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The $p$-curvature conjecture for the non-abelian Gauss-Manin connection
Number Theory · Submitted 12 Dec 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Then, I prove that the analogue of the main technical result of Katz' theorem holds for this connection.
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Originally conjectured unpublished by Grothendieck, then formulated precisely by Katz, the $p$-curvature conjecture is a local-global principle for algebraic differential equations. It is at present open, though various cases are known. Katz subsequently proved this conjecture in a wide range of cases, for differential equations corresponding to the Gauss-Manin connection on algebraic de Rham cohomology. This dissertation addresses the non-abelian analogue of Katz' theorem, in the sense of Simpson's non-abelian Hodge theory, surveyed by Simpson and later developed in characteristic $p$ by Ogus and Vologodsky. Specifically, there is a canonical non-abelian Gauss-Manin connection on $M_{dR}$, the stack of vector bundles with integrable connection, which is the appropriate definition of non-abelian de Rham cohomology. In this dissertation, I introduce this connection and its $p$-curvature; this requires the generalization of the $p$-curvature conjecture due to Bost, Ekedahl and Shepherd-Barron. Then, I prove that the analogue of the main technical result of Katz' theorem holds for this connection.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Originally conjectured unpublished by Grothendieck, then formulated precisely by Katz, the $p$-curvature conjecture is a local-global principle for algebraic differential equations. | background 1.00 | 0.03 | 0.02 | asserted 0.99 |
| s1 | It is at present open, though various cases are known. | background 0.73 | 0.04 | 0.02 | asserted 1.00 |
| s2 | Katz subsequently proved this conjecture in a wide range of cases, for differential equations corresponding to the Gauss-Manin connection on algebraic de Rham cohomology. | background 1.00 | 0.04 | 0.05 | asserted 1.00 |
| s3 | This dissertation addresses the non-abelian analogue of Katz' theorem, in the sense of Simpson's non-abelian Hodge theory, surveyed by Simpson and later developed in characteristic $p$ by Ogus and Vologodsky. | contribution 0.74 | 0.24 | 0.06 | asserted 0.96 |
| s4 | Specifically, there is a canonical non-abelian Gauss-Manin connection on $M_{dR}$, the stack of vector bundles with integrable connection, which is the appropriate definition of non-abelian de Rham cohomology. | background 0.84 | 0.15 | 0.04 | asserted 1.00 |
| s5 | In this dissertation, I introduce this connection and its $p$-curvature; this requires the generalization of the $p$-curvature conjecture due to Bost, Ekedahl and Shepherd-Barron. | contribution 0.92 | 0.91 | 0.07 | asserted 0.99 |
| s6 | Then, I prove that the analogue of the main technical result of Katz' theorem holds for this connection. | result 0.91 | 0.95 | 0.90 | asserted 1.00 |
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Original abstract
Originally conjectured unpublished by Grothendieck, then formulated precisely by Katz, the $p$-curvature conjecture is a local-global principle for algebraic differential equations. It is at present open, though various cases are known. Katz subsequently proved this conjecture in a wide range of cases, for differential equations corresponding to the Gauss-Manin connection on algebraic de Rham cohomology. This dissertation addresses the non-abelian analogue of Katz' theorem, in the sense of Simpson's non-abelian Hodge theory, surveyed by Simpson and later developed in characteristic $p$ by Ogus and Vologodsky. Specifically, there is a canonical non-abelian Gauss-Manin connection on $M_{dR}$, the stack of vector bundles with integrable connection, which is the appropriate definition of non-abelian de Rham cohomology. In this dissertation, I introduce this connection and its $p$-curvature; this requires the generalization of the $p$-curvature conjecture due to Bost, Ekedahl and Shepherd-Barron. Then, I prove that the analogue of the main technical result of Katz' theorem holds for this connection.
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An informal introduction is given in [ref] and a more comprehensive treatment is given in [ref] .
A Simpson correspondence in positive characteristic · Divided powers · 1.00
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