arXiv subjects
Publications and source records attributed to Manjul Bhargava.
We develop geometry-of-numbers methods to count orbits in coregular vector spaces having bounded invariants over any global field. We apply these techniques to bound the average ranks and determine average Selmer group sizes of elliptic curves and Jacobians of hyperelliptic curves over any base global field $F$ of characteristic not $2$, $3$ or $5$.
We develop geometry-of-numbers methods to count orbits in prehomogeneous vector spaces having bounded invariants over any global field. As our primary example, we apply these techniques to determine, for any base global field $F$, the density of discriminants of field extensions of degree at most 5 over $F$.
We show that infinitely many cubic fields have class group of 2-rank 1.
We prove that when totally real (resp., complex) monogenized cubic number fields are ordered by height, the second moment of the size of the $2$-class group is at most $3$ (resp., at most $6$). In the totally real case, we further prove that the second moment of the size of the narrow $2$-class group is at most $9$. This result gives further evidence in support of the general observation, first made in work of Bhargava--Hanke--Shankar and recently formalized into a set of heuristics in work of Siad--Venkatesh, that monogenicity has an altering effect on class group distributions. All of the upper bounds we obtain are tight, conditional on tail estimates.
We show that for any quadratic extension of number fields $K/F$, there exists an abelian variety $A/F$ of positive rank whose rank does not grow upon base change to $K$. This result implies that Hilbert's tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring $\mathcal{O}_K$ of integers of any number field $K$, there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over $\mathcal{O}_K$ has solutions in $\mathcal{O}_K$.
We prove that a positive proportion of integers are expressible as the sum of two rational cubes, and a positive proportion are not so expressible, thus proving a conjecture of Davenport. More generally, we prove that a positive proportion (in fact, at least one sixth) of elliptic curves in any cubic twist family have rank 0, and a positive proportion (in fact, at least one sixth) of elliptic curves with good reduction at 2 in any cubic twist family have rank 1. Our method involves proving that the average size of the 2-Selmer group of elliptic curves in any cubic twist family, having any given root number, is 3. We accomplish this by generalizing a parametrization, due to the second author and Ho, of elliptic curves with extra structure by pairs of binary cubic forms. We then use a novel combination of geometry-of-numbers methods and the circle method that builds on earlier work of Ruth and the first author. In particular, we make use of a new interpretation of the singular integral and series arising in the circle method in terms of real and $p$-adic Haar measures on the relevant group. We prove a uniformity estimate for integral points on the relevant quadric, which along with a sieve allows us to prove that the average size of the 2-Selmer group over the cubic twist family is 3. By suitably partitioning the subset of curves in the family with given root number, we effect a further sieve to show that the root number is equidistributed and that the same average, now taken over only those curves of given root number, is again 3. Finally, we apply the $p$-parity theorem of Dokchitser-Dokchitser and a $p$-converse theorem of Burungale-Skinner to conclude. We also prove the analogue of the above results for the sequence of square numbers: namely, we prove that a positive proportion of square integers are expressible as the sum of two rational cubes, and a positive proportion are not.
Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as may be obtained by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees $n\leq 4$, due to work of van der Waerden and Chow and Dietmann. In this expository article, we outline a proof of van der Waerden's Conjecture for all degrees $n$.
Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$? There are clearly $\gg H^{n-1}$ such polynomials, as may be obtained by setting $a_n=0$. In 1936, van der Waerden conjectured that $O(H^{n-1})$ should in fact also be the correct upper bound for the count of such polynomials. The conjecture has been known previously for degrees $n\leq 4$, due to work of van der Waerden and Chow and Dietmann. The purpose of this paper is to prove van der Waerden's Conjecture for all degrees $n$.
We improve the error terms in the Davenport-Heilbronn theorems on counting cubic fields to $O(X^{2/3 + ε})$. This improves on separate and independent results of the authors and Shankar and Tsimerman. The present paper uses the analytic theory of Shintani zeta functions, and streamlines and simplifies the earlier zeta function proof. We also give a second proof that uses a "discriminant-reducing identity" and translates it into the language of zeta functions. We additionally provide a version of our theorem that counts cubic fields satisfying an arbitrary finite set of local conditions, or even suitable infinite sets of local conditions, where the dependence of the error term on these conditions is described explicitly and significantly improves on our previous works. As we explain, these results lead to quantitative improvements in various arithmetic applications.
We show that an order in a quartic field has fewer than $3000$ essentially different generators as a $\mathbb Z$-algebra (and fewer than $200$ if the discriminant of the order is sufficiently large). This significantly improves the previously best known bound of $2^{72}$. Analogously, we show that an order in a quartic field is isomorphic to the invariant order of at most $10$ classes of integral binary quartic forms (and at most $7$ if the discriminant is sufficiently large). This significantly improves the previously best known bound of $2^{80}$.
