Manjul Bhargava

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Contents 1. Introduction 963 2. A game called í µí±ƒ-orderings 967 2.1. On í µí±Ÿ-removed í µí±ƒ-orderings 968 2.2. On í µí±ƒ-orderings of order ℎ 969 3. Rings of integer-valued polynomials 969 3.1. Polynomials with integer-valued divided differences 970 3.2. Integer-valued polynomials having a given modulus 973 4. Smooth functions on compact subsets of(More)
We prove that when all hyperelliptic curves of genus n ≥ 1 having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2. The method of Chabauty can then be used to obtain an(More)
Using only linear algebra over Z p , we dene a discrete probability distribution on the set of isomorphism classes of short exact sequences of Z p-modules, and then conjecture that as E varies over elliptic curves over a xed global eld k, the distribution of is that one. This single conjecture would explain many of the known theorems and conjectures on(More)