Reductions of some two-dimensional crystalline representations via Kisin modules

Brandon Levin; John Bergdall

Reductions of some two-dimensional crystalline representations via Kisin modules

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For every $k\geq 2$, once $v_p(a_p)>\lfloor (k-1)/p \rfloor$, the semisimple mod $p$ reduction of $V_{k,a_p}$ is $V_{k,0}$.

desk verdict Solid paper: proves a genuine improvement to the Berger–Li–Zhu bound by explicit Kisin-module calculations, and the descent algorithm appears to hold up on a careful read. read the letter →

arxiv 1908.09036 v3 pith:GMCOC7NW submitted 2019-08-23 math.NT

John Bergdall , Brandon Levin This is my paper · ORCID

classification math.NT MSC 11F8011F85

keywords crystallinerepresentationsmodpreductionKisinmodulesmonodromyconditiondescentalgorithmp-adicHodgetheoryGaloislocalconstancy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that, for every prime $p$ and every weight $k\geq 2$, once the slope $v_p(a_p)$ of a two-dimensional crystalline Galois representation of $G_{\mathbb{Q}_p}$ exceeds $\lfloor (k-1)/p \rfloor$, the semisimple reduction modulo $p$ no longer depends on $a_p$: it is the same representation as at $a_p=0$, namely the induction $\operatorname{Ind}_{G_{\mathbb{Q}_{p^2}}}^{G_{\mathbb{Q}_p}}(\omega_2^{k-1}\chi)$. This extends the previous constancy range of [6] from $\lfloor (k-2)/(p-1) \rfloor$ to the larger $\lfloor (k-1)/p \rfloor$. The proof works by constructing explicit integral Kisin modules: for $k\geq 2p+1$ and $v_p(a_p)>\lfloor (k-1)/p \rfloor$, the Frobenius can be put in the simple form $\begin{pmatrix}P&-1\\E^{k-1}&0\end{pmatrix}$ with a single polynomial $P$ of degree at most $k-1$, $P(0)=a_p$, and $P$ integral when the slope is large enough. Reducing that matrix modulo $p$ gives $\begin{pmatrix}0&-1\\u^{k-1}&0\end{pmatrix}$ independently of $a_p$, which forces the constant reduction. The method replaces the usual $p$-adic local Langlands route with a direct descent calculation on Kisin modules, and it is set up to work beyond the crystalline $\mathrm{GL}_2(\mathbb{Q}_p)$ setting.

What carries the argument

The load-bearing objects are Kisin modules—integral $\phi$-modules over $\Lambda[[u]]$ of finite $E$-height—and a descent algorithm for them. The paper starts from a one-parameter family of $\phi$-modules over the rigid-analytic ring $R=O_{F,[0,p^{-1/p}]}$ with Frobenius matrix $C_{a_p}=\begin{pmatrix}a_p(\lambda_-/\lambda_{++})^{h}&-1\\E^h&0\end{pmatrix}$, where $h=k-1$ and $\lambda_\pm$ are explicit infinite products defined from $E(u)=u+p$. It then defines $\gamma$-allowable matrices and four allowed row operations $\alpha_{ij}$ that replace $C$ by $A\ast_\phi C=AC\phi(A)^{-1}$; the key estimate (Proposition 4.3.5(d)) is that each block of operations raises the numerical error $\varepsilon_C$ by at least $\min\{\gamma,p-1\}$, so the errors go to infinity, the infinite product of the $A_m$ converges in $\mathrm{GL}_2(R)$, and the conjugate matrix becomes polynomial with controlled integrality. This polynomial matrix gives the desired Kisin module, and the monodromy condition—checked via Corollary 2.2.5 on a single disc—guarantees it corresponds to the intended crystalline representation.

What would settle it

Take a prime $p$ and a weight $k\geq 2p+1$, choose $a_p$ with $v_p(a_p)$ just above $\lfloor (k-1)/p \rfloor$, and compute the semisimplification of $V_{k,a_p}$ modulo $p$ by an independent method (for instance, the construction of [6]); if the result is not isomorphic to $\operatorname{Ind}_{G_{\mathbb{Q}_{p^2}}}^{G_{\mathbb{Q}_p}}(\omega_2^{k-1}\chi)$, the theorem is false.