We prove an improvement on Schmidt's upper bound on the number of number fields of degree $n$ and absolute discriminant less than X for $6 \leq n \leq 94$. We carry this out by improving and applying a uniform bound on the number of monic integer polynomials, having bounded height and discriminant divisible by a large square, that we proved in a previous work.
In this paper, we resurrect a long-forgotten notion of equivalence for univariate polynomials with integral coefficients introduced by Hermite in the 1850s. We show that the Hermite equivalence class of a polynomial has a very natural interpretation in terms of the invariant ring and invariant ideal associated with the polynomial. We apply this interpretation to shed light on the relationship between Hermite equivalence and more familiar notions of polynomial equivalence, such as ${\rm GL}_2(\mathbb{Z})$- and $\mathbb{Z}$-equivalence. Specifically, we prove that ${\rm GL}_2(\mathbb{Z})$-equivalent polynomials are Hermite equivalent and, for polynomials of degree $2$ or $3$, the converse is also true. On the other hand, for every $n\geq 4$, we give infinite collections of examples of polynomials $f,g\in \mathbb{Z}[X]$ of degree $n$ that are Hermite equivalent but not ${\rm GL}_2(\mathbb{Z})$-equivalent.
We determine average sizes/bounds for the $2$- and $3$-Selmer groups in various families of elliptic curves with marked points, thus confirming several cases of the Poonen--Rains heuristics. As a consequence, we deduce that the average ranks of the elliptic curves in all of these families are bounded. Our proofs are uniform and make use of parametrizations involving various forms of $2 \times 2 \times 2 \times 2$ and $3 \times 3 \times 3$ matrices that we studied in a previous paper. We also deduce that $100\%$ of genus one curves of the form $y^2 = Ax^4 + Bx^2 z^2 + Cz^4$ with $A, B, C \in \mathbb{Z}$, when ordered by $\max\{|B|^2,|AC|\}$, fail the Hasse principle. Other forthcoming applications include proofs that a positive proportion of integers are (respectively, are not) the sum of two rational cubes, and a positive proportion of genus one curves in $\mathbb{P}^1 \times \mathbb{P}^1$ over $\mathbb{Q}$ fail the Hasse principle.
We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we determine the density of integral binary forms that cut out maximal orders in number fields. The latter proves, in particular, an ``arithmetic Bertini theorem'' conjectured by Poonen for $\mathbb{P}^1_\mathbb{Z}$. Our methods also allow us to prove that there are $\gg X^{1/2+1/(n-1)}$ number fields of degree~$n$ having associated Galois group~$S_n$ and absolute discriminant less than $X$, improving the best previously known lower bound of $\gg X^{1/2+1/n}$. Finally, our methods correct an error in and thus resurrect earlier (retracted) results of Nakagawa on lower bounds for the number of totally unramified $A_n$-extensions of quadratic number fields of bounded discriminant.
We determine the probability that a random polynomial of degree $n$ over $\mathbb{Z}_p$ has exactly $r$ roots in $\mathbb{Q}_p$, and show that it is given by a rational function of $p$ that is invariant under replacing $p$ by $1/p$.
In 1848, Hermite introduced a reduction theory for binary forms of degree $n$ which was developed more fully in the seminal 1917 treatise of Julia. This canonical method of reduction made use of a new, fundamental, but irrational $\mathrm{SL}_2$-invariant of binary $n$-ic forms defined over $\mathbb{R}$, which is now known as the Julia invariant. In this paper, for each $n$ and $k$ with $n+k\geq 3$, we determine the asymptotic behavior of the number of $\mathrm{SL}_2(\mathbb{Z})$-equivalence classes of binary $n$-ic forms, with $k$ pairs of complex roots, having bounded Julia invariant. Specializing to $(n,k)=(2,1)$ and $(3,0)$, respectively, recovers the asymptotic results of Gauss and Davenport on positive definite binary quadratic forms and positive discriminant binary cubic forms, respectively.
We determine the density of monic integer polynomials of given degree $n>1$ that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials $f(x)$, such that $f(x)$ is irreducible and $\mathbb Z[x]/(f(x))$ is the ring of integers in its fraction field, is positive, and is in fact given by $ζ(2)^{-1}$. It also follows from our methods that there are $\gg X^{1/2+1/n}$ monogenic number fields of degree $n$ having associated Galois group $S_n$ and absolute discriminant less than $X$, and we conjecture that the exponent in this lower bound is optimal.
In this paper, we prove that when elliptic curves over $\mathbb{Q}$ are ordered by height, the second moment of the size of the $2$-Selmer group is at most $15$. This confirms a conjecture of Poonen and Rains.