Moduli Spaces of Shtukas over the Projective Line
Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 2, pp. 393-418.

On donne des équations explicites pour des espaces de modules de chtoucas de Drinfeld sur la droite projective avec structures de niveau Γ(N), Γ 1 (N) et Γ 0 (N), où N désigne un diviseur effectif sur 1 . Si le degré du diviseur N est suffisamment grand, ces espaces de modules sont des surfaces relatives. On étudie certains invariants de l’espace de modules de chtoucas avec structures de niveau Γ 0 (N) pour plusieurs diviseurs de degré 4 sur 1 .

We provide explicit equations for moduli spaces of Drinfeld shtukas over the projective line with Γ(N), Γ 1 (N) and Γ 0 (N) level structures, where N is an effective divisor on 1 . If the degree of N is high enough, these moduli spaces are relative surfaces. We study some invariants of the moduli space of shtukas with Γ 0 (N) level structure for several degree 4 divisors on  1 .

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DOI : 10.5802/jtnb.1207
Classification : 11G09, 14H60, 11G18
Mots clés : Drinfeld shtukas, moduli spaces, Kodaira types
María Inés de Frutos-Fernández 1

1 Department of Mathematics Imperial College London South Kensington Campus, London SW7 2AZ, UK
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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María Inés de Frutos-Fernández. Moduli Spaces of Shtukas over the Projective Line. Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 2, pp. 393-418. doi : 10.5802/jtnb.1207. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1207/

[1] Michael Artin; H. Peter F. Swinnerton-Dyer The Shafarevich–Tate conjecture for pencils of elliptic curves on K3 surfaces, Invent. Math., Volume 20 (1973), pp. 249-266 | DOI | MR | Zbl

[2] Michael F. Atiyah Vector bundles over an elliptic curve, Proc. Lond. Math. Soc., Volume 7 (1957), pp. 414-452 | DOI | MR | Zbl

[3] Pierre Deligne; Yuval Z. Flicker Counting local systems with principal unipotent local monodromy, Ann. Math., Volume 178 (2013) no. 3, pp. 921-982 | DOI | MR | Zbl

[4] Vladimir G. Drinfeld Proof of the global Langlands conjecture for GL(2) over a function field, Funkts. Anal. Prilozh., Volume 11 (1977), pp. 74-75

[5] Vladimir G. Drinfeld Langlands’ conjecture for GL(2) over functional fields, Proceedings of the International Congress of Mathematicians, 1978, pp. 565-574

[6] Vladimir G. Drinfeld Varieties of modules of F-sheaves, Funct. Anal. Appl., Volume 21 (1987) no. 1-3, pp. 107-122 | DOI

[7] Vladimir G. Drinfeld Cohomology of compactified manifolds of modules of F-sheaves of rank 2, J. Math. Sci., New York, Volume 46 (1989), pp. 1789-1821 | DOI

[8] Noam D. Elkies Elliptic K3 surfaces with a 6–torsion section (2019) (available at http://people.math.harvard.edu/~elkies/6t.pdf)

[9] Robert Friedman Algebraic Surfaces and Holomorphic Vector Bundles, Universitext, Springer, 1998 | DOI

[10] María I. de Frutos-Fernández Modularity of elliptic curves defined over function fields, Ph. D. Thesis, Boston University (USA) (2020) (available at https://hdl.handle.net/2144/41489)

[11] Robin Hartshorne Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer, 1977 | DOI

[12] Kunihiko Kodaira On compact analytic surfaces I, Ann. Math., Volume 71 (1960), pp. 111-152 | DOI | MR | Zbl

[13] Kunihiko Kodaira On compact analytic surfaces II, Ann. Math., Volume 77 (1963), pp. 563-626 | DOI | Zbl

[14] Kunihiko Kodaira On compact analytic surfaces III, Ann. Math., Volume 78 (1963), pp. 1-40 | DOI | MR | Zbl

[15] Laurent Lafforgue Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math., Volume 147 (2002), pp. 1-241 | DOI | MR | Zbl

[16] André Néron Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Publ. Math., Inst. Hautes Étud. Sci., Volume 21 (1964), pp. 361-484 | Numdam | Zbl

[17] Matthias Schütt; Tetsuji Shioda Elliptic surfaces, Algebraic geometry in East Asia – Seoul 2008 (Advanced Studies in Pure Mathematics), Volume 60, Mathematical Society of Japan, 2008, pp. 51-160 | Zbl

[18] Michael Szydlo Elliptic fibers over non–perfect residue fields, J. Number Theory, Volume 104 (2004), pp. 75-99 | DOI | MR | Zbl

[19] Ngo Dac Tuan Introduction to the stacks of shtukas, Algebraic cycles, sheaves, shtukas, and moduli. Impanga lecture notes (Trends in Mathematics), Birkhäuser, 2008, pp. 217-236 | DOI | MR | Zbl

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