V^j(\omega)\) is not in the span of the linearly independent \(\beta_1 , \widetilde{\omega}_1 ,\ldots, \widetilde{\omega}_i\).

Finally, observe that \(U_{-1} \subset U_0 \subset U_1\). If \(j(m)=j(m-1)\), then it is clear by definition that \(U_{m-1}\subset U_m\). Otherwise \(m=2^r\), in which case \(V^{n-r}(\omega_T) \in \mathrm{span}_k\{ \beta_1,\widetilde{\omega}_1,\ldots,\widetilde{\omega}_{m} \}\) since \(\mathrm{ord}_Q(V^{n-r}(\omega_T)) = - 2^{r} - 1\). ◻

Proof of Theorem 29. We claim that \[0 \subset U_{-1} \subset U_0 \subset U_1 \subset \ldots \subset U_{(d-1)/2} = H^0(Y) \subset U_{(d-1)/2-1}^\perp \subset \ldots \subset U_{-1}^\perp \subset \mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(Y)\] is the final filtration for \(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(Y)\). Lemma 17 shows that this is a filtration, that it is preserved by \(V\) and \(\perp\), and furthermore gives the dimensions. As \(\dim_k V(U_i) = \lfloor i/2 \rfloor +2\) for \(i > 0\), we conclude that the final type is \([0,1,2,3,3,\ldots,(d-1)/4+1 ,(d-1)/4+1,(d-1)/4+2 ]\). ◻

Proof of Corollary 1. The first family, constructed as a cover of \(\mathbf{P}^1\), was analyzed in Remark 23. The second family is similar, constructed as a cover of the supersingular elliptic curve \(X\) ramified over the point \(Q\) at infinity, with ramification break \(d\), and relies on Theorem 29. The only non-obvious point is the computation of the dimension of the family. The extension of function fields is generated by adjoining a root of \(z^2 + z = f\), where \(f\) is a rational function on \(X\) which is regular except at infinity. The function \(f\) is unique up to adding functions of the form \(g^2+g\).

By the Riemann-Roch theorem, there exists a rational function on \(X\) which is regular except at infinity and has a pole of order \(n\) at \(Q\) for any \(n >1\). Since \(d>1\), we may assume that the order of \(f\) at \(Q\) is exactly \(d\) by modifying \(f\) by a function of the form \(g^2 + g\). Similarly, we assume that the coefficient of \(t^{-2i}\) in the local expansion of \(f\) at \(Q\) is zero for \(i>1\), and that there is no constant term in the local expansion. Now, by Riemann-Roch, \(\dim_k H^0(X,\mathcal{O}_X(d Q)) = d\), so the space of possible \(f\) which give non-isomorphic extension of the function field has dimension \(d-1-(d-3)/2 = (d+1)/2\). There is a zero-dimensional space of automorphisms of \(X\) which fix the point at infinity, so there is also a \((d+1)/2\)-dimensional family of curves with constant Ekedahl-Oort type. ◻

Proof. We argue by induction on the length of \(w\). The equation holds for \(w=V\), since \(V\) preserves the three summands \(U\), \(Z\) and \(L\), and we have \(V(U)=U\) and \(V(Z)=0\). Now assume it holds for all simple words that have up to \(n\) letters and consider a word \(w\) of \(n+1\) letters. If \(w\) starts with \(V\), write \(w=Vw'\) and observe \[\begin{align} w(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(Y)) &= V(w'(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(Y))) = V\left(U \oplus (w'(L) \cap L) \right) = V(U) \oplus V(w'(L)\cap L) \\ &= U \oplus (V(w'(L)) \cap L) = U \oplus (w(L) \cap L) \end{align}\] as desired. On the other hand, if \(w\) starts with \(\perp\), write \(w=\perp w'\) and observe \[\begin{align} w(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(Y)) &= w'(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(Y))^\perp = (U \oplus (w'(L) \cap L))^\perp = U^\perp \cap (w'(L) \cap L)^\perp \\ &= (U \oplus L) \cap (U\oplus Z \oplus (w'(L)^\perp \cap L)) = U \oplus (w'(L)^\perp \cap L) = U \oplus (w(L) \cap L).\qedhere \end{align}\] ◻