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Extended reading notes

Core claim

The central theorem (Corollary 5.2.3) states that for every $k\geq 2$ and every $a_p$ with $v_p(a_p)>\lfloor (k-1)/p \rfloor$, the semisimple reduction modulo $p$ of the crystalline representation $V_{k,a_p}$ is isomorphic to $V_{k,0}$; explicitly, it is the induction to $G_{\mathbb{Q}_p}$ of the character $\omega_2^{k-1}\chi$ of the unramified quadratic extension of $\mathbb{Q}_p$. The proof goes through an explicit family of rank-two Kisin modules. The base case $a_p=0$ has Frobenius matrix $\begin{pmatrix}0&-1\\E^{k-1}&0\end{pmatrix}$ and an explicit monodromy operator; deforming by $a_p$ gives a one-parameter family of $\phi$-modules on a $p$-adic disc, and the descent algorithm of Section 4 transforms each such module into one whose Frobenius matrix is $\begin{pmatrix}P&-1\\E^{k-1}&0\end{pmatrix}$ with $P$ a polynomial of degree at most $k-1$ and $P(0)=a_p$. When $v_p(a_p)>\lfloor (k-1)/p \rfloor$ and $k\geq 2p+1$, $P$ has integral coefficients, so reducing modulo the maximal ideal yields the $a_p$-independent matrix $\begin{pmatrix}0&-1\\u^{k-1}&0\end{pmatrix}$; the small-weight cases $k<2p+1$ follow from the earlier small-weight theorem of [4].

Load-bearing premise

The central claim rests on the estimate that every allowed row operation raises the numerical error of a $\gamma$-allowable matrix by at least $\min\{\gamma,p-1\}$; if that estimate fails for even one matrix, the descent produces no explicit Kisin module and the calculation of the reduction collapses.

Editorial extensions

If this is right

For every $k\geq 2$, the reduction $\overline{V}_{k,a_p}$ is constant on the slope interval $v_p(a_p)>\lfloor (k-1)/p \rfloor$, and it equals $\overline{V}_{k,0}$.
The constancy range improves from $\lfloor (k-2)/(p-1) \rfloor$ to $\lfloor (k-1)/p \rfloor$, enlarging the region in which the mod $p$ reduction is known explicitly.
In weights $k\geq 2p+1$ the proof produces an explicit integral Kisin module whose Frobenius matrix is $\begin{pmatrix}P&-1\\E^{k-1}&0\end{pmatrix}$ with $P\in m_F[u]$ of degree at most $k-1$ and $P(0)=a_p$; reducing this matrix modulo $p$ gives $\begin{pmatrix}0&-1\\u^{k-1}&0\end{pmatrix}$.
The reduction of $V_{k,a_p}$ modulo $p$ is therefore the same as that of $V_{k,0}$, which is the induction $\operatorname{Ind}_{G_{\mathbb{Q}_{p^2}}}^{G_{\mathbb{Q}_p}}(\omega_2^{k-1}\chi)$.
Because the construction uses Kisin modules rather than $p$-adic local Langlands, the same descent algorithm is available for semi-stable, non-crystalline inputs and for representations beyond $\mathrm{GL}_2(\mathbb{Q}_p)$, as the paper notes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The Section 4 descent estimates are formulated for a general modulus $m$ and the initial setup allows a general Eisenstein polynomial $E(u)$, so the same row-reduction algorithm should produce explicit integral Kisin modules for crystalline representations of unramified extensions of $\mathbb{Q}_p$; checking that is a direct translation of the paper's estimates.
The threshold $\lfloor (k-1)/p \rfloor$ is probably not optimal: the paper itself points to computational and global evidence for $\lfloor (k-1)/(p+1) \rfloor$. A natural test is to run the descent with the choice $a'=h/2-(p-1)/2$ replaced by values tuned to that smaller bound, and see where the error estimate breaks.
Because Theorem 5.2.1 gives an explicit polynomial $P$ for any $v_p(a_p)>0$ over $F$, the algorithm could in principle be implemented symbolically to tabulate Kisin modules and compare reductions with weight-elimination predictions.
The explicit polynomial $P$ is $p$-adically close to the truncation of $a_p(1+u^p/p)^{k-1}$; this proximity might connect the mod $p$ reduction to the slope filtration of overconvergent modular forms, though the paper does not explore that.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

