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

@article{Bhargava2017BoundsO2, title={Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves}, author={Manjul Bhargava and Arul Shankar and Takashi Taniguchi and Frank Thorne and Jacob Tsimerman and Yongqiang Zhao}, journal={arXiv: Number Theory}, year={2017} }

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 $O_{\epsilon}(|{\rm Disc}(K)|^{1/2+\epsilon})$ 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…

## 49 Citations

Bounds for the ℓ ‐torsion in class groups

- Mathematics
- 2017

We prove for each integer ℓ⩾1 an unconditional upper bound for the size of the ℓ ‐torsion subgroup ClK[ℓ] of the class group of K , which holds for all but a zero density set of number fields K of…

On ℓ-torsion in class groups of number fields

- Mathematics
- 2016

For each integer $\ell \geq 1$, we prove an unconditional upper bound on the size of the $\ell$-torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of…

Bounds for the integral points on elliptic curves over function fields

- Mathematics
- 2017

In this paper we give an upper bound for the number of integral points on an elliptic curve E over F_q[T] in terms of its conductor N and q. We proceed by applying the lower bounds for the canonical…

A sharp upper bound for the 2‐torsion of class groups of multiquadratic fields

- MathematicsMathematika
- 2022

Let K be a multiquadratic extension of Q$\mathbb {Q}$ and let Cl+(K)$\text{Cl}^{+}(K)$ be its narrow class group. Recently, the authors (Koymans and Pagano, Int. Math. Res. Not. 2022 (2022), no. 4,…

On a conjecture for $\ell$-torsion in class groups of number fields: from the perspective of moments.

- Mathematics
- 2019

It is conjectured that within the class group of any number field, for every integer $\ell \geq 1$, the $\ell$-torsion subgroup is very small (in an appropriate sense, relative to the discriminant of…

Large families of elliptic curves ordered by conductor

- MathematicsCompositio Mathematica
- 2021

In this paper 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…

Descent on elliptic surfaces and arithmetic bounds for the Mordell-Weil rank

- Mathematics, Computer Science
- 2018

An upper bound for the rank of a non-constant elliptic surface is obtained using the use of p-descent techniques for elliptic surfaces over a perfect field of characteristic not $2$ or $3$ under mild hypotheses.

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

- Mathematics
- 2020

Abstract Elementary abelian groups are finite groups in the form of A = ( ℤ / p ℤ ) r {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer ℓ > 1 {\ell>1} and r > 1 {r>1} , we…

Families of elliptic curves ordered by conductor

- Mathematics, Computer Science
- 2019

It is proved 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, which implies that theaverage rank of these elliptic curve is finite, and bounded by $1.5$.

The average size of $3$-torsion in class groups of $2$-extensions

- Mathematics
- 2021

We determine the average size of the 3-torsion in class groups of G-extensions of a number field when G is any transitive 2-group containing a transposition, for example D4. It follows from the…

## References

SHOWING 1-10 OF 22 REFERENCES

Integral points on elliptic curves and 3-torsion in class groups

- Mathematics
- 2004

We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on…

Brauer-Siegel for arithmetic tori and lower bounds for Galois orbits of special points

- Mathematics
- 2011

In \cite{S}, Shyr derived an analogue of Dirichlet's class number formula for arithmetic Tori. We use this formula to derive a Brauer-Siegel formula for Tori, relating the Discriminant of a torus to…

A bound for the 3-part of class numbers of quadratic fields by means of the square sieve

- Mathematics
- 2006

Abstract We prove a nontrivial bound of O(|D|27/56+ε) for the 3-part of the class number of a quadratic field ℚ(√D) by using a variant of the square sieve and the q-analogue of van der Corput's…

Reflection Principles and Bounds for Class Group Torsion

- Mathematics
- 2007

We introduce a new method to bound -torsion in class groups, combining analytic ideas with reflection principles. This gives, in particular, new bounds for the 3-torsion part of class groups in…

Dirichlet series associated to cubic fields with given quadratic resolvent

- Mathematics
- 2013

Let k be a quadratic field. We give an explicit formula for the Dirichlet series enumerating cubic fields whose quadratic resolvent field is isomorphic to k.
Our work is a sequel to previous work of…

The number of integral points on arcs and ovals

- Mathematics
- 1989

integral lattice points, and that the exponent and constant are best possible. However, Swinnerton–Dyer [10] showed that the preceding result can be substantially improved if we start with a fixed,…

The equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields

- MathematicsCompositio Mathematica
- 2016

For $n=3$, $4$, and 5, we prove that, when $S_{n}$-number fields of degree $n$ are ordered by their absolute discriminants, the lattice shapes of the rings of integers in these fields become…

On the Bombieri-Pila Method Over Function Fields

- Mathematics
- 2015

E. Bombieri and J. Pila introduced a method for bounding the number of integral lattice points that belong to a given arc under several assumptions. In this paper we generalize the Bombieri-Pila…

Linear Growth for Certain Elliptic Fibrations

- Mathematics
- 2015

We prove that the number of rational points of bounded height on certain del Pezzo surfaces of degree 1 defined over Q grows linearly, as predicted by Manin's conjecture.

The density of abelian cubic fields

- Mathematics
- 1954

In the following note we show that the abelian cubic fields are rare in relation to all cubic fields over the rationals. This is no surprise since an irreducible cubic equation generates an abelian…