Proof. The proof is similar to the proof of Lemma 11. For the first claim let \(\widetilde{\omega}_{i,j} \in W_w\). Then by Proposition 18 and Lemma 4 we have \(\pi_W(V(\widetilde{\omega}_{i,j}))=\widetilde{\omega}_{i,j/2}\) for \(j\) even and \(0\) for \(j\) odd. Thus,

\[\pi_W(V(W_w)) = \pi_W\left( V\left( \bigoplus_{i=1}^m W_{i,\phi(d_i,w)} \right) \right) = \bigoplus_{i=1}^m W_{i,\left \lfloor \phi(d_i,w)/2 \right \rfloor} = \bigoplus_{i=1}^m W_{i,\phi(d_i,Vw)} = W_{Vw} .\]

The second claim follows from Proposition 20 and Lemma 4. More precisely, we obtain \[\pi_W(W_w^\perp) = \pi_W\left( \bigcap_{i=1}^m W_{i,\phi(d_i,w)}^\perp \right) = \bigoplus_{i=1}^m W_{i,d_i-1-\phi(d_i,w)} = \bigoplus_{i=1}^m W_{i,\phi(d_i,\perp w)} = W_{\perp w}. \qedhere\] ◻

Proof. We begin by noting that \[\label{eq:32upper32to32lower322} V^n( w(L))^\perp \supseteq V^n(M \oplus T_0 \oplus W_w^U)^\perp = V^n(M \oplus W_w^U)^\perp \cap V^n(T_0)^\perp.\tag{22}\] Now \(V^n(M \oplus W_w) \subseteq M_0 \oplus W_{V^n w}\) and the inclusion \(V^n(M \oplus W_w) \subseteq V^n(M \oplus W_w^U)\) has codimension at most \(c\). Hence there exists a space \(W_{V^n w}^U\) containing \(W_{V^n w}\) with codimension at most \(c\) such that \(V^n(M \oplus W_w^U) \subseteq M_0 \oplus W_{V^nw}^U\). Taking symplectic complements gives \[V^n(M \oplus W_w^U)^\perp \supseteq (M_0 \oplus W_{V^nw}^U)^\perp = M_0^\perp \cap W_{V^n w}^{U ,\perp}.\] By Lemma 5, we know \(M_0^\perp \supseteq M_0 \oplus W \oplus T_0\). Furthermore, \(W_{V^n w}^{U ,\perp}\) is contained in \(W_{V^n w}^\perp\) with codimension at most \(c\). Projecting to both spaces to \(W\), we see that \(\pi_W(W_{V^n w}^{U ,\perp})\) is contained in \(\pi_W(W_{V^n w}^\perp)= W_{\perp V^n w}\) with codimension at most \(c\). Thus \(W_{\perp V^n w}^L\colonequals \pi_W(W_{V^n w}^{U ,\perp} ) \cap V^n(T_0)^\perp\) satisfies Equation 21 and has codimension at most \(c+\dim_k(V^n(T_0)) \leq c+l_X\) inside \(W_{\perp V^n w}\). Since \(W_{V^n w}^U\) and \(V^n(T_0)\) both consist of regular differentials, it follows that \(W_V \subseteq \pi_W(W_{V^n w}^{U ,\perp} ) \cap V^n(T_0)^\perp = W_{\perp V^n w}^L\). ◻

Proof. We begin by defining \[\label{eq:def32wU} W^U_{\bot V^nw} \colonequals \pi_{T_1 \oplus W}(V^n(M_0 \oplus T_0 \oplus W_w^L)^\perp) + W_{\bot V^n w} .\tag{24}\] Observe that

• \(W^U_{\bot V^nw} = \left( \pi_{T_1 \oplus W}(V^n(M_0 \oplus W_w^L)^\perp) \cap \pi_{T_1 \oplus W}(V^n(T_0)^\perp) \right) + W_{\bot V^n w}\);

• \(W_{\bot V^n w} \subseteq W^U_{\bot V^n w}\); and

• \(W^U_{\bot V^nw} \subseteq T_1 \oplus W\).

