Characteristic-Zero Lifts of Partial Hasse Invariants

• Characteristic-zero lifts of partial Hasse invariants extend classical modular invariants from characteristic p to zero, unifying crystalline, modular, and differential approaches.
• They are constructed via crystalline Dieudonné theory and explicit computational techniques, which yield refined arithmetic stratifications in Shimura varieties and Hilbert modular settings.
• These lifts detect μ-ordinary and non-ordinary loci, providing practical tools for analyzing Newton strata and guiding the reduction of twisted modular forms.

Characteristic-zero lifts of partial Hasse invariants refer to the canonical extension of modular invariants, originally defined in characteristic $p$, to characteristic zero. These lifts are crucial in relating the geometry and arithmetic of Shimura varieties, $p$-divisible groups, and Hilbert modular varieties across various characteristics. They generalize the classical Hasse invariant and provide a fine arithmetic stratification, particularly detecting the $\mu$-ordinary and non-ordinary loci. Central to modern advances, their construction unifies crystalline, modular, and differential approaches, with explicit computational techniques and enumerative formulas.

1. Definition and Local Construction of Partial Hasse Invariants

Let $p$ be a prime and $F/\mathbb{Q}_p$ a finite unramified extension with ring of integers $\mathcal{O} = \mathcal{O}_F$ and residue field $\kappa = \mathcal{O}/p$. For a $p$-divisible group $G$ over a $\kappa$-scheme $S$ equipped with an $\mathcal{O}$-action, the conormal sheaf of the Cartier dual $G^D$ admits a decomposition

$$\omega_{G^D} = \bigoplus_{\tau \in \operatorname{Hom}(\kappa, \kappa)}\omega_{G^D, \tau}$$

with each summand locally free of rank $q_\tau$. The signature $(p_\tau,q_\tau)$, determined by ranks of the respective conormal and cotangent sheaves, specifies the nature of the group.

Partial Hasse invariants are constructed using the crystalline Dieudonné theory. Specifically, for each $\tau_0$ with $q_{\tau_0} = q$,

$$\widetilde{\mathrm{Ha}}_{\tau_0}: \wedge^{q}\omega_{G^D,\tau_0} \to \wedge^q\omega_{G^D,{\tau_0}}^{(p^f)} \simeq \det(\omega_{G^D,\tau_0})^{\otimes p^f}$$

divides the wedge of Frobenius by an explicit $p$-power, and twists yield

$$\widetilde{\mathrm{Ha}}_{\tau_0}(G) \in H^0\Big(S, \det(\omega_{G^D,\tau_0})^{\otimes(p^f-1)}\Big).$$

These invariants detect the locus where the Newton and Hodge polygons meet at $q_\tau$; their vanishing defines divisors that stratify the moduli space according to slopes and $p$-rank strata (Hernandez, 2016).

2. Properties: Functoriality, Cartier Divisors, and Product Structure

Partial Hasse invariants possess extensive functoriality and compatibility properties:

• Base Change: Their formation commutes with any morphism of smooth bases.
• Duality: There is a canonical identification:

$$\det(\omega_{G,\tau})^{\otimes(p^f-1)} \simeq \det(\omega_{G^D,\tau})^{\otimes(p^f-1)}$$

with the invariants coinciding under dualization.

• Products: For $G = G_1 \times G_2$ and $k_\tau(G) = k_\tau(G_1) + k_\tau(G_2)$,

$$\widetilde{\mathrm{Ha}}_\tau(G_1 \times G_2) = \widetilde{\mathrm{Ha}}_\tau(G_1) \otimes \widetilde{\mathrm{Ha}}_\tau(G_2).$$

• Cartier Divisor: The vanishing locus of each $\widetilde{\mathrm{Ha}}_\tau$ defines a reduced, effective Cartier divisor on the moduli stack, with the complementary $\mu$-ordinary locus Zariski dense by Wedhorn's purity.

The global $\mu$-ordinary Hasse invariant is defined as

$$\widetilde{^\mu\mathrm{Ha}}(G) = \bigotimes_{\tau} \widetilde{\mathrm{Ha}}_\tau(G)$$

and cuts out the non-$\mu$-ordinary locus in codimension 1, guaranteeing the divisors are reduced (Hernandez, 2016).

3. Characteristic-Zero Lifting: Crystalline and Modular Approaches

Given a smooth $\mathcal{O}$-scheme $\widetilde{S}$ with special fiber $S$, and a $p$-divisible $\mathcal{O}$-group $G/S$ lifting uniquely to $\widetilde{G}/\widetilde{S}$, the corresponding Dieudonné crystal extends and carries the same partial Hasse structures. By uniqueness, each $\widetilde{\mathrm{Ha}}_\tau(G)$ lifts canonically to a global section over $\widetilde{S}$: $$H^0\!\big(\widetilde{S},\,\det(\omega_{\widetilde{G}^D,\tau})^{\otimes(p^f-1)}\big).$$ This provides a characteristic-zero lift, unifying and generalizing classical Eisenstein series-type constructions in modular and Siegel settings (Hernandez, 2016).

