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Infinitesimal deformation of p-adic differential equations on Berkovich curves

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Abstract

We show that if a differential equation \(\mathscr {F}\) over a quasi-smooth Berkovich curve X has a certain compatibility condition with respect to an automorphism \(\sigma \) of X, then \(\mathscr {F}\) acquires a semi-linear action of \(\sigma \) (i.e. lifting that on X). The compatibility condition forces the automorphism \(\sigma \) to be close to the identity of X, so the above construction applies to a certain class of automorphisms called infinitesimal. This generalizes André and Di Vizio (Astérisqué 1(296):55–111, 2004) and Pulita (Compos. Math. 144(4):867–919, 2008). We also obtain an application to Morita’s p-adic Gamma function, and to related values of p-adic L-functions.

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Notes

  1. The terminology quasi-Smooth is that of [20], this corresponds to rig-smooth curves in the rigid analytic setting.

  2. We mean that \(\Lambda _x\subseteq \Gamma _S\) is a simply connected neighborhood of x in \(\Gamma _S\) (no loops), and moreover that \(\Lambda _x\) is a finite disjoint union of segments [x, y[ out of x, all incident upon x, such that ]x, y[ is the skeleton of a virtual open annulus in X (this is possible since \(\Gamma _S\) is locally finite).

  3. An example is given by an elliptic curve X with good reduction. In this case a weak triangulation of X is given by an individual point \(S=\{x\}\) which is the unique point of X without neighborhoods isomorphic to an analytic domain of the affine line. In this case \(\Gamma _S=S=\{x\}=\Lambda _x\), and the unique open of the \(\Gamma _S\)-covering is \(Y_x=X\). The same happens for \(\mathbb {P}^{1,\mathrm {an}}_K\) with a triangulation \(S=\{x\}\), with x of type 2 or 3.

  4. In this article we do not consider the G-topology (cf. [7, Section 3.3]). Opens of X are subsets of X that are open with respect to the Berkovich topology, coverings are collections of opens of X whose union is X, and sheaves on X are genuine sheaves over this topological space X.

  5. Notice that \(D^+(t,\sigma )\) is allowed to be equal to the individual point \(\{t\}\).

  6. If the residual field \(\widetilde{K}\) has characteristic 0, then \(p=1\) and this condition is always satisfied.

  7. As an example, if \(X=C^-(0;]R_1,R_2[)\) is an open annulus with empty weak triangulation, then condition (ii) asks that there exist unspecified \(\varepsilon _1,\varepsilon _2>0\) such that (4.4) holds for all \(x\in ]x_{0,R_2-\varepsilon _2},x_{0,R_2}[\) and all \(x\in ]x_{0,R_1},x_{0,R_1+\varepsilon _1}[\).

  8. Here \(\Sigma _\Omega :=\{\sigma _\Omega \}_{\sigma \in \Sigma }\), where as usual \(\sigma _\Omega =\sigma \widehat{\otimes } \mathrm {Id}_\Omega \).

  9. This is equivalent to saying that for all point \(g:\mathscr {H}(g)\rightarrow G\) the equation \(\mathscr {F}_{\mathscr {H}(g)}\) is \(\sigma _g\)-compatible.

  10. We recall that this happens if and only if \(\mathcal {R}_{\emptyset }(x_{0,\rho },\sigma )< \mathcal {R}_{\emptyset ,1}(x_{0,\rho },\mathscr {F})\), for all \(\rho \in ]1-\varepsilon ,1[\).

  11. It is a full subcategory of the category of all differential equations.

  12. By an abuse in this statement by the boundary of an open disk we mean its relative boundary in \(\mathbb {A}^{1,\mathrm {an}}_\Omega \), while by the boundary of a closed disk we mean as usual its Shilov Boundary.

  13. Namely the intersections are quasi-Stein, and the matrix of \(\nabla \) is given by \(G=d/dT_1(Y_\chi )\cdot Y_\chi ^{-1}\) (cf. (2.7)). Over an intersection the two matrices of the stratifications differs by multiplication by a matrix killed by \(d/dT_1\), so they furnishes the same G.

  14. Conjecturally every connected analytic domain of \(\mathbb {A}^{1,\mathrm {an}}_K\) is quasi-Stein.

  15. Notice that \(G_{[1]}\), and also \(G_{[n]}\), has a denominator. It belong however to \(M_n(\mathcal {O}(X))\) because \(a\notin X\).

  16. Recall that \(\mathcal {R}^{\sigma _{q,h}}(x)= \mathcal {R}_{S}(x,\sigma _{q,h})\cdot \rho _{S,T}(x)= |(q-1)T+h|(x)\), as in the proof of Lemma 3.3.3.

  17. \(X_{\widehat{K^{\mathrm {alg}}}}\) is a disjoint union of affinoid domains of the type \(Y=D^+(c_0,R_0)-\cup _{i=1}^sD^-(c_i,R_i)\), for which \(\min _{x\in Y} \{\text {Radius of }(D(x,S))\}= \min (R_0,R_1,\ldots ,R_s)\).

  18. By Lemma 6.1.2 this means \(|(q-1)t_x+h|<r(x)\), where r(x) is the radius of D(x) (cf. Definition 1.1.9).

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Acknowledgments

We are grateful to D. Barsky for suggesting and helpful discussions, and for guidance and advice in the formulas of Sect. 7.4. We also thank Yves André, Gilles Christol, Jeröme Poineau, Bernard Le Stum, Michel Gros, and Bertrand Toen for helpful discussions.

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    Andrea Pulita

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Pulita, A. Infinitesimal deformation of p-adic differential equations on Berkovich curves. Math. Ann. 368, 111–164 (2017). https://doi.org/10.1007/s00208-016-1417-y

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