On the canonical representation of curves in positive characteristic (Q374040)

If \(X\) is a smooth, projective curve over an algebraically closed field \(k\) with genus \(g\) and automorphism group \(G\), then the so-called canonical representation of \(G\) is given by the action of \(G\) on the \(g\)-dimensional vector space global differentials on \(X\). This paper concerns itself with the question of when the canonical representation is irreducible. In characteristic zero, the Hurwitz bound on automorphisms and standard facts about representation theory combine to show that in order for the canonical representation to be irreducible, \(g\) must be no greater than 82. The main result of this paper is that, in characteristic \(p\), there exist arbitrarily high \(g\) such that there exist curves of genus \(g\) with irreducible canonical representation. In particular, the curve \(X\) given by the smooth projective completion of \(y^2 = x^p - x\) has genus \(g = (p-1)/2\) and irreducible canonical representation. The proof involves explicitly relating the canonical representation to the representation of \(PSL_2(\mathbb{F}_p)\) on \(\mathrm{Sym}^{g-1}(k^2)\), which is well-known to be irreducible.NEWLINENEWLINEThe author also studies the action of \(G\) on \(H^1_{dR}(X/k)\), which fits into the following exact sequence of \(k[G]\)-modules: NEWLINE\[NEWLINE0 \to H^0(X, \Omega^1_{X/k}) \to H^1_{dR}(X/k) \to H^1(X, \mathcal{O}_X) \to 0.NEWLINE\]NEWLINE The main result here is that, unlike in characteristic zero, this exact sequence does not split. The proof of this proceeds by an explicit analysis of Čech cocycles for the various cohomology groups, along with the expression of Serre duality via the residue map.