Let X be quasi-isometric to either the mapping class group equipped with the word metric, or to Teichmüller space equipped with either the Teichmüller metric or the Weil-Petersson metric. We introduce a unified approach to study the coarse geometry of these spaces. We show that the quasi-Lipschitz image in X of a box in R is locally near a standard model of a flat in X . As a consequence, we sh...

Raoul Bott has inspired many of us by the magnificence of his ideas, by the way he approaches and explains mathematics, and by his warmth, friendship, and humor. In celebration of Raoul’s eightieth birthday we offer this brief article in which we will explain how the recent cohomological ideas of Jan Nekovár̆ [N2] imply (under mild hypotheses plus the Shafarevich-Tate conjecture) systematic grow...

Determining all rational points on a curve of genus at least 2 can be difficult. Chabauty's method (1941) is to intersect, for prime number p, in the p-adic Lie group jacobian, closure Mordell-Weil with curve. If rank less than then this has never failed. Minhyong Kim's non-abelian Chabauty programme aims remove condition rank. The simplest case, called quadratic Chabauty, was developed by Bala...

Recall that a Q-Fano variety is by definition a normal projective variety X such that the anticanonical divisor class −K = −K X is Q-Cartier and ample. For such X we define the (Weil) index i = i(X) to be the largest integer such that K X /i exists as a Weil divisor (see [R] for a discussion of Weil divisors and reflexive sheaves, and also Lemma 2 below; NB i differs from the (industry-standard...

Let E/Q be an elliptic curve defined over Q of conductor N and let Gal(Q/Q) be the absolute Galois group of an algebraic closure Q of Q. For an automorphism σ ∈ Gal(Q/Q), we let Q be the fixed subfield of Q under σ. We prove that for every σ ∈ Gal(Q/Q), the Mordell-Weil group of E over the maximal Galois extension of Q contained in Q σ has infinite rank, so the rank of E(Q σ ) is infinite. Our ...

The canonical height ĥ on an abelian variety A defined over a global field k is an object of fundamental importance in the study of the arithmetic of A. For many applications it is required to compute ĥ(P ) for a given point P ∈ A(k). For instance, given generators of a subgroup of the Mordell-Weil group A(k) of finite index, this is necessary for most known approaches to the computation of gen...

An extension of subgroups $H\leqslant K\leqslant F_A$ the free group rank $|A|=r\geqslant 2$ is called onto when, for every ambient basis $A'$, Stallings graph $\Gamma_{A'}(K)$ a quotient $\Gamma_{A'}(H)$. Algebraic extensions are and converse implication was conjectured by Miasnikov-Ventura-Weil, resolved in negative, first Parzanchevski-Puder $r=2$, recently Kolodner general rank. In this not...

A technique of descent via 4-isogeny is developed on the Jacobian of a curve of genus 2 of the form: Y 2 = q1(X)q2(X)q3(X), where each qi(X) is a quadratic defined over Q. The technique offers a realistic prospect of calculating rank tables of Mordell-Weil groups in higher dimension. A selection of worked examples is included as illustration.

Given two elliptic curves defined over a number field K, not both with j-invariant zero, we show that there are infinitely many D ∈ K × with pairwise distinct image in K × /K × 2 , such that the quadratic twist of both curves by D have positive Mordell-Weil rank. The proof depends on relating the values of pairs of cubic polynomials to rational points on another elliptic curve, and on a fiber p...