Manjul Bhargava

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In our first article [2] we developed a new view of Gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms. Moreover, we showed that the groups arising from these composition laws were closely related to the class groups of orders in quadratic number fields, while the spaces underlying those(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)
In 1801 Gauss laid down a remarkable law of composition on integral binary quadratic forms. This discovery, known as Gauss composition, not only had a profound influence on elementary number theory but also laid the foundations for ideal theory and modern algebraic number theory. Even today, Gauss composition remains one of the best ways of understanding(More)
Given a subset X of a Dedekind domain D, and a polynomial F # D[x], the fixed divisor d(X, F ) of F over X is defined to be the ideal in D generated by the elements F(a), a # X. In this paper we derive a simple expression for d(X, F ) explicitly in terms of the coefficients of F, using a generalized notion of ``factorial'' introduced by the author in a(More)
In a previous paper [9], we showed that the average rank of all elliptic curves, when ordered by height, is finite. This was accomplished by proving that the average size of the 2-Selmer group of elliptic curves, when ordered by height, is exactly 3; it then followed from the latter result that (the limsup of) the average rank of all elliptic curves is(More)
The abelian group E(Q) of rational points on E is finitely generated [Mor22]. Hence E(Q) ' Z ⊕ T for some nonnegative integer r (the rank) and some finite abelian group T (the torsion subgroup). The torsion subgroup is well understood, thanks to B. Mazur [Maz77], but the rank remains a mystery. Already in 1901, H. Poincaré [Poi01, p. 173] asked what is the(More)