To check 23 , note that since \(w(L) \supseteq M_0 \oplus T_0 \oplus W_w^L\) we know that \[\label{equation:32Vnwlperp} V^n(w(L))^\perp \subseteq V^n(M_0 \oplus T_0 \oplus W_w^L)^\perp = V^n(M_0 \oplus W_w^L)^\perp \cap V^n(T_0)^\perp.\tag{25}\] Now recalling Equation 12 , observe that \[\label{eq:first32projection} V^n(M_0 \oplus W_w^L)^\perp = M_0 \oplus T_0 \oplus \pi_{M_1 \oplus T_1 \oplus W} (V^n(M_0 \oplus W_w^L)^\perp) \subseteq M \oplus T_0 \oplus \pi_{T_1 \oplus W} (V^n(M_0 \oplus W_w^L)^\perp) .\tag{26}\] Similarly as \(V^n(T_0)^\perp \supseteq M_0 \oplus T_0\) we see \[V^n(T_0)^\perp = M_0 \oplus T_0 \oplus \pi_{M_1 \oplus T_1 \oplus W}(V^n(T_0)^\perp) \subseteq M \oplus T_0 \oplus \pi_{T_1 \oplus W}(V^n(T_0)^\perp).\] Therefore Equation 25 gives that \(V^n(w(L))^\perp \subseteq M \oplus T_0 \oplus W^U_{\bot V^nw}\) as desired.

It remains to bound the codimension of \(W_{\bot V^n w} \subseteq W_{\bot V^n w}^U\). We begin by noting \(V^n(M_0)\) has dimension \(l_X-a_X^n\) by the rank-nullity theorem. Moreover, using Lemma 19 yields \[\dim_k(\pi_W(V^n(W_w))) = \dim_k (W_{V^nw}) = \sum_{i=1}^m \phi(d_i,V^n w)=\phi(V^n w).\] It follows that \[\begin{align} \dim_k (V^n(M_0 \oplus W_w^L)) &\geq (l_X-a_X^n) + \phi(V^n w)-c, \intertext{so that} \dim_k ( V^n(M_0 \oplus W_w^L)^\perp) &\leq 2l_Y - (l_X-a_X^n) - \phi(V^n w)+c \\ &= 4l_X + \sum_{i=1}^m (d_i-1) - (l_X-a_X^n) - \sum_{i=1}^m \phi(d_i,V^n w)+c \\ &= 3l_X + a_X^n + \phi(\bot V^n w) + c. \end{align}\] The last step uses that \(\phi(d_i,\bot V^n w) = d_i-1-\phi(d_i,V^n w)\). Then Equation 26 gives that \[\label{eq:32upper32bound32general32case} \dim_k (\pi_{T_1 \oplus W}(V^n(M_0 \oplus W_w^L)^\perp)) \leq l_X + a_X^n + \phi(\bot V^n w) + c\tag{27}\] Letting \(c'\) be the codimension of \(\pi_{T_1 \oplus W}(V^n(M_0 \oplus W_w^L)^\perp) \cap V^n(T_0)^\perp \subseteq \pi_{T_1 \oplus W}(V^n(M_0 \oplus W_w^L)^\perp)\), we conclude that \[\dim_k (\pi_{T_1 \oplus W} (V^n(M_0 \oplus T_0 \oplus W_w^L)^\perp)) \leq \phi(\perp V^n w) + l_X + a_X^n +c - c'.\] Furthermore, as \(W_{\bot V^n w} \subseteq \pi_{T_1 \oplus W}(V^n(M_0 \oplus W_w^L)^\perp)\) we see that \[\mathrm{codim}_k\left(W_{\bot V^n w} \cap V^n(T_0)^\perp \subseteq W_{\bot V^n w} \right) \leq c'\] Letting \(Z = \pi_{T_1 \oplus W} (V^n(M_0 \oplus T_0 \oplus W_w^L)^\perp)\), we estimate \[\begin{align} \mathrm{codim}_k(W_{\bot V^n w} \subseteq W_{\bot V^n w}^U) &\leq \dim_k Z - \dim_k( W_{\bot V^n w} \cap Z)\\ &= \dim_k Z - \dim_k( W_{\bot V^n w} \cap V^n(T_0)^\perp)\\ &\leq \left( \phi(\bot V^n w) + l_X + a_X^n +c - c' \right)\\ & - \left( \phi(\bot V^n w) - \mathrm{codim}_k\left(W_{\bot V^n w} \cap V^n(T_0)^\perp \subseteq W_{\bot V^n w} \right) \right) \\ &\leq c+l_X+a_X^n. \end{align}\] This gives the general claim about the codimension.