In the context of Hilbert modular varieties, Bogo–Li establish a construction of characteristic-zero lifts for genus-zero, non-compact curves $Y \hookrightarrow \mathcal{M}_F$ via solutions $F_j(t)$ to Picard–Fuchs differential equations, with the reduction mod $p$ congruent to the polynomial defining the non-ordinary locus. They define explicit holomorphic twisted modular forms $h_{p,j}(\tau)$ with $p$-integral $t$-expansions that reduce to the partial Hasse polynomials $\mathrm{ph}_{p,j}(t)$, providing uniqueness and modularity in the lift (Bogo et al., 23 Jan 2026).

4. Explicit Computations and Enumerative Formulas

On genus-zero curves in Hilbert modular varieties, pulling back the partial Hasse invariants yields monic polynomials in a Hauptmodul $t$. The construction via Picard-Fuchs theory shows that for each partial Hasse polynomial $\mathrm{ph}_{p,j}(t)$ corresponding to divisor $D_j$,

$$\deg \mathrm{ph}_{p,j}(t) = \dim_{\mathbb{C}} M_{w_j}(\Gamma, \varphi) - 1,$$

with explicit weights $w_j = (p-1,0)$ or $(0,p-1)$ (split) and $w_j = (-1,p)$ or $(p,-1)$ (inert). Consequently, the number of non-ordinary points is given by

$$\#\{ Y_{t_0} \text{ non-ordinary}\} = \frac{1}{p-1}\left(\dim M_k(\Gamma,\varphi) - \text{correction}\right)$$

and exhibits linear growth in $p$ modulo corrections, recapitulating and generalizing Deuring’s formula in the modular curve context (Bogo et al., 23 Jan 2026).

For families such as the Teichmüller curve $W_{13}$, precise formulas for the degrees and counts of supersingular fibers emerge, dependent on the congruence class of $p$ and the topology of $W_{13}$.

5. Reduction of Modular and Twisted Modular Forms Modulo $p$

The reductions of twisted modular forms, particularly those with $p$-integral $t$-expansions, are governed by rational functions whose denominators are powers of the partial Hasse polynomials and additional factors from elliptic fixed points. Explicitly, for $f_j(\tau) \in M_{(-k, kp)}(\Gamma, \varphi)$ with sufficiently large $p$, one obtains

$$f_j(t) \equiv \frac{P_f(t)}{\mathrm{ph}_{p,j}(t)^k\prod_i (t(e_i)-t)^{\alpha_{j,i}}} \pmod{p}$$

where $P_f(t)$ lies in the ring of integers of the coefficient field. Conversely, every such rational function arises from some twisted modular form, providing a description of the mod $p$ reduction behavior (Bogo et al., 23 Jan 2026).

6. Hypergeometric and Differential-Equation-Based Constructions

Several family-specific constructions utilize hypergeometric series and explicit determinants, notably for the triangle curve $\Delta(2,5,\infty)$. The Jacobian family

$C_\eta: y^2 = x^5 - 5x^3 + 5x - 2\eta$

yields modular forms $Q(\tau), R(\tau)$ and twisted modular forms $B(\tau)$, with recursion

$$(1-5Qx^4 + 5Q^2x^8 - 2Rx^{10})^{-1/2} = \sum_{n\ge 0} d_n(Q,R)x^n,$$

providing explicit characteristic-zero lifts for both inert and split cases: \begin{align*} h_{p,1}(\tau) &= B^p(\tau) d_{p-3}(Q(\tau),R(\tau)), \ h_{p,2}(\tau) &= B^{-1}(\tau) d_{3p-1}(Q(\tau), R(\tau)), & p \text{ inert}, \ h_{p,1}(\tau) &= d_{3p-3}(Q(\tau), R(\tau)), \ h_{p,2}(\tau) &= B^{p-1}(\tau) d_{p-1}(Q(\tau), R(\tau)), & p \text{ split}, \end{align*} which are rewritten in terms of classical hypergeometric functions. In each good prime case, squares of these forms reduce mod $p$ to constants, ensuring their modular interpretations and lifting properties (Bogo et al., 23 Jan 2026).

7. Applications: $\mu$-Ordinary Locus and Shimura Varieties

In integral models of Shimura varieties of PEL type, especially those reducing to $\mathrm{GL}_h(\mathcal{O}_F)$ at $p$, the partial Hasse invariants detect Newton strata, with the $\mu$-ordinary stratum given by the non-vanishing locus. The construction recovers established invariants on the special fiber and aligns with explicit calculations via Serre–Tate theory and display formulas. In these moduli spaces, the explicit nature and functoriality of characteristic zero lifts afford concrete tools for analyzing geometric and arithmetic structures in mixed characteristic settings (Hernandez, 2016).

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For further details and explicit constructions of characteristic-zero lifts of partial Hasse invariants, see (Hernandez, 2016) and (Bogo et al., 23 Jan 2026).