On the Davenport–Heilbronn theorems and second order terms

By Manjul Bhargava, A. Shankar, Jacob Tsimerman. 2010

150 citations recorded by the source corpus.

We give simple proofs of the Davenport–Heilbronn theorems, which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3-torsion elements in the class groups of quadratic fields having bounded discriminant. We also establish second main terms for these theorems, thus proving a conjecture of Roberts. Our arguments provide natural interpretations for the various constants appearing in these theorems in terms of local masses of cubic rings.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c214ebd64bccf643d4e3d71b40f57a0e089467df

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Citations in the corpus

Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces (2015, unknown): “Then as N p → ∞ , µ τ ( p ) is bounded above by a constant c τ := 1 / #Aut( τ ) which is strictly between 0 and 1; this follows from [19, Lemma 18], [3, Lemma 21], [4, Lemma 20] (see [5] for the definition of #Aut( τ )).”
Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves (2010, methodology): “rongly maximal) at all p. We denote the set of elements in UZ that are maximal at pby Up(U) and the set of elements in WZ that are strongly maximal at pby Vp(W). We then have the following lemma (see [9], [4]). Lemma 4.18 We have µp(Up(U)) = (p3 −1)(p2 −1)/p5; µp(Vp(W)) = (p3 −1)2(p2 −1)2 /p10. 40 4”
The geometric sieve and the density of squarefree values of invariant polynomials (2014, methodology): “d the identical arguments there show that the same formula holds also in the cases n = 3 and n = 4. We have proven Theorem 1.3. Remark 4.3 Although we do not carry this out here, using the methods of [1, 15, 39], it is possible to also estimate the o(Xm/d) in these various deductions and thus obtain”
Secondary terms in counting functions for cubic fields (2011, background): “roach nevertheless owes a great deal to theirs. Remark. Roberts’ conjecture has also been proved, with an error term of O(X13/16+ǫ), in recent and independent work of Bhargava, Shankar, and Tsimerman [7]. Their proof also allows local speciﬁcations, as in our Theorem 1.3. Our methods extend to the r”