We now prove the sharper upper bound under the assumption \(\sum_{i=1}^m (d_i+1) > 4g_X-4\). Under this assumption, Lemma 6 yields \[M \subseteq (M_0 \oplus W_V)^\perp \subseteq V^n(M_0 \oplus W_w^L)^\perp.\] As a result, the inclusion in Equation 26 becomes an equality: \[V^n(M_0 \oplus W_w^L)^\perp = M \oplus T_0 \oplus \pi_{T_1 \oplus W} (V^n(M_0 \oplus W_w^L)^\perp).\] After computing the dimension of the left-hand side in the same way as before, we now get \[\begin{align} \dim_k (\pi_{T_1 \oplus W}(V^n(M_0 \oplus W_w^L)^\perp)) &= \dim_k (V^n(M_0 \oplus W_w^L)^\perp) - \dim_k (M \oplus T_0) \\ &= \dim_k (V^n(M_0 \oplus W_w^L)^\perp) - 3l_X \\ &\leq a_X^n + \phi(\bot V^n w) + c. \end{align}\] Comparing this to Equation 27 , the bound has decreased by \(l_X\) using the assumption \(\sum_{i=1}^m (d_i+1) >4g_X -4\). From there on, the same argument building on Equation 27 shows that \[\mathrm{codim}_k (W_{\bot V^n w} \subseteq W_{\bot V^n w}^U) \leq c+a_X^n.\qedhere\] ◻

Proof. The proof is by induction on \(t\).

To make the base case easier, we write \(\bar{w}\colonequals \bot V^{n_t} \bot \ldots V^{n_1-1} \bot V\). Note that \(\bar{w}(L) = w(L)\) since \(V(L) = V(L)^\perp=L_0\). This reduces us to the base case \(w=\bot V\): \[w(L) = V(L)^\perp = V(L) = L_0 = M_0 \oplus T_0 \oplus W_V.\] Thus for \(w=\bot V\) we may set \(W_w^L = W_w^U = W_w\), so both codimensions equal zero.

For the induction step, assume the statement holds for a simple word \(w\) containing \(t\) instances of \(\bot\), and let \(n>0\). Using \(W^U_w\), Lemma 20 gives a space \(W_{V^n \bot w}^L\), which has codimension at most \(\left \lceil \frac{3t}{2} \right \rceil l_X + l_X = \left \lfloor \frac{3(t+1)}{2} \right \rfloor l_X\) inside \(W_{\bot V^n w}\). If we additionally assume \(\sum_{i=1}^m (d_i+1) > 4g_X-4\), then the codimension is at most \(tl_X + l_X = (t+1)l_X\).

Analogously, using Lemma 21 gives a space \(W_{\bot V^n w}^U \subseteq T_1 \oplus W\), in which \(W_{\perp V^n w}\) has codimension at most \[\left \lfloor \frac{3t}{2} \right \rfloor l_X + l_X + a_X^n \leq \left \lfloor \frac{3t}{2} \right \rfloor l_X + 2l_X = \left \lceil \frac{3(t+1)}{2} \right \rceil l_X,\] as desired. If we additionally assume \(\sum_{i=1}^m (d_i+1) > 4g_X -4\), then the codimension is at most \(tl_X + a^n_X \leq t l_X + l_X = (t+1)l_X\). ◻

