# Torsion conjecture

In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for abelian varieties states that the order of the torsion group of an abelian variety over a number field can be bounded in terms of the dimension of the variety and the number field. A stronger version of the conjecture is that the torsion is bounded in terms of the dimension of the variety and the degree of the number field. The torsion conjecture has been completely resolved in the case of elliptic curves.

## Elliptic curves

From 1906 to 1911, Beppo Levi published a series of papers investigating the possible finite orders of points on elliptic curves over the rationals. He showed that there are infinitely many elliptic curves over the rationals with the following torsion groups:
* C$_n$ with 1 ≤ n ≤ 10, where C$_n$ denotes the cyclic group of order n;
* C$_{12}$;
* C$_{2n}$ × C$_2$ with 1 ≤ n ≤ 4, where × denotes the direct sum.
At the 1908 International Mathematical Congress in Rome, Levi conjectured that this is a complete list of torsion groups for elliptic curves over the rationals. The torsion conjecture for elliptic curves over the rationals was independently reformulated by Trygve Nagell (1952) and again by Andrew Ogg (1971), with the conjecture becoming commonly known as Ogg's conjecture.
Andrew Ogg (1971) drew the connection between the torsion conjecture for elliptic curves over the rationals and the theory of classical modular curves. In the early 1970s, the work of Gérard Ligozat, Daniel Kubert, Barry Mazur, and John Tate showed that several small values of n do not occur as orders of torsion points on elliptic curves over the rationals. Barry Mazur (1977, 1978) proved the full torsion conjecture for elliptic curves over the rationals. His techniques were generalized by Kamienny (1992) and Kamienny & Mazur (1995), who obtained uniform boundedness for quadratic fields and number fields of degree at most 8 respectively. Finally, Loïc Merel (1996) proved the conjecture for elliptic curves over any number field. He proved for K a number field of degree $d=[K:\mathbb {Q} ]$ and an elliptic curve $E/K$ that there is a bound on the order of the torsion group depending only on the degree $|E(K)_{\text{tors}}|\leq B(d)$. Furthermore if $P\in E(K)_{\text{tors}}$ is a point of prime order $p$ we have $p\leq d^{3d^{2}}.$
An effective bound for the size of the torsion group in terms of the degree of the number field was given by Parent (1999). Parent proved that for $P\in E(K)_{\text{tors}}$ a point of prime power order $p^{n}$ we have $p^{n}\leq B(d,p)={\begin{cases}129(3^{d}-1)(3d)^{6}&{\text{if }}p=2,\\65(5^{d}-1)(2d)^{6}&{\text{if }}p=3,\\65(3^{d}-1)(2d)^{6}&{\text{if }}p>3.\end{cases}}$ Setting $B_{\text{max}}(d)=129(5^{d}-1)(3d)^{6}$ we get from the structure result behind the Mordell-Weil theorem, i.e. there are two integers $n_{1},n_{2}$ such that $E(K)_{\text{tors}}\cong \mathbb {Z} /n_{1}\mathbb {Z} \times \mathbb {Z} /n_{2}\mathbb {Z}$, a coarse but effective bound $B(d)=\left(B_{\text{max}}(d)^{B_{\text{max}}(d)}\right)^{2}.$
Joseph Oesterlé gave in private notes from 1994 a slightly better bound for points of prime order $p$ of $p\leq (3^{d/2}+1)^{2}$, which turns out to be useful for computations over fields of small order, but alone is not enough to yield an effective bound for $|E(K)_{\text{tors}}|$. Derickx et al. (2023) provide a published version of Oesterlé's result.
For number fields of small degree more refined results are known (Sutherland 2012). A complete list of possible torsion groups has been given for elliptic curves over $\mathbb {Q}$ (see above) and for quadratic and cubic number fields. In degree 1 and 2 all groups that arise occur infinitely often. The same holds for cubic fields except for the group C$_{21}$ which occurs only in a single elliptic curve over $K=\mathbb {Q} (\zeta _{9})^{+}$. For quartic and quintic number fields the torsion groups that arise infinitely often have been determined. The following table gives the set of all prime numbers $S(d)$ that actually arise as the order of a torsion point $P\in E(K)_{\text{tors}}$ where ${\text{Primes}}(q)$ denotes the set of all prime numbers at most q (Derickx et al. (2023) and Khawaja (2024)).
Primes that occur as orders of torsion points in small degree $d$ :
| $d$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| $S(d)$ | ${\text{Primes}}(7)$ | ${\text{Primes}}(13)$ | ${\text{Primes}}(13)$ | ${\text{Primes}}(17)$ | ${\text{Primes}}(19)$ | ${\text{Primes}}(19)\cup \{37\}$ | ${\text{Primes}}(23)$ | ${\text{Primes}}(23)$ |
The next table gives the set of all prime numbers $S'(d)$ that arise infinitely often as the order of a torsion point (Derickx et al. (2023)).
Primes that occur infinitely often as orders of torsion points in small degree $d$ :
| $d$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| $S'(d)$ | ${\text{Primes}}(7)$ | ${\text{Primes}}(13)$ | ${\text{Primes}}(13)$ | ${\text{Primes}}(17)$ | ${\text{Primes}}(19)$ | ${\text{Primes}}(19)$ | ${\text{Primes}}(23)$ | ${\text{Primes}}(23)$ |
Barry Mazur gave a survey talk on the torsion conjecture on the occasion of the establishment of the Ogg Professorship at the Institute for Advanced Study in October 2022.