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 $K$ be a number field.  

Theorem (Mordell-Weil): If $E$ is an elliptic curve over $K$, then $E(K)$ is a finitely generated abelian group.

Thus $E(K)_{\rm tor}$ is a finite group. 

Problem:  Which finite abelian groups $E(K)_{\rm tor}$ occur, as we vary over all elliptic curves $E/K$?

Observation: $E(K)_{\rm tor}$ is a finite subgroup of $\CC/\Lambda$, so $E(K)_{\rm tor}$ is cyclic or a product of two cyclic groups.

An Old Conjecture

Conjecture (Levi around 1908; re-made by Ogg in 1960s): 

  When $K=\QQ$, the groups $E(\QQ)_{\rm tor}$, as we vary over all $E/\QQ$, are the following 15 groups:

    $\ZZ/m\ZZ$                            for $m\leq 10$ or $m=12$

    $(\ZZ/2\ZZ) \times (\ZZ/2v\ZZ)$    for $v\leq 4$.

Note:

1. This is really a conjecture about rational points on certain curves of (possibly) higher genus (title of Michael Stoll's talk today)...
2. Or, it's a conjecture in arithmetic dynamics about periodic points.

Modular Curves

The modular curves $Y_0(N)$ and $Y_1(N)$:

• Let $Y_0(N)$ be the affine modular curve over $\QQ$ whose points parameterize isomorphism classes of pairs $(E,C)$, where $C \subset E$ is a cyclic subgroup of order $N$.
• Let $Y_1(N)$ be ...  of pairs $(E,P)$, where $P\in E(\overline{\QQ})$ is a point of order $N$.

Let $X_0(N)$ and $X_1(N)$ be the compactifications of the above affine curves.

Observation: There is an elliptic curve $E/K$ with $p \mid \#E(K)$ if and only if $Y_1(p)(K)$ is nonempty.

Also, $Y_0(N)$ is a quotient of $Y_1(N)$, so if $Y_0(N)(K)$ is empty, then so is $Y_0(N)$. 

Mazur's Theorem (1970s)

Theorem (Mazur) If $p \mid \#E(\QQ)_{\rm tor}$ for some elliptic curve $E/\QQ$, then $p>13$.

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 $E(\QQ)_{\rm tor}$.

Here are representative curves by the way (there are infinitely many for each $j$-invariant):

Mazur's Method

Theorem (Mazur) If $p \mid \#E(\QQ)_{\rm tor}$ for some elliptic curve $E/\QQ$, then $p>13$.

Basic idea of the proof:  

1. Find a rank zero quotient $A$ of $J_0(p)$ such that...
2. ... the induced map $f:X_0(p) \to A$ 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). 
3. Then consider the point $x \in Y_0(p)$ corresponding to a pair $(E,\langle P \rangle)$, where $P$ has order $p$.  
4. If $E$ has potentially good reduction at $3$, get contradiction by injecting $p$-torsion mod $3$ since $p>13$, so $E$ has multiplicative reduction, hence we may assume $x$ reduces to the cusp $\infty$. 
5. The image of $x$ in $A(\QQ)$ is thus in the kernel of the reduction map mod $3$.     But this kernel of reduction is a formal group, hence torsion free.  But $A(\QQ)=A(\QQ)_{\rm tor}$ is finite, so image of $x$ is 0. 
6. Use that $f$ is a formal immersion at infinity along with step 5, to show that $x=\infty$, which is a contradiction since $x\in Y_0(p).$

Mazur uses for $A$ the Eisenstein quotient of $J_0(p)$ because he is able to prove -- way back in the 1970s! -- that this quotient has rank $0$ by doing a $p$-descent.   This is long before much was known toward the BSD conjecture.  More recently one can:

• Merel 1995: use the winding quotient of $J_0(p)$, which is the maximal analytic rank $0$ 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 $J_1(p)$, which leads to a similar argument as above.  This quotient has rank 0 by Kato's theorem.  

Kamienny-Mazur

A prime $p$ is a torsion prime for degree $d$ if there is a number field $K$ of degree $d$ and an elliptic curve $E/K$ such that $p \mid \#E(K)_{\rm tor}$. 

