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Arul Shankar

Publications and source records attributed to Arul Shankar.

36 records · Page 2Linked to original sources
arXiv2022

Squarefree values of polynomial discriminants II

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.

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arXiv2021

Squarefree values of polynomial discriminants I

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.

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arXiv2020

Heuristics for the asymptotics of the number of $S_n$-number fields

We give a heuristic argument supporting conjectures of Bhargava on the asymptotics of the number of $S_n$-number fields having bounded discriminant. We then make our arguments rigorous in the case $n=3$ giving a new elementary proof of the Davenport-Heilbronn theorem. Our basic method is to count elements of small height in $S_n$-fields while carefully keeping track of the index of the monogenic ring that they generate.

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arXiv2019

Families of elliptic curves ordered by conductor

In this article, we study the family of elliptic curves $E/\mathbb{Q}$, having good reduction at $2$ and $3$, and whose $j$-invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the ratio $Δ(E)/C(E)$ is squarefree; and second, the set of elliptic curves $E$ such that $Δ(E)/C(E)$ is bounded by a small power $(<3/4)$ of $C(E)$. Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor. Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the $2$-Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is $3$. This implies that the average rank of these elliptic curves is finite, and bounded by $1.5$.

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arXiv2018

Odd degree number fields with odd class number

For every odd integer $n \geq 3$, we prove that there exist infinitely many number fields of degree $n$ and associated Galois group $S_n$ whose class number is odd. To do so, we study the class groups of families of number fields of degree $n$ whose rings of integers arise as the coordinate rings of the subschemes of $\mathbb{P}^1$ cut out by integral binary $n$-ic forms. By obtaining upper bounds on the mean number of $2$-torsion elements in the class groups of fields in these families, we prove that a positive proportion (tending to $1$ as $n$ tends to $\infty$) of such fields have trivial $2$-torsion subgroup in their class groups and narrow class groups. Conditional on a tail estimate, we also prove the corresponding lower bounds and obtain the exact values of these averages, which are consistent with the heuristics of Cohen-Lenstra-Martinet-Malle and Dummit-Voight. Additionally, for any order $\mathcal{O}_f$ of degree $n$ arising from an integral binary $n$-ic form $f$, we compare the sizes of $\mathrm{Cl}_2(\mathcal{O}_f)$, the $2$-torsion subgroup of ideal classes in $\mathcal{O}_f$, and $\mathcal{I}_2(\mathcal{O}_f)$, the $2$-torsion subgroup of ideals in $\mathcal{O}_f$. For the family of orders arising from integral binary $n$-ic forms and contained in fields with fixed signature $(r_1,r_2)$, we prove that the mean value of the difference $|\mathrm{Cl}_2(\mathcal{O}_f)| - {2^{1-r_1-r_2}}|\mathcal{I}_2(\mathcal{O}_f)|$ is equal to $1$, generalizing a result of Bhargava and the third-named author for cubic fields. Conditional on certain tail estimates, we also prove that the mean value of $|\mathrm{Cl}_2(\mathcal{O}_f)| - {2^{1-r_1-r_2}}|\mathcal{I}_2(\mathcal{O}_f)|$ remains $1$ for certain families obtained by imposing local splitting and maximality conditions.

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arXiv2017

The number of quartic $D_4$-fields ordered by conductor

We consider families of number fields of degree 4 whose normal closures over $\mathbb{Q}$ have Galois group isomorphic to $D_4$, the symmetries of a square. To any such field $L$, one can associate the Artin conductor of the corresponding 2-dimensional irreducible Galois representation with image $D_4$. We determine the asymptotic number of such quartic $D_4$-fields ordered by conductor, and compute the leading term explicitly as a mass formula, verifying heuristics of Kedlaya and Wood. Additionally, we are able to impose any local splitting conditions at any finite number of primes (sometimes, at an infinite number of primes), and as a consequence, we also compute the asymptotic number of order 4 elements in class groups and narrow class groups of quadratic fields ordered by discriminant. Traditionally, there have been two approaches to counting quartic fields, using arithmetic invariant theory in combination with geometry-of-number techniques, and applying Kummer theory together with L-function methods. Both of these strategies fall short in the case of $D_4$-fields ordered by conductor since counting quartic fields containing a quadratic subfield with large discriminant is difficult. However, when ordering by conductor, we utilize additional algebraic structure arising from the outer automorphism of $D_4$ combined with both approaches mentioned above to obtain exact asymptotics.

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arXiv2017

Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves

We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields $K$ (the trivial bound being $O_ε(|{\rm Disc}(K)|^{1/2+ε})$ by Brauer--Siegel). This yields corresponding improvements to: 1) bounds of Brumer and Kramer on the sizes of 2-Selmer groups and ranks of elliptic curves; 2) bounds of Helfgott and Venkatesh on the number of integral points on elliptic curves; 3) bounds on the sizes of 2-Selmer groups and ranks of Jacobians of hyperelliptic curves; and 4) bounds of Baily and Wong on the number of $A_4$-quartic fields of bounded discriminant.