Proof. This is an adaptation in characteristic \(2\) of the proof of [29], with the difference that we no longer require \(\eta\) to be regular. The decomposition \(\eta=\omega_0 + \omega_1 y\) has implicitly fixed an equation \(y^2+y=\psi\) defining \(Y\). Now observe \[V(\eta) = V(\omega_0) + V(\omega_1 y) = V(\omega_0) + V(\omega_1(y^2+\psi)) = V(\omega_0) + V(\psi \omega_1) + y V(\omega_1) = 0,\] so that \(V(\omega_1)=0\) and \(V(\omega_0) = V(\psi \omega_1)\). Since \(\mathrm{ord}_{P_i}(\omega_0) \geq -n\), it follows that \[\label{eq:32BC8463} \mathrm{ord}_{P_i}(V(\psi \omega_1)) = \mathrm{ord}_{P_i}(V(\omega_0)) \geq - \left \lfloor \frac{n+1}{2} \right \rfloor.\tag{29}\] Now, we may pick a uniformizer \(z_i \in \widehat{\mathcal{O}}_{X,P_i}\) such that \(\psi = c^2z_i^{-d_i}\) for some \(c \in k^\times\). Since \(V(\omega_1)=0\), there exists a function \(f_1\in \widehat{\mathcal{O}}_{X,P_i} \cong k[\![z]\!]\) such that \(\omega_1 = f_1^2 dz_i\). Then we obtain \[V(\psi \omega_1) = V(c^2f_1^2 z_i^{-d_i}z_i) = cf_1z_i^{-\frac{d_i+1}{2}} dz_i.\] Combining this with Equation 29 yields \[\begin{align} \mathrm{ord}_{P_i}(f_1) &\geq \frac{d_i+1}{2} - \left \lfloor \frac{n+1}{2} \right \rfloor = \left \lceil \frac{d_i-n}{2} \right \rceil, \intertext{and therefore} \mathrm{ord}_{P_i}(\omega_1) &= \mathrm{ord}_{P_i}(f_1^2 dz_i) \geq 2 \left \lceil \frac{d_i-n}{2} \right \rceil. \qedhere \end{align}\] ◻

Proof. Since \((\eta,(f_i))\) is in \(M_0 \oplus T_0 \oplus W_w\), we can write \(\eta=\omega_0 + \omega_1 y\), where \(\omega_1\) is regular. Recalling Definitions 9 and 17, note \(\mathrm{ord}_{P_i}(\omega_0) \geq -\phi(d_i,w) - 1\) for every \(i\). Then Lemma 22 yields that \[\mathrm{ord}_{P_i}(\omega_1) \geq 2 \left \lceil \frac{d_i-1-\phi(d_i,w)}{2} \right \rceil = 2 \left \lceil \frac{\phi(d_i,w')}{2} \right \rceil.\] Thus, if we consider the effective divisor \[E= \sum_{i=1}^m 2 \left \lceil \frac{\phi(d_i,w')}{2} \right \rceil [P_i],\] we see \(\pi_*(\eta) = \omega_1 \in H^0(X,\Omega_X^1(-E))=0\) since \(\deg(E)>2g_X-2\) by Equation 30 . ◻

Proof. We use the lower bound from Proposition 31 to the word \(\bot w'\), which implies \[w(L) = V^r\left(w'(L)^\perp\right) \supseteq V^r\left(M_0 \oplus W_{\perp w'}^L \oplus T_0 \right)= V^r(M_0 \oplus W_{\perp w'}^L) + V^r(T_0).\] where the subspace \(W_{\perp w'}^L\) satisfies \[\mathrm{codim}_k(W_{\perp w'}^L \subseteq W_{\perp w'}) \leq \begin{cases} \left \lfloor \frac{3t}{2} \right \rfloor l_X &if \sum_{i=1}^m (d_i+1) \leq 4g_X -4\\ tl_X &if \sum_{i=1}^m (d_i+1) > 4g_X-4 , \end{cases}\] so that \(\dim_k (\pi_W(V^r(W_{\perp w'}^L))) \geq L_2(X,\pi,w)\) by Lemma 19. Note that \(\dim_k(V^r(M_0)) =l_X-a_X^r\). If furthermore \(\deg(\lfloor D/2^r\rfloor ) \geq n(X)\) then Tango’s theorem (see [28] and [4]) implies that \[M_0 \subseteq V^r(M_0 \oplus W_V) \subseteq V^r(M_0 \oplus W_{\perp w'}^L),\] so that in either case \(\dim_k (V^r(M_0 \oplus W_{\perp w'}^L)) \geq L_1(X,\pi,w) + L_2(X,\pi,w).\)

