Decomposing representations of finite groups on Riemann-Roch spaces

Joyner, David; Ksir, Amy

doi:10.1090/S0002-9939-07-08967-8

Decomposing representations of finite groups on Riemann-Roch spaces
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by David Joyner and Amy Ksir PDF

Proc. Amer. Math. Soc. 135 (2007), 3465-3476 Request permission

Abstract:

If $G$ is a finite subgroup of the automorphism group of a projective curve $X$ and $D$ is a divisor on $X$ stabilized by $G$, then we compute a simplified formula for the trace of the natural representation of $G$ on the Riemann-Roch space $L(D)$, under the assumption that $L(D)$ is “rational”, $D$ is nonspecial, and the characteristic is “good”. We discuss the partial formulas that result if $L(D)$ is not rational.

References

Niels Borne, Une formule de Riemann-Roch équivariante pour les courbes, Canad. J. Math. 55 (2003), no. 4, 693–710 (French, with French summary). MR 1994069, DOI 10.4153/CJM-2003-029-2
Ya. G. Berkovich and E. M. Zhmud′, Characters of finite groups. Part 1, Translations of Mathematical Monographs, vol. 172, American Mathematical Society, Providence, RI, 1998. Translated from the Russian manuscript by P. Shumyatsky [P. V. Shumyatskiĭ] and V. Zobina. MR 1486039, DOI 10.1090/mmono/172
C. Chevalley, A. Weil, “Über das Verhalten der Integrale erster Gattung bei Automorphismen des Funktionenkörpers,” Abh. Math. Sem. Univ. Hamburg 10 (1934), 358-361.
Ron Y. Donagi, Seiberg-Witten integrable systems, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 3–43. MR 1492533
Ron Donagi, Spectral covers, Current topics in complex algebraic geometry (Berkeley, CA, 1992/93) Math. Sci. Res. Inst. Publ., vol. 28, Cambridge Univ. Press, Cambridge, 1995, pp. 65–86. MR 1397059
Noam D. Elkies, Wiles minus epsilon implies Fermat, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993) Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 38–40. MR 1363494
G. Ellingsrud and K. Lønsted, An equivariant Lefschetz formula for finite reductive groups, Math. Ann. 251 (1980), no. 3, 253–261. MR 589254, DOI 10.1007/BF01428945
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.3; 2002, (http://www.gap-system.org).
David Joyner and Amy Ksir, Modular representations on some Riemann-Roch spaces of modular curves $X(N)$, Computational aspects of algebraic curves, Lecture Notes Ser. Comput., vol. 13, World Sci. Publ., Hackensack, NJ, 2005, pp. 163–205. MR 2182040, DOI 10.1142/9789812701640_{0}012
D. Joyner and W. Traves, “Representations of finite groups on Riemann-Roch spaces,” preprint. math.AG/0210408
Ernst Kani, The Galois-module structure of the space of holomorphic differentials of a curve, J. Reine Angew. Math. 367 (1986), 187–206. MR 839131, DOI 10.1515/crll.1986.367.187
Bernhard Köck, Computing the equivariant Euler characteristic of Zariski and étale sheaves on curves, Homology Homotopy Appl. 7 (2005), no. 3, 83–98. MR 2205171
Amy E. Ksir, Dimensions of Prym varieties, Int. J. Math. Math. Sci. 26 (2001), no. 2, 107–116. MR 1836786, DOI 10.1155/S016117120101153X
Sh\B{o}ichi Nakajima, Galois module structure of cohomology groups for tamely ramified coverings of algebraic varieties, J. Number Theory 22 (1986), no. 1, 115–123. MR 821138, DOI 10.1016/0022-314X(86)90032-6
Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380