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.
Ogg's conjecture | |
Field: | Number theory |
Conjectured By: | Beppo Levi |
Conjecture Date: | 1908 |
First Proof By: | Barry Mazur |
First Proof Date: | 1977–1978 |
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:
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 and again by, with the conjecture becoming commonly known as Ogg's conjecture.
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. proved the full torsion conjecture for elliptic curves over the rationals. His techniques were generalized by and, who obtained uniform boundedness for quadratic fields and number fields of degree at most 8 respectively. Finally, proved the conjecture for elliptic curves over any number field. He proved for a number field of degree
d=[K:Q]
E/K
|E(K)tors|\leqB(d)
P\inE(K)tors
p
p\leq
3d2 | |
d |
.
An effective bound for the size of the torsion group in terms of the degree of the number field was given by . Parent proved that for
P\inE(K)tors
pn
Bmax(d)=129(5d-1)(3d)6
n1,n2
E(K)tors\congZ/n1Z x Z/n2Z
B(d)=\left(Bmax
Bmax(d) | |
(d) |
\right)2.
Joseph Oesterlé gave in private notes from 1994 a slightly better bound for points of prime order
p
p\leq(3d/2+1)2
|E(K)tors|
For number fields of small degree more refined results are known . A complete list of possible torsion groups has been given for elliptic curves over
Q
+ | |
K=Q(\zeta | |
9) |
S(d)
P\inE(K)tors
Primes(q)
d | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
S(d) | Primes(7) | Primes(13) | Primes(13) | Primes(17) | Primes(19) | Primes(19)\cup\{37\} | Primes(23) | Primes(23) |
The next table gives the set of all prime numbers
S'(d)
d | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
S'(d) | Primes(7) | Primes(13) | Primes(13) | Primes(17) | Primes(19) | Primes(19) | Primes(23) | Primes(23) |
Barry Mazur gave a survey talk on the torsion conjecture[1] on the occasion of the establishment of the Ogg Professorship[2] at the Institute for Advanced Study in October 2022.
q
. Trygve Nagell . Problems in the theory of exceptional points on plane cubics of genus one . Den 11te Skandinaviske Matematikerkongress, Trondheim 1949, Oslo . . 71–76 . 1952. 608098404.