Let K be the fraction field of a strictly Henselian DVR of characteristic p ≥ 0 with algebraic closureK, and let α1, ..., α d ∈ P 1 K (K). In this paper, we give explicit generators and relations for the prime-to-pétale fundamental group of P 1 K {α1, ..., α d } that depend (solely) on their intersection behavior. This is done by a comparison theorem that relates this situation to a topological one. Namely, let a1, ..., a d be distinct power series in C[[x]] with the same intersection behavior as the αi's, converging on an open disk centered at 0, and choose a point z0 = 0 lying in this open disk. We compare the natural action of Gal(K/K) on the prime-to-pétale fundamental group of PK {α1, ..., α d } to the topological action of looping z0 around the origin on the fundamental group of P 1 C {a1(z0), ..., a d (z0)}. This latter action is, in turn, interpreted in terms of Dehn twists. A corollary of this result is that every prime-to-p G-Galois cover of P 1 K {α1, ..., α d } satisfies that its field of moduli (as a G-Galois cover) has degree over K dividing the exponent of G/Z(G).

Let k be a subfield of C which contains all 2-power roots of unity, and let K = k(α1, α2, ..., α2g+1), where the αi's are independent and transcendental over k, and g is a positive integer. We investigate the image of the 2-adic Galois action associated to the Jacobian J of the hyperelliptic curve over K given by y 2 = 2g+1 i=1 (x − αi). Our main result states that the image of Galois in Sp(T2(J)) coincides with the principal congruence subgroup Γ(2) ✁ Sp(T2(J)). As an application, we find generators for the algebraic extension K(J[4])/K generated by coordinates of the 4-torsion points of J.

Let k be a field of characteristic 0, and let α1, α2, and α3 be algebraically independent and transcendental over k. Let K be the transcendental extension of k obtained by adjoining the elementary symmetric functions of the αi's. Let E be the elliptic curve defined over K which is given by the equation . by giving recursive formulas for the generators of each, and a gen-Moreover, if we assume that k contains all 2-power roots of unity, for each n, we show that K(E[2 n ]) contains K ′ n and is contained in a certain quadratic extension of K ′ n+1 .

Let K be a field of characteristic different from 2 and let E be an elliptic curve over K, defined either by an equation of the form y 2 = f (x) with degree 3 or as the Jacobian of a curve defined by an equation of the form y 2 = f (x) with degree 4. We obtain generators over K of the 8-division field K(E[8]) of E given as formulas in terms of the roots of the polynomial f , and we explicitly describe the action of a particular automorphism in Gal(K(E[8])/K).

Let A be a two-dimensional abelian variety defined over a number field K. Fix a prime number $\ell $ and suppose $\#A({\mathbf {F}_{\mathfrak {p}}}) \equiv 0 \pmod {\ell ^2}$ for a set of primes ${\mathfrak {p}} \subset {\mathcal {O}_{K}}$ of density 1. When $\ell =2$ Serre has shown that there does not necessarily exist a K-isogenous $A'$ such that $\#A'(K)_{{tor}} \equiv 0 \pmod {4}$ . We extend those results to all odd $\ell $ and classify the abelian varieties that fail this divisibility principle for torsion in terms of the image of the mod- $\ell ^2$ representation.