Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 071, 17 pages      arXiv:1712.02437
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

The Chevalley-Weil Formula for Orbifold Curves

Luca Candelori
Department of Mathematics, University of Hawaii at Manoa, Honolulu, HI, USA

Received December 08, 2017, in final form July 02, 2018; Published online July 17, 2018

In the 1930s Chevalley and Weil gave a formula for decomposing the canonical representation on the space of differential forms of the Galois group of a ramified Galois cover of Riemann surfaces. In this article we prove an analogous Chevalley-Weil formula for ramified Galois covers of orbifold curves. We then specialize the formula to the case when the base orbifold curve is the (reduced) modular orbifold. As an application of this latter formula we decompose the canonical representations of modular curves of full, prime level and of Fermat curves of arbitrary exponent.

Key words: orbifold curves; automorphisms; modular curves; Fermat curves.

pdf (426 kb)   tex (23 kb)


  1. Barraza P., Rojas A.M., The group algebra decomposition of Fermat curves of prime degree, Arch. Math. (Basel) 104 (2015), 145-155.
  2. Behrend K., Noohi B., Uniformization of Deligne-Mumford curves, J. Reine Angew. Math. 599 (2006), 111-153, math.AG/0504309.
  3. Candelori L., Franc C., Vector-valued modular forms and the modular orbifold of elliptic curves, Int. J. Number Theory 13 (2017), 39-63, arXiv:1506.09192.
  4. Candelori L., Franc C., Kopp G.S., Generating weights for the Weil representation attached to an even order cyclic quadratic module, J. Number Theory 180 (2017), 474-497, arXiv:1606.07844.
  5. Candelori L., Hartland T., Marks C., Yépez D., Indecomposable vector-valued modular forms and periods of modular curves, Res. Number Theory 4 (2018), Art. 17, 24 pages, arXiv:1707.01693.
  6. Chevalley C., Weil A., Hecke E., Über das Verhalten der Integrale 1. Gattung bei Automorphismen des Funktionenkörpers, Abh. Math. Sem. Univ. Hamburg 10 (1934), 358-361.
  7. Furuta M., Steer B., Seifert fibred homology $3$-spheres and the Yang-Mills equations on Riemann surfaces with marked points, Adv. Math. 96 (1992), 38-102.
  8. Kani E., The Galois-module structure of the space of holomorphic differentials of a curve, J. Reine Angew. Math. 367 (1986), 187-206.
  9. Karpilovsky G., Group representations, Vol. 3, North-Holland Mathematics Studies, Vol. 180, North-Holland Publishing Co., Amsterdam, 1994.
  10. Lang S., Introduction to algebraic and abelian functions, Graduate Texts in Mathematics, Vol. 89, 2nd ed., Springer-Verlag, New York - Berlin, 1982.
  11. Le Bruyn L., Dense families of $B_3$-representations and braid reversion, J. Pure Appl. Algebra 215 (2011), 1003-1014.
  12. Mason G., 2-dimensional vector-valued modular forms, Ramanujan J. 17 (2008), 405-427.
  13. Mehta V.B., Seshadri C.S., Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), 205-239.
  14. Naeff R., The Chevalley-Weil formula, M.Sc. Thesis, University of Amsterdam, 2005.
  15. Nakajima S., On Galois module structure of the cohomology groups of an algebraic variety, Invent. Math. 75 (1984), 1-8.
  16. Nasatyr B., Steer B., Orbifold Riemann surfaces and the Yang-Mills-Higgs equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), 595-643, alg-geom/9504015.
  17. Phillips R., Sarnak P., The spectrum of Fermat curves, Geom. Funct. Anal. 1 (1991), 80-146.
  18. Serre J.-P., Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York - Heidelberg, 1977.
  19. Streit M., Homology, Belyǐ functions and canonical curves, Manuscripta Math. 90 (1996), 489-509.
  20. Tuba I., Wenzl H., Representations of the braid group $B_3$ and of ${\rm SL}(2,{\bf Z})$, Pacific J. Math. 197 (2001), 491-510, math.RT/9912013.
  21. Tzermias P., The group of automorphisms of the Fermat curve, J. Number Theory 53 (1995), 173-178.
  22. Weil A., Über Matrizenringe auf Riemannschen Flächen und den Riemann-Rochsehen Satz, Abh. Math. Sem. Univ. Hamburg 11 (1935), 110-115.
  23. Yui N., On the Jacobian variety of the Fermat curve, J. Algebra 65 (1980), 1-35.

Previous article  Next article   Contents of Volume 14 (2018)