\(p\)-torsion monodromy representations of elliptic curves over geometric function fields
Let \(E\) be an elliptic curve defined over a function field \(K=k(B)\) (\(k\) an algebraically closed field of characteristic 0 and \(B\) a complex curve) and, for any prime \(p\), let \(E[p]\) be its \(p\)-torsion. Let \(\rho_{E,p}\) be the Galois representation associated to \(E[p]\): the paper proves that non-isotrivial elliptic curves \(E_{/K}\) are classified (modulo isogenies) by their \(\rho_{E,p}\) as long as \(p\) is larger than a constant only depending on the gonality of \(B\), i.e., the smallest degree \(d\) for which there exists a degree \(d\) map \(B\rightarrow \mathbb{P}^1\).NEWLINENEWLINE\noindent The authors consider the compactification \(X(p)\) of the coarse moduli scheme \(Y(p)\) representing pairs \((E_{/K},\varphi:(\mathbb{Z}/p)^2\buildrel\sim\over\rightarrow E[p])\) (\(\varphi\) an isomorphism) and focus their attention on \(Z(p)\), the quotient of \(X(p)\times X(p)\) parametrizing triples \((E_{/K},E'_{/K},\psi:E[p]\buildrel\sim\over\rightarrow E'[p])\). The main result of the paper is obtained as a consequence of a conjecture of \textit{E. Kani} and \textit{W. Schanz} [Math. Z. 227, No. 2, 337--366 (1998; Zbl 0996.14012)] predicting the non-existence of non-Hecke rational and elliptic curves in \(Z(p)\) for large \(p\). To prove the conjecture the authors lift a curve \(B\) of gonality \(n\) in \(Z(p)\) to a curve \(C\rightarrow (X(p)\times X(p))^n\) (using the gonality map \(B\rightarrow \mathbb{P}^1\)) and provide a lower and upper bound on the genus of \(C\) using the Riemann-Hurwitz formula. The lower bound just uses the canonical divisor of \((X(p)\times X(p))^n\), while the upper one uses estimates on the ramification of \(C\rightarrow \mathbb{P}^1\) which involve crucial generalizations of results of \textit{J.-M. Hwang} and \textit{W.-K. To} [Am. J. Math. 124, No. 6, 1221--1246 (2002; Zbl 1024.32013); ibid. 134, No. 1, 259--283 (2012; Zbl 1241.53032)] on volumes of complex analytic subvarieties (some valid for any curve and some specific for non-Hecke curves). Those bounds are not compatible for non-Hecke curves, hence provide a contradiction which proves the Kani-Schanz conjecture.
scientific article; zbMATH DE number 6647930
scientific article