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arXiv2014

Average size of the 2-Selmer group of Jacobians of monic even hyperelliptic curves

In [5], Manjul Bhargava and Benedict Gross considered the family of hyperelliptic curves over $\Q$ having a fixed genus and a marked rational Weierstrass point. They showed that the average size of the 2-Selmer group of the Jacobians of these curves, when ordered by height, is 3. In this paper, we consider the family of hyperelliptic curves over $\Q$ having a fixed genus and a marked rational non-Weierstrass point. We show that when these curves are ordered by height, the average size of the 2-Selmer group of their Jacobians is 6. This yields an upper bound of 5/2 on the average rank of the Mordell-Weil group of the Jacobians of these hyperelliptic curves. Finally using an equidistribution result, we modify the techniques of [16] to conclude that as $g$ tends to infinity, a proportion tending to 1 of these monic even-degree hyperelliptic curves having genus $g$ have exactly two rational points - the marked point at infinity and its hyperelliptic conjugate.

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arXiv2013

The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1

In this article, we prove that the average rank of elliptic curves over $\mathbb{Q}$, when ordered by height, is less than $1$ (in fact, less than $.885$). As a consequence of our methods, we also prove that at least four fifths of all elliptic curves over $\mathbb{Q}$ have rank either 0 or 1; furthermore, at least one fifth of all elliptic curves in fact have rank 0. The primary ingredient in the proofs of these theorems is a determination of the average size of the $5$-Selmer group of elliptic curves over $\mathbb{Q}$; we prove that this average size is $6$. Another key ingredient is a new lower bound on the equidistribution of root numbers of elliptic curves; we prove that there is a family of elliptic curves over $\mathbb{Q}$ having density at least $55\%$ for which the root number is equidistributed.

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arXiv2013

The average number of elements in the 4-Selmer groups of elliptic curves is 7

We prove that when all elliptic curves over $\mathbb{Q}$ are ordered by height, the average size of their 4-Selmer groups is equal to 7. As a consequence, we show that a positive proportion (in fact, at least one fifth) of all 2-Selmer elements of elliptic curves, when ordered by height, do not lift to 4-Selmer elements, and thus correspond to nontrivial 2-torsion elements in the associated Tate--Shafarevich groups.

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arXiv2013

Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0

We prove an asymptotic formula for the number of ${\rm SL}_3({\mathbb Z})$-equivalence classes of integral ternary cubic forms having bounded invariants. We use this result to show that the average size of the 3-Selmer group of all elliptic curves, when ordered by height, is 4. This implies that the average rank of all elliptic curves, when ordered by height, is less than 1.17. Combining our counting techniques with a recent result of Dokchitser and Dokchitser, we prove that a positive proportion of all elliptic curves have rank 0. Assuming the finiteness of the Tate-Shafarevich group, we also show that a positive proportion of elliptic curves have rank 1. Finally, combining our counting results with the recent work of Skinner and Urban, we show that a positive proportion of elliptic curves have analytic rank 0; i.e., a positive proportion of elliptic curves have non-vanishing $L$-function at $s=1$. It follows that a positive proportion of all elliptic curves satisfy BSD.

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arXiv2013

Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves

We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and cubic cases, respectively. Our techniques are quite general, and may be applied to counting integral orbits in other representations of algebraic groups. We use these counting results to prove that the average rank of elliptic curves over $\mathbb{Q}$, when ordered by their heights, is bounded. In particular, we show that when elliptic curves are ordered by height, the mean size of the 2-Selmer group is 3. This implies that the limsup of the average rank of elliptic curves is at most 1.5.

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arXiv2013

Counting $S_5$-fields with a power saving error term

We show how the Selberg $Λ^2$-sieve can be used to obtain power saving error terms in a wide class of counting problems which are tackled using geometry of numbers. Specifically, we give such an error term for the counting function of $S_5$-quintic fields.

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arXiv2013

Abelian surfaces over finite fields with prescribed groups

Let A be an abelian surface over F_q, the field of q elements. The rational points on A/\F_q form an abelian group A(\F_q) \simeq \Z/n_1\Z \times \Z/n_1 n_2 \Z \times \Z/n_1 n_2 n_3\Z \times\Z/n_1 n_2 n_3 n_4\Z. We are interested in knowing which groups of this shape actually arise as the group of points on some abelian surface over some finite field. For a fixed prime power q, a characterization of the abelian groups that occur was recently found by Rybakov. One can use this characterization to obtain a set of congruences modulo the integers $n_1, n_2, n_3, n_4$ on certain combinations of coefficients of the corresponding Weil polynomials. We use Rybakov's criterion to show that groups \Z/n_1\Z \times \Z/n_1 n_2 \Z \times \Z/n_1 n_2 n_3\Z \times\Z/n_1 n_2 n_3 n_4\Z do not occur if n_1 is very large with respect to n_2, n_2, n_4 (Theorem \ref{splitbound}), and occur with density zero in a wider range of the variables (Theorem \ref{splitbound-average}).

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arXiv2012

On the Davenport-Heilbronn theorems and second order terms

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.

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