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A differential approach to Ax-Schanuel, I

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arxiv 2102.03384 v5 pith:TZXHTAV3 submitted 2021-02-05 math.NT math.AGmath.LO

David Blázquez-Sanz , Guy Casale , James Freitag , Joel Nagloo This is my paper · ORCID
classification math.NTmath.AGmath.LO
keywords ax-schanueldifferentialderivativesuniformizersgeneralgiveresultstechniques
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In this paper, we prove several Ax-Schanuel type results for uniformizers of geometric structures; our general results describe the differential algebraic relations between the solutions of the partial differential equations satisfied by the uniformizers. In particular, we give a proof of the full Ax-Schanuel Theorem with derivatives for uniformizers of simple projective structure on curves including unifomizers of any Fuchsian group of the first kind and any genus. Combining our techniques with those of Ax, we give a strong Ax-Schanuel result for the combination of the derivatives of the j-function and the exponential function. In the general setting of Shimura varieties, we obtain an Ax-Schanuel theorem for the derivatives of uniformizing maps. Our techniques combine tools from differential geometry, differential algebra and the model theory of differentially closed fields.

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Cited by 4 Pith papers

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    math.AG 2026-07 accept novelty 7.0 of 10

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  2. Bialgebraic geometry of B\"ottcher coordinates

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    Classifies f-bialgebraic sets for Böttcher coordinates of polynomials and proves Ax-Lindemann-Weierstrass and Ax-Schanuel analogs when the Julia set is disconnected.

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  4. Unlikely intersections in Shimura varieties and beyond: a survey

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