A positive proportion of locally soluble hyperelliptic curves over ℚ have no point over any odd degree extension
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This article is published in Journal of the American Mathematical Society.The article was published on 2016-07-27 and is currently open access. It has received 42 citations till now. The article focuses on the topics: Hyperelliptic curve & Hyperelliptic curve cryptography.read more
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Squarefree values of polynomial discriminants I
TL;DR: In this paper, it was shown that the lower density of monic integer polynomials with square-free discriminant is positive and that the exponent in this lower bound is optimal.
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Odd degree number fields with odd class number
Wei Ho,Arul Shankar,Ila Varma +2 more
TL;DR: In this article, the authors studied the class groups of families of number fields of degree n whose rings of integers arise as the coordinate rings of the subschemes of integral binary $n$-ic forms.
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Odd degree number fields with odd class number
Wei Ho,Arul Shankar,Ila Varma +2 more
TL;DR: In this paper, the authors studied the class groups of families of number fields of degree n whose rings of integers arise as the coordinate rings of the subschemes of P1 cut out by integral binary n-ic forms and showed that a positive proportion of such fields have trivial 2-torsion subgroups in their class groups and narrow class groups.
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Quadratic Twists of Abelian Varieties With Real Multiplication
TL;DR: In this article, it was shown that a positive proportion of the quadratic twists of the Jacobian of a hyperelliptic curve have no rational points other than those fixed by the hypereliptic involution.
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Cycle class maps and birational invariants
Brendan Hassett,Yuri Tschinkel +1 more
TL;DR: In this paper, the authors introduce new obstructions to rationality for geometrically rational three-folds arising from the geometry of curves and their cycle maps, and introduce new obstacles to rationality.
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La descente sur les variétés rationnelles, II
Book
Torsors and Rational Points
TL;DR: A detailed exposition of the general theory of torsors with key examples, the relation of descent to the Manin obstruction, and applications of descent: to conic bundles, to bielliptic surfaces, and to homogenous spaces of algebraic groups is given in this article.
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Duality Theorems for Curves over P-adic Fields
TL;DR: In this paper, it was shown that if k is a p-adic field, so that Br(k) is canonically isomorphic to Q/7I, then these pairings induce isomorphisms of H 1 (G, Pic0(X) ) (resp. Pico(X), Br(X)), with the dual abelian groups of H 2 (X, G, Pie(X)).
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The Cassels-Tate pairing on polarized abelian varieties
Bjorn Poonen,Michael Stoll +1 more
TL;DR: In this paper, it was shown that a positive proportion of hyperelliptic curves of even genus g. 2 over a global field k have a Jacobian with nonsquare #.. (if finite).