John,

You should know two things before you spend time on this paper. First, the main theorem is real progress: it improves the fifteen-year-old Berger–Li–Zhu bound on slope stability for mod p reductions of two-dimensional crystalline representations from floor((k-2)/(p-1)) to floor((k-1)/p), and in that range identifies the reduction explicitly as the induction of omega_2^{k-1} chi. Second, the proof is a direct calculation, not a black box: the authors construct rational Kisin modules explicitly, then run a descent algorithm to get integral modules and read off the reduction. I read Section 4.3 carefully, and the error-growth estimate in Proposition 4.3.5(d) is load-bearing but appears sound. The reader's report and the stress-test both went through the same calculation and found no gap.

What is actually new: the explicit family of phi-modules in Section 3 and the descent algorithm in Section 4. Previous explicit Kisin-module computations (Caruso–David–Mézard; Le–Le Hung–Levin–Morra) were limited to small weights. Here the descent works uniformly in the weight, with careful p-adic estimates. The integrality step (Prop 5.2.2) is clean, and the application to the reduction is immediate.

Soft spots, in proportion: the descent algorithm is intricate and not machine-checked. The estimates in Lemma 4.3.4 have an exceptional case for alpha_11 that the authors handle with part (c); it is delicate but does not look wrong. The paper relies on external inputs—Kisin's theory for the equivalence, Berger's theorem for small weights h < 2p, and [25, Prop 5.3] for the monodromy criterion. The last is by one of the authors, but it is a published result in Inventiones and the use is legitimate; self-citation is not a problem here. The theorem itself is a bound, not a full classification, and the authors note that Arsovski later improved it under extra conditions. That does not undercut the result; the method here is different and gives the explicit reduction for all a_p in the range.

Who this is for: anyone working on Serre weight conjectures, p-adic local Langlands, or explicit integral p-adic Hodge theory. It deserves a serious referee. My recommendation: send it to review. The mathematics is sound, and the paper would benefit from minor expository additions in Section 4.3, but I would not desk-reject.

Referee Report

0 major / 4 minor

Summary. The paper studies the semisimple mod p reductions of the two-dimensional irreducible crystalline representations V_{k,a_p} of G_{Q_p} with Hodge–Tate weights 0 and k−1 and Frobenius characteristic polynomial X^2 − a_p X + p^{k−1}. The main theorem (Theorem 1.1.1, Corollary 5.2.3) states that for every k≥2 and every a_p with v_p(a_p)>⌊(k−1)/p⌋, the semisimple reduction V_{k,a_p} is isomorphic to V_{k,0}, and it is explicitly the induction Ind_{G_{Q_{p^2}}}^{G_{Q_p}}(ω_2^{k−1}χ). The proof is built on explicit Kisin modules: Section 3 determines the family of ϕ-modules satisfying the monodromy relation (Proposition 3.0.4), Section 4 develops a descent algorithm from a p-adic disc to the formal power series ring (Theorem 4.3.7) with quantitative error growth (Proposition 4.3.5), and Section 5 applies the algorithm to obtain an integral Kisin module under the stated slope condition (Proposition 5.2.2) and then reads off the reduction.

Significance. If correct, the main theorem is a genuine improvement over the Berger–Li–Zhu bound δ_p(k)≤⌊(k−2)/(p−1)⌋ and reaches the range suggested by global and computational evidence up to a known gap. The paper is also valuable as one of the few fully explicit calculations with Kisin modules beyond small Hodge–Tate weights. I found the central argument coherent: the monodromy relation is verified by direct calculation, the descent is reduced to a precise error-growth statement whose exceptional case (Lemma 4.3.4(c)) is handled correctly, and the integrality step (Lemma 5.1.3 and Proposition 5.2.2) uses only elementary coefficient estimates. The reduction constancy is not presupposed: the constructed Kisin module is produced independently of the desired isomorphism, and the appeal to [4] is confined to the small-weight range h<2p. The paper therefore appears to meet the standard for publication once the presentation issues below are addressed.

