Torsion conjecture

The torsion conjecture is an important idea in mathematics, especially in areas called algebraic geometry and number theory. It talks about something called the torsion group of special mathematical shapes known as abelian varieties.

The conjecture suggests that the size of this torsion group can be limited based on two things: how complex the shape is (its dimension of the variety) and which number system, or number field, it comes from (number field).

One version of the torsion conjecture says that the size of the torsion group depends only on the complexity of the shape and the degree of the number field. This means we could predict the maximum size of the torsion group without needing to know every detail about the specific shape or number field.

So far, mathematicians have solved the torsion conjecture for simpler shapes called elliptic curves. This progress shows how deep and challenging this problem is, and it helps us understand more about the hidden structures in numbers and shapes. The conjecture is linked to other important ideas, like the uniform boundedness conjecture.

Elliptic curves

From 1906 to 1911, Beppo Levi looked at how points can move and repeat on special math shapes called elliptic curves. He discovered many ways these points could repeat.

Later, mathematicians tried to figure out all the possible ways points could repeat on these curves. Barry Mazur proved an important guess called the torsion conjecture for these curves. Other mathematicians expanded his work to show it worked for more complex number systems too.

Primes that occur as orders of torsion points in small degree d {\displaystyle d}

d {\displaystyle d}	1	2	3	4	5	6	7	8
S ( d ) {\displaystyle S(d)}	Primes ( 7 ) {\displaystyle {\text{Primes}}(7)}	Primes ( 13 ) {\displaystyle {\text{Primes}}(13)}	Primes ( 13 ) {\displaystyle {\text{Primes}}(13)}	Primes ( 17 ) {\displaystyle {\text{Primes}}(17)}	Primes ( 19 ) {\displaystyle {\text{Primes}}(19)}	Primes ( 19 ) ∪ { 37 } {\displaystyle {\text{Primes}}(19)\cup \{37\}}	Primes ( 23 ) {\displaystyle {\text{Primes}}(23)}	Primes ( 23 ) {\displaystyle {\text{Primes}}(23)}

Primes that occur infinitely often as orders of torsion points in small degree d {\displaystyle d}

d {\displaystyle d}	1	2	3	4	5	6	7	8
S ′ ( d ) {\displaystyle S'(d)}	Primes ( 7 ) {\displaystyle {\text{Primes}}(7)}	Primes ( 13 ) {\displaystyle {\text{Primes}}(13)}	Primes ( 13 ) {\displaystyle {\text{Primes}}(13)}	Primes ( 17 ) {\displaystyle {\text{Primes}}(17)}	Primes ( 19 ) {\displaystyle {\text{Primes}}(19)}	Primes ( 19 ) {\displaystyle {\text{Primes}}(19)}	Primes ( 23 ) {\displaystyle {\text{Primes}}(23)}	Primes ( 23 ) {\displaystyle {\text{Primes}}(23)}