Galois structure of the holomorphic polydifferentials of curves
Abstract
Suppose $X$ is a smooth projective geometrically irreducible curve over a perfect field $k$ of positive characteristic $p$. Let $G$ be a finite group acting faithfully on $X$ over $k$ such that $G$ has nontrivial, cyclic Sylow $p$subgroups. In this paper we show that for $m > 1$, the decomposition of $\mathrm{H}^0(X,\Omega_X^{\otimes m})$ into a direct sum of indecomposable $kG$modules is uniquely determined by the divisor class of a canonical divisor of $X/G$ together with the lower ramification groups and the fundamental characters of the closed points of $X$ that are ramified in the cover $X\to X/G$. This extends to arbitrary $m > 1$ the $m = 1$ case treated by the first author with T. Chinburg and A. Kontogeorgis. We discuss some applications to congruences between modular forms in characteristic $0$, to the tangent space of the global deformation functor associated to $(X,G)$, and to the $kG$module structure of RiemannRoch spaces associated to divisors on $X$.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.10789
 Bibcode:
 2021arXiv211010789B
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory;
 Primary 11G20;
 Secondary 14H05;
 14G17;
 20C20
 EPrint:
 45 pages. arXiv admin note: text overlap with arXiv:1707.07133