minor comments (4)

[§4.3 (Theorem 4.3.7)] The notation A:=∏_m A_m is ambiguous: the inductive definition C^{(m)}=A_m *_φ C^{(m−1)} means that the relevant limit is the right-to-left product lim_{n→∞} A_n⋯A_1, so the product ordering should be stated explicitly.
[§4.3 (Lemma 4.3.4)] The proof of Lemma 4.3.4 handles the diagonal cases (1,1) and (2,2) in detail and leaves the off-diagonal cases to the translations in Remark 4.3.3; since Lemma 4.3.4 underpins Proposition 4.3.5(d), adding one sentence recording the parameter choices for α_{12} and α_{21} would make the verification of the off-diagonal bounds easier to check.
[§5.2 (Theorem 5.2.1 and Proposition 5.2.2)] The displayed lower bound for v_R(P−T_{≤N}(a_p(λ_−/λ_++)^h)) is missing a closing parenthesis in both Theorem 5.2.1 and the proof of Proposition 5.2.2; the parenthesized expression should read T_{≤N}(a_p(λ_−/λ_++)^h).
[§1.1] The sentence about Arsovski's work contains a double negative: 'they do not recover neither the more specific Theorem 1.2.1 nor Theorem 5.2.1' should be 'they do not recover either the more specific Theorem 1.2.1 or Theorem 5.2.1'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the mod p reduction is computed from an explicitly constructed Kisin module, not fitted or imported from a self-citation chain.

full rationale

The central claim (Theorem 1.1.1 / Corollary 5.2.3) is that V_{k,a_p}^ss is isomorphic to V_{k,0}^ss whenever v_p(a_p) > floor((k-1)/p). The proof constructs, in Proposition 5.2.2, an integral Kisin module whose Frobenius matrix is ((P,-1),(E^{k-1},0)) with P in m_F[u] and P(0)=a_p; reducing modulo m_F gives the ap-independent matrix ((0,-1),(u^{k-1},0)). The polynomial P is produced by the descent algorithm of Section 4 (Theorem 4.3.7), whose convergence relies on explicit error-growth estimates (Lemma 4.3.4, Proposition 4.3.5). None of these steps presupposes the constancy of the reduction: the ap-independence of the mod p matrix is an output of the construction, not an input. The one-parameter family of Section 3 is derived from the monodromy relation (Proposition 3.0.4), not fitted to the desired reduction. The small-weight case h < 2p is imported from Berger's external theorem [4, Theoreme 3.2.1]. The self-citations are not load-bearing: Proposition 2.2.4, although said to be based on [25, Prop. 5.3], is given a complete proof in the text, and [3] appears only as an introductory remark about applications. The derivation is therefore self-contained with respect to the claimed constancy result.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The proof introduces no new physical or mathematical entities beyond the explicit families of phi-modules and their matrices. The free parameters are auxiliary choices in the descent estimates (a' and the disc radius m); they are not fit to data and drop out of the final theorem. All substantive background is imported from cited literature, most heavily Kisin's theory [23], the monodromy criterion from [25], and Berger's small-weight theorem [4].

free parameters (2)

a' (auxiliary exponent) = h/2 - (p-1)/2 in Proposition 5.2.2
Introduced in Theorem 5.2.1 as a flexible choice constrained by p v_p(a_p) > a' and later specialized to a' = h/2 - (p-1)/2 to make the valuation jump exceed h. It is a proof device; the final theorem is independent of the exact choice.
m (radius parameter) = m = p in Section 5
Defines the disc R = O_{F,[0,p^{-1/p}]} via m=p; chosen because λ_- and λ_{++} converge on this disc. It is a normalization, not a fitted constant, and affects only the proof's estimates.