Finally, we analyze the contribution from \(T_0\). By Lemma 4 we have \(\dim_k (\pi_{T_0}(V^r(T_0))) = l_X-a_X^r\), which contributes to lower bound on \(\dim_k(w(L)).\) If additionally \(\sum_{i=1}^m \left \lceil \frac{\phi(d_i,w')}{2} \right \rceil \geq g_X\), then Corollary 7 shows that \(V(\eta)=0\) implies \(\pi_*(\eta)=0\), so that \[\ker\left (V : M_0 \oplus T_0 \oplus W_{\perp w'}^L \to M_0 \oplus T_0 \oplus W_{\perp w'}^L \right) \subseteq M_0 \oplus W_{\perp w'}^L .\] In this case the contribution of \(T_0\) to the lower bound is \(l_X\). Thus in either case the contribution from \(T_0\) is at least \(L_3(X,\pi,w)\). Therefore we conclude \[\begin{align} \dim_k (w(L)) & \geq \dim_k (V^r(M_0 \oplus W_{\perp w'}^L \oplus T_0)) \\ &\geq L_1(X,\pi,w) + L_2(X,\pi,w) + L_3(X,\pi,w) = L(X,\pi,w). \qedhere \end{align}\] ◻

Proof. By Proposition 31, we have \(w'(L)^\perp \subseteq M \oplus W_{\perp w'}^U \oplus T_0\) with \[\mathrm{codim}_k(W_{\perp w'}\subseteq W_{\perp w'}^U) \leq \begin{cases} \left \lceil \frac{3t}{2} \right \rceil l_X &if \sum_{i=1}^m (d_i+1) \leq 4g_X -4 \\ tl_X &if \sum_{i=1}^m (d_i+1) > 4g_X-4.\end{cases}\] By Proposition 18, we have \[\begin{align} V^r(M \oplus W_{\perp w'}) &\subseteq M_0 \oplus W_{V^r \perp w'} = M_0 \oplus W_w \intertext{so that} \dim_k(V^r(M \oplus W_{\perp w'}^U)) &\leq l_X + \phi(w) + \mathrm{codim}_k(W_{\perp w'}\subseteq W_{\perp w'}^U). \end{align}\] Moreover, one observes \(\dim_k(V^r(T_0)) \leq \dim_k(T_0) = l_X\) and deduces \[\dim_k(w(L)) \leq \dim_k(V^r(M \oplus W_{\perp w'}^U \oplus T_0)) \leq \dim_k(V^r(M \oplus W_{\perp w'}^U)) + \dim_k(V^r(T_0)). \qedhere\] ◻

Proof. Combine Lemma 23 and Lemma 24. ◻

Proof. Let \(n=\dim_k (w(L))\), so \(L(X,\pi,w) \leq n \leq U(X,\pi,w)\) follows immediately from Proposition 32. By Lemma 18, the space \(U \oplus w(L)\) occurs in the canonical filtration of \(\mathop{\mathrm{\mathrm{H}}}_{\textrm{dR}}^1(Y)\), and therefore it occurs in any final filtration. The dimension of \(U \oplus w(L)\) is \(f_Y+n\). By definition, we have \[\nu_{f_Y+n} = \dim_k(V(U \oplus w(L))) = \dim_k(U \oplus V(W(L))) = f_Y + \dim_k(V(w(L))).\] Then the corollary follows from applying Proposition 32 to the word \(Vw\). ◻

Proof. We focus first on the lower bound. By definition, we have \(\varepsilon^r= \delta_U^r-c_2^r\), so \(c_2^r=\varepsilon^r-\delta_U^r\). Next, we observe \(c_2^{r+1} \leq c_2^r\), since there is a surjection \[V^r(M\oplus T_0 \oplus W_{\widehat{w}}^U) / V^{r}(\widehat{w}(L)) \to V^{r+1}(M\oplus T_0 \oplus W_{\widehat{w}}^U) / V^{r+1}(\widehat{w}(L)).\] This gives \[\varepsilon^{r+1} = \delta_U^{r+1} - c_2^{r+1} \geq \delta_U^{r+1} - c_2^r = \delta_U^{r+1} - (\delta_U^r - \varepsilon^r) = \varepsilon^r + (\delta_U^{r+1}-\delta_U^r).\] The upper bound is established in an analogous fashion using \(\widehat{w}(L)\) and \(M_0 \oplus T_0 \oplus W_{\widehat{w}}^L\). ◻

