arXiv subjects

Pavel Čoupek

Publications and source records attributed to Pavel Čoupek.

6 recordsLinked to original sources

arXiv2026

Exactness property of Breuil-Kisin functors and Bloch-Kato Selmer groups

Let $K$ be a $p$-adic field and $T$ a lattice in a semistable representation of $\mathrm{Gal}(\overline{K}/K)$ with Hodge-Tate weights in $[0, r]$. Assuming $0\leq r<p-1$, we prove that for a semistable extension of $\mathbb{Z}_p$ by $T$, the corresponding sequence of strongly divisible modules is exact. Analogous statements are proved for Breuil-Kisin modules and for prismatic $F$-crystals for all $r\geq 0$. In the crystalline case, we deduce that the integral Bloch-Kato Selmer group $H^1_f(K, T)$ is computed by $\mathrm{Ext}^1$ in the category of crystalline strongly divisible modules. Using further exactness results, we define a tensor product of strongly divisible modules, which commutes with the functors to Galois representations. As an application, we show that for abelian varieties $A_1, A_2$ over $K$ with good reduction, the cup product map $δ_1\cupδ_2:A_1(K)\otimes A_2(K)\rightarrow H^2(K, T_p(A_1)\otimes T_p(A_2))$ induced by the Kummer sequences of $A_1, A_2$ factors through an $\mathrm{Ext}^2$ group of strongly divisible modules.

math.NT↗

arXiv2026

Ramification bounds via Wach modules and q-crystalline cohomology

Let $K$ be an absolutely unramified $p$-adic field. We establish a ramification bound, depending only on the given prime $p$ and an integer $i$, for mod $p$ Galois representations associated with Wach modules of height at most $i$. Using an instance of $q$-crystalline cohomology (in its prismatic form), we thus obtain improved bounds on the ramification of $\mathrm{H}^{i}_{et}(X_{\mathbb{C}_K}, \mathbb{Z}/p\mathbb{Z})$ for a smooth proper $p$-adic formal scheme $X$ over $\mathcal{O}_K$, for arbitrarily large degree $i$.

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arXiv2025

Constructing vector-valued automorphic forms on unitary groups

We introduce a method for producing vector-valued automorphic forms on unitary groups from scalar-valued ones. As an application, we construct an explicit example. Our strategy employs certain differential operators. It is inspired by work of Cléry and van der Geer in the setting of Siegel modular forms, but it also requires overcoming challenges that do not arise in the Siegel setting.

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arXiv2022

Crystalline condition for $A_{\mathrm{inf}}$-cohomology and ramification bounds

For a prime $p>2$ and a smooth proper $p$-adic formal scheme $X$ over $\mathcal{O}_K$ where $K$ is a $p$-adic field, we study a series of conditions ($\mathrm{Cr}_s$), $s\geq 0$ that partially control the $G_K$-action on the image of the associated Breuil-Kisin prismatic cohomology $\mathrm{R}Γ_Δ(X/\mathfrak{S})$ inside the $A_{\mathrm{inf}}$-prismatic cohomology $\mathrm{R}Γ_Δ(X_{A_{\mathrm{inf}}}/A_{\mathrm{inf}})$. The condition ($\mathrm{Cr}_0$) is a criterion for a Breuil-Kisin-Fargues $G_K$-module to induce a crystalline representation used by Gee and Liu, and thus leads to a proof of crystallinity of $\mathrm{H}^i_{\text{ét}}(X_{\overlineη}, \mathbb{Q}_p)$ that avoids the crystalline comparison. The higher conditions ($\mathrm{Cr}_s$) are used to adapt the strategy of Caruso and Liu in order to establish ramification bounds for the mod $p$ representations $\mathrm{H}^{i}_{\text{ét}}(X_{\overlineη}, \mathbb{Z}/p\mathbb{Z}),$ for arbitrary $e$ and $i$, which extend or improve existing bounds in various situations.

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arXiv2022

Geometric quadratic Chabauty over number fields

This article generalizes the geometric quadratic Chabauty method, initiated over $\mathbb{Q}$ by Edixhoven and Lido, to curves defined over arbitrary number fields. The main result is a conditional bound on the number of rational points on curves that satisfy an additional Chabauty type condition on the Mordell-Weil rank of the Jacobian. The method gives a more direct approach to the generalization by Dogra of the quadratic Chabauty method to arbitrary number fields.

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arXiv2019

Cotilting Sheaves on Noetherian Schemes

We develop theory of (possibly large) cotilting objects of injective dimension at most one in general Grothendieck categories. We show that such cotilting objects are always pure-injective and that they characterize the situation where the Grothendieck category is tilted using a torsion pair to another Grothendieck category. We prove that for Noetherian schemes with an ample family of line bundles a cotilting class is closed under injective envelopes if and only if it is invariant under twists by line bundles, and that such cotilting classes are parametrized by specialization closed subsets disjoint from the associated points of the scheme. Finally, we compute the cotilting sheaves of the latter type explicitly for curves as products of direct images of indecomposable injective modules or completed canonical modules at stalks.

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