assumptions (7)

standard math Kisin's equivalence between finite height phi-modules over O_F satisfying the monodromy condition and effective filtered phi-modules, and the functor D preserving weakly admissible modules (Theorem 2.3.1).
Foundational result from [23] invoked throughout, especially in Theorem 3.0.5 and Corollary 5.2.3.
standard math The monodromy criterion of Corollary 2.2.5: a phi-module satisfies the monodromy condition iff N_∇ has no pole at u=-p (based on [25, Proposition 5.3]).
Used to verify that the constructed modules M_{a_p} correspond to weakly admissible filtered phi-modules; if false the comparison to V_{h+1,a_p} fails.
standard math The classification of irreducible crystalline representations V_{h+1,a_p} by HT weights and Frobenius polynomial, and Breuil's description of V_{h+1,0} as Ind(ω_2^h χ) [10, Proposition 3.2].
Sets up the target representations and the explicit form of V_{h+1,0} used in Corollary 5.2.3.
standard math Berger's theorem [4, Théorème 3.2.1] giving the reduction for small weights h < 2p.
The proof of Corollary 5.2.3 defers the entire range h < 2p to this external result; the main theorem's validity in that range depends on it.
standard math Serre's theorem that semisimple mod p representations of G_{Q_p} are tamely ramified, hence restriction to the totally wildly ramified G_∞ is fully faithful [29, Proposition 4].
Used in Corollary 2.3.2 to pass between Kisin modules and Galois representations of G_∞.
standard math Rigid analytic Banach algebra properties of R = O_{F,[0,p^{-1/p}]} and the valuation v_R estimates in Lemma 4.1.1.
Technical foundation for the descent algorithm's truncation and convergence arguments.
standard math Unique factorization in F[[u]] used to conclude r=1 in the determinant comparison in Theorem 5.2.1.
Ensures f = E^h after comparing determinants; relies on E(u)=u+p having a single root u=-p.

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Pith. "Pith review of Reductions of some two-dimensional crystalline representations via Kisin modules." pith.science (2026). https://pith.science/paper/GMCOC7NW

@misc{pith2026190809036,
  author       = {Pith},
  title        = {Pith review of: Reductions of some two-dimensional crystalline representations via Kisin modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GMCOC7NW}},
  note         = {Machine review of arXiv:1908.09036}
}

abstract

We determine rational Kisin modules associated with two-dimensional, irreducible, crystalline representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ of Hodge-Tate weights $0, k-1$. If the slope is larger than $\lfloor \frac{k-1}{p} \rfloor$, we further identify an integral Kisin module, which we use to calculate the semisimple reduction of the Galois representation. In that range, we find that the reduction is constant, thereby improving on a theorem of Berger, Li, and Zhu.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [4]

L. Berger. Repr´ esentations modulaires de GL 2(Qp) et repr´ esentations galoisiennes de dimension 2. Ast´ erisque, (330):263–279, 2010

work page 2010

[2] [1]

Reduction modulo p of two-dimensional crystalline representations of G_{Q_p} of slope less than three

B. Arsovski. Reduction modulo p of two-dimensional crystalline representations of GQp of slope less than three. Preprint, 2015. Available at arXiv:1503.08309

work page Pith review arXiv 2015

[3] [2]

Limiting measures of supersingularities

B. Arsovski. Limiting measures of supersingularities. Preprint, 2020. Available at arXiv:1911.12220

work page Pith review arXiv 2020

[4] [3]

Reductions of $2$-dimensional semi-stable representations with large $\mathcal L$-invariant

J. Bergdall, B. Levin, and T. Liu. Reductions of 2-dimens ional semi-stable representations with large L-invariant. Preprint, 2020. Available at arXiv:2006.16294

work page Pith review arXiv 2020

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L. Berger. Local constancy for the reduction mod p of 2-dimensional crystalline representations. Bull. Lond. Math. Soc. , 44(3):451–459, 2012

work page 2012

[6] [6]

Berger, H

L. Berger, H. Li, and H. J. Zhu. Construction of some famil ies of 2-dimensional crystalline representations. Math. Ann., 329(2):365–377, 2004

work page 2004

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Bhattacharya and E

S. Bhattacharya and E. Ghate. Reductions of Galois repre sentations for slopes in (1 , 2). Doc. Math., 20:943–987, 2015

work page 2015

[8] [8]

Bhattacharya, E

S. Bhattacharya, E. Ghate, and S. Rozensztajn. Reductio ns of Galois representations of slope 1. J. Algebra , 508:98–156, 2018

work page 2018