Topological rigidity of maps in positive characteristic and anabelian geometry

Achinger, Piotr; Stix, Jakob

doi:10.1090/proc/16858

Topological rigidity of maps in positive characteristic and anabelian geometry
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by Piotr Achinger and Jakob Stix;

Proc. Amer. Math. Soc. 152 (2024), 4171-4186

DOI: https://doi.org/10.1090/proc/16858

Published electronically: August 26, 2024

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Abstract:

We study pairs of non-constant maps between two integral schemes of finite type over two (possibly different) fields of positive characteristic. When the target is quasi-affine, Tamagawa showed that the two maps are equal up to a power of Frobenius if and only if they induce the same homomorphism on their étale fundamental groups. We extend Tamagawa’s result by adding a purely topological criterion for maps to agree up to a power of Frobenius.

References

Piotr Achinger, Wild ramification and $K(\pi , 1)$ spaces, Invent. Math. 210 (2017), no. 2, 453–499., DOI 10.1007/s00222-017-0733-5
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). MR 238860
David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008. With appendices by C. P. Ramanujam and Yuri Manin; Corrected reprint of the second (1974) edition. MR 2514037
Jakob Stix, Affine anabelian curves in positive characteristic, Compositio Math. 134 (2002), no. 1, 75–85. MR 1931963, DOI 10.1023/A:1020290619656
Akio Tamagawa, The Grothendieck conjecture for affine curves, Compositio Math. 109 (1997), no. 2, 135–194. MR 1478817, DOI 10.1023/A:1000114400142
Akio Tamagawa, Fundamental groups and geometry of curves in positive characteristic, Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999) Proc. Sympos. Pure Math., vol. 70, Amer. Math. Soc., Providence, RI, 2002, pp. 297–333. MR 1935411, DOI 10.1090/pspum/070/1935411