Proof. We first prove the inequality \[\label{eq:32delta94r} \phi(V^r \widehat{w}) + 2(l_X-a_X^r) \leq \delta^r \leq \phi(V^r \widehat{w}) + 2l_X.\tag{36}\] The proof of the lower bound in Equation 36 is analogous to the proof of Lemma 23. Namely, we observe \[\begin{align} \delta^r = \dim_k (V^r(M_0 \oplus T_0 \oplus W_{\widehat{w}})) &\geq \dim_k (\pi_{M_0}(V^r(M_0))) + \dim_k (\pi_{T_0}(V^r(T_0))) + \dim_k (\pi_W(V^r(W_{\widehat{w}}))) \\ &= 2(l_X-a_X^r) + \phi(V^r\widehat{w}). \end{align}\] For the upper bound of Equation 36 , we observe that \(V^r(M_0 \oplus W_{\widehat{w}}) \subseteq M_0 \oplus W_{V^r\widehat{w}}\). Therefore, \[\begin{align} \delta^r = \dim_k (V^r(M_0 \oplus T_0 \oplus W_{\widehat{w}})) &\leq \dim_k(M_0 \oplus W_{V^r\widehat{w}}) + \dim_k (T_0) \\ &=2l_X+\phi(V^r \widehat{w}). \end{align}\]

The lower bound in the lemma follows immediately from subtracting Equation 36 from its analog for \(\delta^{r+1}\). For the upper bound, observe that \[\begin{align} \delta^{r+1} &= \dim_k (V^{r+1} (M_0 \oplus W_{\widehat{w}} \oplus T_0)) \\ &\leq \dim_k (V^{r+1}(M_0 \oplus W_{\widehat{w}})) + l_X \\ &= \phi(V^{r+1}\widehat{w}) + \dim_k(\pi_{M_0}( V^{r+1}(M_0 \oplus W_{\widehat{w}}))) + l_X \end{align}\] and that \[\delta^r \geq \phi(V^r\widehat{w}) + \dim_k(\pi_{M_0}( V^{r}(M_0 \oplus W_{\widehat{w}}))) + (l_X-a_X^r).\] Then the lemma follows as \(\dim_k(\pi_{M_0}( V^{r+1}(M_0 \oplus W_{\widehat{w}}))) \leq \dim_k(\pi_{M_0}(V^r(M_0 \oplus W_{\widehat{w}})))\). ◻

Proof. This follows from Lemmas 25 and  26 and the observation \(0 \leq e_j^r-e_j^{r+1} \leq e_j^0\). ◻

Proof. For ease of exposition, we assume \(\sum_{i=1}^m (d_i+1) > 4g_X -4\), so that Lemma 6 applies and the bounds in Proposition 32 are sharper. The case \(\sum_{i=1}^m (d_i+1) \leq 4g_X -4\) works analogously. Throughout this proof, we use the simplification \(a_X^r \leq l_X\) for every \(r\).

Let \(n=\dim_k(w(L))\) and let \(t=s-1\). Then Proposition 32 implies \[\phi(w) - tl_X \leq L(X,\pi,w) \leq n =\dim_k(w(L)) \leq U(X,\pi,w) \leq \phi(w)+(t+2)l_X.\] Recalling Lemma 18, note that \(\nu_{f_Y+n}=f_Y + \dim_k(V(w(L)))\). We consider two cases.

• Assume \(f_Y+n\leq l\). Then Proposition 32 yields \[\nu_{l} \geq \nu_{f_Y+n} \geq f_Y + L(X,\pi,Vw) \geq f_Y+\phi(Vw)-tl_X = (f_Y+\phi(Vw)+l_X)-sl_X,\] which proves the lower bound of the theorem. For the upper bound, we use Proposition 33 and Equation 33 : \[\begin{align} \nu_{f_Y+n} &\leq f_Y+ n+\phi(Vw)-\phi(w)+l_X + tl_X. \intertext{Now using that \nu_j\leq \nu_{j+1}\leq \nu_j+1 we see } \nu_{l} &\leq l +\phi(Vw) - \phi(w) + (t+1) l_X . \intertext{Finally recalling that l=f_Y+\phi(w)+2l_X yields} \nu_{l} &\leq f_Y+ \phi(Vw) + (t+3)l_X = (f_Y+\phi(Vw)+l_X)+(s+1)l_X. \end{align}\] Thus the assertion is proven when \(f_Y+n\leq l\).

