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Diophantine problems and p-adic period mappings

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Abstract

We give an alternative proof of Faltings’s theorem (Mordell’s conjecture): a curve of genus at least two over a number field has finitely many rational points. Our argument utilizes the set-up of Faltings’s original proof, but is in spirit closer to the methods of Chabauty and Kim: we replace the use of abelian varieties by a more detailed analysis of the variation of p-adic Galois representations in a family of algebraic varieties. The key inputs into this analysis are the comparison theorems of p-adic Hodge theory, and explicit topological computations of monodromy. By the same methods we show that, in sufficiently large dimension and degree, the set of hypersurfaces in projective space, with good reduction away from a fixed set of primes, is contained in a proper Zariski-closed subset of the moduli space of all hypersurfaces. This uses in an essential way the Ax–Schanuel property for period mappings, recently established by Bakker and Tsimerman.

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Notes

  1. In §4 of [43] the assumption is stated that K is algebraically closed, but this is not used in the proof of Proposition 4.2. Alternately [2, Theorem 5.8] can be used to pass from K to \({\bar{K}}\).

  2. Here and in Sect. 10, the symbol p is used abusively to refer to the indexing on a Hodge filtration. We hope this will not cause confusion.

  3. Here, and in other contexts, we will write \(V_{\mathrm {dR}}\) even though we are using the crystalline functor, because in our applications it will be helpful to think of it in terms of de Rham cohomology.

  4. We use \({{\mathbf {F}}}_q^+\) to denote the additive group \({{\mathbf {F}}}_q\).

  5. A priori, the map is defined up to conjugation by the normalizer of \(\mathrm {Aff}(q)\) in \(\mathrm {Sym}({\mathbf {F}}_q)\). This normalizer is equal to \(\mathrm {Aff}(q)\).

  6. One can also give an algebro-geometric argument, as follows. Suppose to the contrary. Now, as in Sect. 7, there is an associated finite covering \(Y' \rightarrow Y\) such that the various \(Z_i\) fit together into a curve fibration \({\mathsf {Z}} \rightarrow Y'\). If (a) were false, the theorem of the fixed part means that the Hodge structure of the fibers of \({\mathsf {Z}} \rightarrow Y'\) are constant, at least over one component of \(Y'\). By Torelli, this means that all the fibers are actually isomorphic. This contradicts de Franchis’s theorem.

  7. It seems likely that we could also deduce the p-adic transcendence result from the complex transcendence result using the Seidenberg embedding theorem, as in [31, Section 2.5]. We thank the referee for bringing this to our attention.

  8. As in Sect. 2.5, we are abusing the symbol p to refer to the indexing on a Hodge filtration.

  9. This is an unrealistically strong assumption. We include this statement simply to make clear the importance of this problem—controlling monodromy drop along subvarieties—for our method.

  10. In Sect. 10.2only, the symbol d represents the degree of a hypersurface, and \(n-1\) its dimension.

  11. We outline how this is done. We may describe the Zariski closure \({\mathbf {Z}}\) in question as the Tannakian group associated to the neutral Tannakian category of \(G_{{{\mathbf {Q}}}_p}\)-modules generated by \({\mathsf {V}}_y \otimes {{\mathbf {Q}}}_p\) (i.e., the automorphisms of the natural fiber functor). By the theory of Fontaine–Laffaile, there is another fiber functor on this category, arising from passing to filtered \(\phi \)-modules; in particular, this gives rise to another Tannakian group \({\mathbf {Z}}' \), which acts on the (primitive part of the) de Rham cohomology of \(X_y \times _{{{\mathbf {Q}}}} {{\mathbf {Q}}}_p\). These two fiber functors become isomorphic over \(\overline{{{\mathbf {Q}}}_p}\) (cf. [11, §3]); in particular there is an isomorphism of \({\mathsf {V}}_y \otimes \overline{{{\mathbf {Q}}}_p}\) with the de Rham cohomology of \(X_y \otimes _{{{\mathbf {Q}}}} \overline{{{\mathbf {Q}}}_p}\), which can be taken to preserve the respective intersection forms, and which carries \({\mathbf {Z}}_{\overline{{{\mathbf {Q}}}_p}}\) to \({\mathbf {Z}}'_{\overline{{{\mathbf {Q}}}_p}}\).

