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Torsion Points on Elliptic Curves over Quartic Fields
William Stein
(this is joint work with Sheldon Kamienny)
University of Washington
May 2010
Motivating Problem
Let be a number field.
Theorem (Mordell-Weil): If is an elliptic curve over , then is a finitely generated abelian group.
Thus is a finite group.
Problem: Which finite abelian groups occur, as we vary over all elliptic curves ?
Observation: is a finite subgroup of , so is cyclic or a product of two cyclic groups.
An Old Conjecture
(Z/2Z) x (Z/2vZ) for v<=4.
Conjecture (Levi around 1908; re-made by Ogg in 1960s):
When , the groups , as we vary over all , are the following 15 groups:
for or
for .
Note:
- This is really a conjecture about rational points on certain curves of (possibly) higher genus (title of Michael Stoll's talk today)...
- Or, it's a conjecture in arithmetic dynamics about periodic points.
Modular Curves
The modular curves and :
- Let be the affine modular curve over whose points parameterize isomorphism classes of pairs , where is a cyclic subgroup of order .
- Let be ... of pairs , where is a point of order .
Let and be the compactifications of the above affine curves.
Observation: There is an elliptic curve with if and only if is nonempty.
Also, is a quotient of , so if is empty, then so is .
Mazur's Theorem (1970s)
Theorem (Mazur) If for some elliptic curve , then .
Combined with previous work of Kubert and Ogg, one sees that Mazur's theorem implies Levi's conjecture, i.e., a complete classification of the finite groups .
Here are representative curves by the way (there are infinitely many for each -invariant):
Mazur's Method
Theorem (Mazur) If for some elliptic curve , then .
Basic idea of the proof:
- Find a rank zero quotient of such that...
- ... the induced map is a formal immersion at infinity (this means that the induced map on complete local rings is surjective, or equivalently, that the induced map on cotangent spaces is surjective).
- Then consider the point corresponding to a pair , where has order .
- If has potentially good reduction at , get contradiction by injecting -torsion mod since , so has multiplicative reduction, hence we may assume reduces to the cusp .
- The image of in is thus in the kernel of the reduction map mod . But this kernel of reduction is a formal group, hence torsion free. But is finite, so image of is 0.
- Use that is a formal immersion at infinity along with step 5, to show that , which is a contradiction since
Mazur uses for the Eisenstein quotient of because he is able to prove -- way back in the 1970s! -- that this quotient has rank by doing a -descent. This is long before much was known toward the BSD conjecture. More recently one can:
- Merel 1995: use the winding quotient of , which is the maximal analytic rank quotient. This makes the arguments easier, and we know by Kolyvagin-Logachev et al. or by Kato that the winding quotient has rank 0.
- Parent 1999: use the winding quotient of , which leads to a similar argument as above. This quotient has rank 0 by Kato's theorem.
Kamienny-Mazur
A prime is a torsion prime for degree if there is a number field of degree and an elliptic curve such that .
Let . For example, .
Finding all possible torsion structure over all fields of degree often involves determining , then doing some additional work (which we won't go into). E.g.,
Theorem (Frey, Faltings): If is finite, then the set of groups , as varies over all elliptic curves over all number fields of degree , is finite.
Kamienny and Mazur: Replace by the symmetric power and gave an explicit criterion in terms of independence of Hecke operators for to be a formal immersion at . A point , where has degree , then defines a point , etc.
Theorem (Kamienny and Mazur):
- ,
- is finite for ,
- has density 0 for all .
Corollary (Uniform Boundedness): There is a fixed constant such that if is an elliptic curve over a number field of degree , then .
(Very surprising!)
Torsion Structures over Quadratic Fields
Theorem (Kenku, Momose, Kamienny, Mazur): The complete list of subgroups that appear over quadratic fields is:
Z/mZ for m<=16 or m=18 (Z/2Z) x (Z/2vZ) for v<=6. (Z/3Z) x (Z/3vZ) for v=1,2 (Z/4Z) x (Z/4ZZ)
and each occurs for infinitely many -invariants.
What is ?
Kamienny, Mazur: "We expect that , but it would simply be too embarrassing to parade the actual astronomical finite bound that our proof gives."
But soon, Merel in a tour de force, proves (by using the winding quotient and a deep modular symbols argument about independence of Hecke operators):
Theorem (Merel, 1996): , for .
thus proving the full Universal Boundedness Conjecture, which is a huge result.
Shortly thereafter Oesterle modifies Merel's argument to get a much better upper bound:
Theorem (Oesterle): .
Parent's Method: Nailing Down S(3)
By Oesterle, we know that .
In 1999, Parent made Kamienny's method applied to explicit and computable, and used this to bound explicitly, showing that . This makes crucial use of Kato's theorem toward the Birch and Swinnerton-Dyer conjecture!
In subsequent work, Parent rules out finally giving the answer:
The list of groups that occur for cubic is still unknown. However, using the notion of trigonality of modular curves (having a degree 3 map to ), Jeon, Kim, and Schweizer showed that the groups that appear for infinitely many -invariants are:
Z/mZ for m<=16, 18, 20 Z/2Z x Z/2vZ for v<=7
What about Degree 4?
By Oesterle, we know that .
Recently, Jeon, Kim, and Park (2006), again used gonality (and big computations with Singular), to show that the groups that appear for infinitely many -invariants for curves over quartic fields are:
Z/mZ for m<=18, or m=20, m=21, m=22, m=24 Z/2Z x Z/2vZ for v<=9 Z/3Z x Z/3vZ for v<=3 Z/4Z x Z/4vZ for v<=2 Z/5Z x Z/5Z Z/6Z x Z/6Z
Question (Kamienny to me): Is
Explicit Kamienny-Parent for
To attack the above unsolved problem about , we made Parent's (1999) approach very explicit in case and (he gives a general criterion for any ...). One arrives that the following (where is a certain explicitly computed element of the Hecke algebra):
NOTES:
- This looks pretty crazy, but this is really just a way of expressing the condition that a certain map is a formal immersion.
- As gets large, there are a LOT of 4-tuples of elements of the Hecke algebra to test for independence mod 2.
- Here is code that implements this algorithm: code.sage
Running the Algorithm
After a few days we find that the criterion is not satisfied for , but it is for .
Conclusion:
Theorem (Kamienny, Stein): .
It's unclear to me, but Kamienny seems to also have a proof that rules out , which would nearly answer the big question for degree .
Future Work
- Kamienny (unpublished): "Moreover 29, 31, 41 , and 59 can't occur over any quartic field... I've known an easy geometric proof for a long time, but I simply forgot about it..."
- Kamienny (unpublished): "For 19 and 23 we only get the result for fields in which at least one of 2, 3 doesn't remain prime. We can try dealing with 19 and 23 by looking (later) at equations for the modular curves if that's computable."
- Alternatively, deal with 19 and 23 in a way similar to how Parent dealt with for , which was the one case he couldn't address using his criterion. (His paper on looks very painful though!)
- Make the algorithm for showing that more efficient. Right now it takes way too long.
- Given 3, repeat my calculations, but for and hope to replace the Oesterle bound of by