Let $S(d) = \{ \text{torsion primes for degree } \leq d \}$.  For example, $S(1) = \{2,3,5,7\}$. 

Finding all possible torsion structure over all fields of degree $\leq d$ often involves determining $S(d)$, then doing some additional work (which we won't go into).  E.g.,

Theorem (Frey, Faltings): If $S(d)$ is finite, then the set of groups $E(K)_{\rm tor}$, as $E$ varies over all elliptic curves over all number fields $K$ of degree $\leq d$, is finite. 

Kamienny and Mazur: Replace $X_0(p)$ by the symmetric power $X_0(p)^{(d)}$ and gave an explicit criterion in terms of independence of Hecke operators for $f_d: X_0(p)^{(d)} \to J_0(p)$ to be a formal immersion at $(\infty, \infty,\ldots,\infty)$.   A point $y\in X_0(p)(K)$, where $K$ has degree $d$, then defines a point $\tilde{y} \in X_0(p)^{(d)}(\QQ)$, etc.

Theorem (Kamienny and Mazur):

• $S(2) = \{2,3,5,7,11,13\}$,
• $S(d)$ is finite for $d\leq 8$,
• $S(d)$ has density 0 for all $d$.

Corollary (Uniform Boundedness): There is a fixed constant $B$ such that if $E/K$ is an elliptic curve over a number field of degree $\leq 8$, then $\# E(K)_{\rm tor} \leq B$.

(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 $j$-invariants.

What is $S(d)$?

Kamienny, Mazur: "We expect that $max(S(3)) \leq 19$, 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):  $\max(S(d)) < d^{3 d^2}$, for $d\geq 2$.

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): $\max(S(d)) < (3^{d/2}+1)^2$.

Parent's Method: Nailing Down S(3)

By Oesterle, we know that $\max(S(3)) \leq 37$.  

In 1999, Parent made Kamienny's method applied to $J_1(p)$ explicit and computable, and used this to bound $S(3)$ explicitly, showing that $\max(S(3)) \leq 17$.   This makes crucial use of Kato's theorem toward the Birch and Swinnerton-Dyer conjecture!  

In subsequent work, Parent rules out $17$ finally giving the answer:

$$  S(3) = \{2,3,5,7,11,13\}  $$

The list of groups $E(K)_{\rm tor}$ that occur for $K$ cubic is still unknown.  However, using the notion of trigonality of modular curves (having a degree 3 map to $P^1$), Jeon, Kim, and Schweizer showed that the groups that appear for infinitely many $j$-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 $\max(S(4)) \leq 97$.

Recently, Jeon, Kim, and Park (2006), again used gonality (and big computations with Singular), to show that the groups that appear for infinitely many $j$-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 $S(4) = \{2,3,5,7,11,13,17\}?$

Explicit Kamienny-Parent for $d=4$

To attack the above unsolved problem about $S(4)$, we made Parent's (1999) approach very explicit in case $d=4$ and $\ell=2$ (he gives a general criterion for any $d$...).  One arrives that the following (where $t$ is a certain explicitly computed element of the Hecke algebra):

NOTES:

1. This looks pretty crazy, but this is really just a way of expressing the condition that a certain map is a formal immersion. 
2. As $p$ gets large, there are a LOT of 4-tuples of elements of the Hecke algebra to test for independence mod 2.
3. 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 $p=29,31$, but it is for $37\leq p \leq 97$. 

Conclusion:

Theorem (Kamienny, Stein):  $\max(S(4)) \leq 31$. 

It's unclear to me, but Kamienny seems to also have a proof that rules out $29,31$, which would nearly answer the big question for degree $4$. 

Future Work

1. 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..."  
2. 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."
3. Alternatively, deal with 19 and 23 in a way similar to how Parent dealt with $p=17$ for $d=3$, which was the one case he couldn't address using his criterion.  (His paper on $p=17$ looks very painful though!)
4. Make the algorithm for showing that $\max(S(4)) \leq 31$ more efficient.  Right now it takes way too long.
5. Given 3, repeat my calculations, but for $d=5$ and hope to replace the Oesterle bound of $\max(S(5)) \leq 271$ by $$\max(S(5)) \leq 43 \quad\text{  (or something close)}$$