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p-adic families of modular forms and p-adic Abel-Jacobi maps

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Abstract

We show that p-adic families of modular forms give rise to certain p-adic Abel-Jacobi maps at their p-new specializations. We introduce the concept of differentiation of distributions, using it to give a new description of the Coleman-Teitelbaum cocycle that arises in the context of the \(\mathcal {L}\)-invariant.

Résumé

Nous associons certaines applications p-adiques d’Abel-Jacobi aux familles analytiques de formes modulaires à ses poids nouveaux en p. Nous introduisons le concept de la dérivée d’une distribution. Utilisant ce concept, nous donnons une nouvelle perspective sur le cocycle de Coleman-Teitelbaum dans le contexte de l’invariant \(\mathcal {L}\).

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Notes

  1. Indeed let K be the finite kernel, so that we may write \(\mathbf {q} _{\Gamma _{N^{+},N^{-}}}=K\times \mathbf {q}_{\Gamma _{N^{+},N^{-}}}^{\prime } \) with \(\mathbf {q}_{\Gamma _{N^{+},N^{-}}}^{\prime }\subset \mathbf {q} _{\Gamma _{N^{+},N^{-}}}\) such that \({{\mathrm{ord}}}\) is injective on \(\mathbf {q} _{\Gamma _{N^{+},N^{-}}}^{\prime }\) and such that the image is a lattice. Then, according to [18, §6.4], \(\frac{\mathbf {T}_{\Gamma _{N^{+},N^{-}}}}{\mathbf {q}_{\Gamma _{N^{+},N^{-}}}^{\prime }}\) is an analytic torus. But since K is finite, \(\frac{\mathbf {T}_{\Gamma _{N^{+},N^{-}}}}{\mathbf {q}_{\Gamma _{N^{+},N^{-}}}}=( \frac{\mathbf {T} _{\Gamma _{N^{+},N^{-}}}}{\mathbf {q}_{\Gamma _{N^{+},N^{-}}}^{\prime }}) ^{K}\) exists (again by [18, §6.4]) and it is also an analytic torus.

  2. Indeed, we have \(I_{\ell }^{0}\circ \iota =\ell \circ I_{\times }^{0}\) for \( \ell ={{\mathrm{ord}}}\) or \(\log _{0}\) by the analogue of (32) It follows that the \(\mathcal {L}\)-invariant of [28] considered in [35] is the \(\mathcal {L}\)-invariant of the monodromy module attached to \(J_{\Gamma _{N^{+},N^{-}}}\), i.e. the \(\mathcal {L}\) -invariant of \(J_{\Gamma _{N^{+},N^{-}}}\). Hence, the main result of [35] gives the required lift of the main result of [16].

  3. We write \(V\otimes _{\iota }W\) (resp. \(V\otimes W\)) to denote \(V\otimes _{K}W\) with the inductive (resp. projective) tensor topology. Also, a morphism \(V\rightarrow W\) is a continuous, K-linear if V and W are locally convex spaces. Completion means Hausdorff completion.

  4. Since the multiplication law \(\mathcal {D}\left( T\right) \times \mathcal {D} \left( T\right) \rightarrow \mathcal {D}\left( T\right) \) is only separately continuous, \(\mathcal {D}\left( T\right) \) is not a locally convex algebra. When T is compact the multiplication law is continuous and \({{\mathrm{Hom}}}_{\mathcal { L}}\left( \mathcal {D}\left( T\right) ,\mathcal {O}\right) \) is the space of morphisms of locally convex algebras.

  5. We remark that [40, Theorem 2.2] can be refined to a bicontinuous isomorphism and extended to the case where the manifold is strictly paracompact.

  6. Suppose that \(\left( F_{i}\right) \rightarrow F\) is a net in \(\mathcal {A}_{ \mathbf {k}}\left( X\right) \) converging to \(F\in \mathcal {A}\left( X, \mathcal {O}\right) \). If \(t\in T\) let \(t:X\rightarrow X\) be the locally analytic map given by left multiplication by t, so that \(t^{*}\left( F_{i}\right) \rightarrow t^{*}\left( F\right) \) Since the multiplication in \(\mathcal {O}\) is separately continuous, \(G\mapsto t^{ \mathbf {k}_{2}}G\) is continuous and \(t^{\mathbf {k}_{2}}F_{i}\rightarrow t^{ \mathbf {k}_{2}}F\). Since \(t^{*}\left( F_{i}\right) =t^{\mathbf {k} _{2}}F_{i}\) we deduce that \(t^{*}\left( F\right) =t^{\mathbf {k}_{2}}F\) for every \(t\in T\), meaning that \(F\in \mathcal {A}_{\mathbf {k}}\left( X\right) \).

  7. According to [32, PartII, Ch.III §12 Theorem1] there exists at most one p-adic manifold structure on \(\overline{X}\) making \(\pi :X\rightarrow \overline{X}\) a topological quotient in such a way that \(\pi \) is a submersion. See [32, PartII, Ch.III §12 Theorem 2] for natural conditions granting the existence of such a quotient.

  8. We have indeed.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada

    Matthew Greenberg

  2. Dipartimento di Matematica, Università di Milano, Via Cesare Saldini 50, 20133, Milan, Italy

    Marco Adamo Seveso

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  1. Matthew Greenberg
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Correspondence to Marco Adamo Seveso.

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MG’s research is supported by NSERC of Canada.

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Greenberg, M., Seveso, M.A. p-adic families of modular forms and p-adic Abel-Jacobi maps. Ann. Math. Québec 40, 397–434 (2016). https://doi.org/10.1007/s40316-016-0060-z

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