#### Filter Results:

- Full text PDF available (30)

#### Publication Year

1997

2015

- This year (0)
- Last 5 years (16)
- Last 10 years (21)

#### Publication Type

#### Co-author

#### Journals and Conferences

#### Key Phrases

Learn More

- Manjul Bhargava
- The American Mathematical Monthly
- 2000

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you… (More)

In the first two articles of this series, we investigated various higher analogues of Gauss composition, and showed how several algebraic objects involving orders in quadratic and cubic fields could be explicitly parametrized. In particular, a central role in the theory was played by the parametrizations of the quadratic and cubic rings themselves. These… (More)

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)

- Manjul Bhargava
- 2006

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)

- Manjul Bhargava
- Discrete Mathematics
- 1997

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)