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Gonality of modular curves in characteristic \(p\). (English) Zbl 1138.14016

Let \(k\) be an algebraically closed field of positive characteristic \(p\). Let \(N\geq1\) be an integer prime to \(p\) and \(e\geq1\) an integer. Denote by \(X(p^e;N)\) the modular curve parametrizing Igusa structures of level \(p^e\) [N. Katz and B. Mazur, Arithmetic Moduli of Elliptic Curves, Princeton Univ. Press (1985; Zbl 0576.14026), chapter 12] and full level \(N\) structure.
In this paper a modular curve means any quotient of \(X(p^e;N)\) by a subgroup of \(((\mathbb{Z}/p^e\mathbb{Z})^*\times\text{SL}_2(\mathbb{Z}/N\mathbb{Z}))/\{\pm1\}\). The author proves that in any sequence of distinct modular curves their \(k\)-gonality tends to infinity. As an application he obtains the following nice result on the image of Galois representations. Let \(E\) be an ordinary elliptic curve defined over a field \(K\) of characteristic \(p\geq0\), \(K^s\) a separable closure of \(K\), \(\text{Gal}(K^s/K)\) the absolute Galois group of \(K\). Let \(e\geq1\) be an integer, \(F^e:E\to E^{(p^e)}\) the \(e\)-th iterate of the absolute Frobenius morphism and \(V_e:E^{(p^e)}\to E\) its dual isogeny. Then \(\text{Gal}(K^s/K)\) acts on both \(\varprojlim_e\ker(V_e)\) and \(\prod_{\ell\neq p}T_{\ell}(E)\), yielding a homomorphism \(\rho_E:\text{Gal}(K^s/K)\to\mathbb{Z}_p^*\times\prod_{\ell\neq p}\text{GL}_2(\mathbb{Z}_{\ell})\). Let \(S:=\mathbb{Z}_p^*\times\prod_{\ell\neq p}\text{SL}_2(\mathbb{Z}_{\ell})\).
His second result is that given an integer \(d\geq1\), there exists a constant \(N_{p,d}\) depending only on \(p\) and \(d\) such that for any field \(k\) of characteristic \(p\geq0\), and any field \(K\) of degree at most \(d\) over \(k(t)\), and any elliptic curve \(E\) defined over \(K\) whose \(j\)-invariant is not algebraic over \(k\), the index \((S:\rho_E(\text{Gal}(K^s/K))\cap S)\) is at most \(N_{p,d}\).

MSC:

14G35 Modular and Shimura varieties
11G18 Arithmetic aspects of modular and Shimura varieties
14H51 Special divisors on curves (gonality, Brill-Noether theory)

Citations:

Zbl 0576.14026
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