    The Hodge filtration gives this fiber functor the structure of a filtered fiber functor; it gives a parabolic subgroup \({\mathbf {P}}' \subset {\mathbf {Z}}'\). Now Wintenberger’s canonical splitting of the Hodge filtration provides a character \(\varphi _W{:}\,{\mathbf {G}}_m \rightarrow {\mathbf {P}}'\).

    Now pass to \({{\mathbf {C}}}\) by means of an isomorphism \(\overline{{{\mathbf {Q}}}_p} \simeq {{\mathbf {C}}}\); then \({\mathbf {Z}}'_{{{\mathbf {C}}}}\) acts on the cohomology of \(X_y \otimes _{{{\mathbf {Q}}}} {{\mathbf {C}}}\), as does \(\varphi _0\). We claim that \(\varphi _0\) and \(\varphi _W|_{S^1}\) are conjugate inside \(\mathrm {GAut}(H^d(X_y \otimes _{{{\mathbf {Q}}}} {{\mathbf {C}}})^{\mathrm {prim}})\); but they both preserve the Hodge filtration and induce the same scalar on the successive quotients; the conjugacy then follows by Lemma 2.5.

  12. In more detail: in our reasoning to date, instead of using the finiteness of conjugacy classes of possible \(\rho _y^{\mathrm {ss}}{:}\,G_{{{\mathbf {Q}}}} \rightarrow {\mathbf {G}}'\), we could instead use the stronger finiteness provided by the last sentence of Lemma 2.6. Namely, we fix for each y a parabolic subgroup \({\mathbf {Q}}_y\) containing the image of \(\rho _y\), such that the projection of \(\rho _y\) to its Levi gives the semisimplification, and then use the finiteness up to conjugacy of possible pairs \(({\mathbf {Q}}_y, \rho _y^{\mathrm {ss}})\).

  13. We say here that \((P, \phi _M, F_M)\) is conjugate to \((P', \phi _{M'}, F_{M'})\) when there is \(g \in G\) such that \(\mathrm {Ad}(g) P = P'\), and the induced isomorphism of Levi quotients carries \((\phi _M, F_M)\) to \((\phi _{M'}, F_{M'})\).

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Acknowledgements

This paper owes, of course, a tremendous debt to the work of Faltings—indeed, all the main tools come from his work. Some of the ideas originated in a learning seminar run at Stanford University on Faltings’s proof [14]. The 2017 Stanford Ph.D. thesis [26] of B.L. contained an earlier version of the arguments of this paper. In particular, that thesis presented a proof of the Mordell conjecture conditional on an assumption about monodromy, and verified that assumption for a certain Kodaira–Parshin family in genus 2. We thank Brian Conrad for many helpful conversations and suggestions. A.V. would like to thank Benjamin Bakker, Andrew Snowden and Jacob Tsimerman for interesting discussions. B.L. would like to thank Zeb Brady, Lalit Jain, Daniel Litt, and Johan de Jong. We received helpful comments and feedback from several people about earlier versions of this paper. We would like to thank, in particular, Dan Abramovich, Pedro A. Castillejo, Raymond Cheng, Brian Conrad, Ulrich Goertz, Sergey Gorchinskiy, Kiran Kedlaya, Aaron Landesman, Siyan Daniel Li, Lucia Mocz, Bjorn Poonen, Jack Sempliner, Will Sawin, and Bogdan Zavyalov. We similarly would like to thank the anonymous referee for his or her time and effort. We thank Brian Conrad for pointing out the proof of Lemma 2.4, and for simplifying the proof of Lemma 9.3. We thank Jordan Ellenberg for an interesting discussion about monodromy. During much of the work on this paper, B.L. was supported by a Hertz fellowship and an NSF fellowship and A.V. was supported by an NSF grant. During the final stages of writing A.V. was an Infosys member at the Institute for Advanced Study. We thank all these organizations for their support of our work.

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    Brian Lawrence

  2. Institute for Advanced Study, Princeton, NJ, USA

    Akshay Venkatesh

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Lawrence, B., Venkatesh, A. Diophantine problems and p-adic period mappings. Invent. math. 221, 893–999 (2020). https://doi.org/10.1007/s00222-020-00966-7

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