• Assume \(f_Y+n>l\). Then Proposition 32 immediately yields the upper bound \[\nu_{l} \leq \nu_{f_Y+n} \leq f_Y + U(X,\pi,Vw) \leq f_Y+\phi(Vw)+(t+2)l_X = (f_Y+\phi(Vw)+l_X)+sl_X.\] For the lower bound, Proposition 33 and Equation 33 show \[\begin{align} \nu_{f_Y+n} &\geq f_Y + n + \phi(Vw) - \phi(w)-2l_X - (t+1)l_X. \intertext{Then we conclude that } \nu_{l} &\geq l +\phi(Vw)-\phi(w) - 2l_X - (t+1)l_X\\ & =f_Y + \phi(Vw) - (t+1)l_X = (f_Y+\phi(Vw)+l_X) -(s+1)l_X, \end{align}\] which finishes the proof.

 ◻

Proof. Let \(M_t\) be the maximal gap between consecutive values of \(\phi(d,w) \leq \frac{d-1}{2}\) for simple words \(w\) with at most \(t\) instances of \(\bot\). (By consecutive we mean that there does not exist a word \(w_3\) of the same form such that \(\phi(d,w_1) < \phi(d,w_3) < \phi(d,w_2)\).) We will prove that \(M_t\) decreases at the appropriate rate as \(t\) increases.

Let \(i=\phi(d,w_1)\) and \(j=\phi(d,w_2)\). Then recalling Definition 12 we see \[\left \lfloor \frac{|i-j|}{2^n} \right \rfloor \leq |\phi(d,V^n\bot w_1) - \phi(d,V^n\bot w_2) | \leq \left \lceil \frac{|i-j|}{2^n} \right \rceil.\] This implies \(M_{t+1} \leq \left \lceil \frac{M_t}{2} \right\rceil\). Note that \(M_0=\frac{d-1}{2}\), so that \(M_s=1\) when \(s\geq \left \lceil \log_2 \left( \frac{d-1}{2} \right) \right \rceil\). ◻

Proof. For \(1 \leq l \leq f_Y\), the statement follows immediately from the fact \(f_Y=\max\{j \; | \; \nu_j=j\}\).

This fact also implies \(\nu_{f_Y+1}=f_Y\). Combining this with the rule \(\nu_j\leq \nu_{j+1} \leq \nu_j +1\) gives the statement for \(f_Y<l<f_Y+2l_X\).

Finally, assume \(f_Y+2l_X \leq l \leq g_Y = f_Y+2l_X + \frac{d-1}{2}\). Let \(m:=l-f_Y-2l_X\), so that \(0 \leq m \leq \frac{d-1}{2}\). Then Lemma 27 implies that there exists a word \(w=V^{n_s} \perp \ldots \perp V^{n_1}\) with \(s\leq \left \lceil \log_2\left(\frac{d-1}{2}\right) \right \rceil\) such that \(m=\phi(d,w)\) and hence \(l=f_Y+\phi(d,w)+2l_X\). We apply Theorem 34 to this word \(w\), yielding \[\left| \nu_l - (f_Y + \phi(Vw) + l_X) \right| \leq \begin{cases} \frac{3\lceil \log_2(d-1) \rceil}{2} l_X &if d \leq 4g_X -5\\ \lceil \log_2(d-1) \rceil l_X &if d > 4g_X -5. \end{cases}\] Here we have used \(\log_2\left(\frac{d-1}{2}\right)+1 = \log_2(d-1)\). The proof is concluded by observing \[\phi(d,Vw)=\left \lfloor \frac{\phi(d,w)}{2} \right \rfloor = \left \lfloor \frac{l-f_Y-2l_X}{2} \right \rfloor = \left \lfloor \frac{l-f_Y}{2} \right \rfloor- l_X. \qedhere\] ◻