Torsion Points on Elliptic Curves over Quartic Fields

William Stein, Univ of Washington

(this is joint work with Sheldon Kamienny and Michael Stoll)

Sage Days 23, July 2010 at the 

New Contribution

We prove that there are no elliptic curves over a number field $K$ of degree 4 that have a $K$-rational point of prime order $>17$.

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.

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

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

"Theorem": $E(K)_{\rm tor}$ is cyclic or a product of two cyclic groups

(Proof: $E(K)_{\rm tor}$ is a finite subgroup of $\CC/\Lambda$.)

An Old Conjecture

Conjecture (Bepo Levi, 1908 [see Schoof-Schappacher 2004]; remade by Nagel in 1952; then remade by Ogg in 1960's): 

  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:

• This is a conjecture about rational points on certain curves of (possibly) higher genus 
• Levi wrote three papers on this problem.    He viewed it as classifying the ways in which the chord and tangent method can fail to generate infinitely many points on an elliptic curve.   He proved some amazing results!   "... the group A = Z/nZ does not occur for n = 14, 16 and n = 20 and A =Z/nZ × Z/2Z does not occur for n = 10 and n = 12. In these cases he must study some thorny diophantine equations, defining plane curves of genus 1 or 2. He concludes by infinite descent, very much in the spirit of Fermat. In the jargon of the contemporary arithmetic of elliptic curves the infinite descent involves a 2-descent [Levi 1906–08, 17]."

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: If $Y_1(p)(K)$ is empty, then there is no elliptic curve $E/K$ with $p \mid \#E(K)_{\rm tor}$.

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

Mazur's Theorem (1970s)

Theorem (Mazur) We have $Y_1(p)(\QQ)=\emptyset$ for $p>13$.  Thus if $p \mid \#E(\QQ)_{\rm tor}$ for some elliptic curve $E/\QQ$, then $p\leq 13$.

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 (it turns out that 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 $p\leq 13$.

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). 
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 good or additive reduction at $3$, get contradiction by injecting the $p$-torsion mod $3$ since $p>13$. Thus may assume $E$ has multiplicative reduction, and with some work may assume that $x$ reduces (mod 3) 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 contained in a formal group, hence torsion free.  But $A(\QQ)=A(\QQ)_{\rm tor}$ is finite, so image of $x$ in $A(\QQ)$ 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 $q$-descent, for primes $q$ that divide the numerator of $(p-1)/12$.   This is long before much was known toward the Birch and Swinnerton-Dyer conjecture.   Now, much more is known.  More recently, one can:

• Merel 1995: use the bigger winding quotient of $J_0(p)$, which is the maximal analytic rank $0$ quotient.  This makes the arguments above easier and more generalizable.  We know by Kolyvagin-Logachev et al. or by Kato that the winding quotient has rank 0.  (For $p=67$ the winding and Eisenstein quotients already differ, since 67a has trivial torsion and rank 0.)
  
  
• Parent 1999: use instead the winding quotient of $J_1(p)$, which leads to a similar argument as above.  This quotient has rank 0 by Kato's theorem (Kolyvagin-Logachev does not apply).

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.   (Not effective.)

Idea of Kamienny and Mazur: Replace $X_0(p)$ by the symmetric power $X_0(p)^{(d)}$ and gave an explicit criterion in terms of linear independence of Hecke operators (or $q$-expansions of modular forms) 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 $d$.

Abromovich soon proved that $S(d)$ is finite for $d\leq 14$. 

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/4vZ)   for v = 1,2.

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 in all degrees, a huge result.

Shortly thereafter Oesterle modifies Merel's argument to get a much better upper bound:

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

Remark (Merel, personal communication, 2010-05-10)

1. The known bounds for $S(d)$ are exponential in $d$.  However, a polynomial bound on $S(d)$ in $d$ is expected. Therefore, one can not expect to computationally determine the exact list of torsion primes in degree for many more $d$'s. 
2. The bound is obtained by considering (essentially) two cases (according to the type of reduction modulo $\ell$ of your elliptic curve) : in one case it is easily seen to be exponential in $d$, the hard case finally yields a bound which is polynomial in $d$ (something like $O(d^8)$ in my paper, $O(d^6)$ after Oesterlé, I suspect one can lower it to $O(d^2)$). Unsatisfying!
3. If you want a bound depending on the field $K$ (instead of just the degree of $K$), you can obtain a bound like O(size of the residue field of $K$ of smallest order).

Parent's Kamienny 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 $17$ (using an involved argument with formal groups) 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, 2004] 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

Remark: Parent also gave an explicit bound for the torsion of order powers of prime numbers in his thesis...

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: Is $S(4) = \{2,3,5,7,11,13,17\}?$

Explicit Kamienny-Parent for $d=4$

To attack the above unsolved problem about $S(4)$, I made Parent's (1999) approach very explicit in case $d=4$ and $\ell=2$ (he gives a general criterion for any $d$ and $\ell$, which he calls "Kamienny's Criterion"...).  One arrives that the following (where $t$ is a certain explicitly computable element of the Hecke algebra).    With $\ell=2, d=4$, we have $(1+\ell^{d/2})^2=25$.

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 several hours, 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):  $\max(S(4)) \leq 31$. 

This is of course a "theorem that relies on a big computer calculation" that involves substantial code.  

$p\leq 31$: A complete solution Stoll + Hess + [CES]

I talked about this problem in Vancouver recently.  Michael Stoll was there, and afterwords we worked on the case of smaller $p$...

Theorem (Kamienny, Stein, Stoll): $S(4) = \{2,3,5,7,11,13,17\}$

Proof uses that ${\rm rank}(J_1(p))=0$ for $p\leq 31$ (and more!), (see [Conrad-Edixhoven-Stein] about the arithmetic of $J_1(p)$ for small $p$), so one can use much more direct geometric arguments.    This also involves some large computations with Magma on explicit algebraic curves, e.g., Riemann-Roch spaces, enumerating and reducing divisors, etc., built on top of Florian Hess's function fields package.  Stoll: "Finding the degree 4 points takes about 3 hours [...]  The other problem is that Magma crashes once in a while when turning a point into a place. This will be fixed in the next release, but for now, one may have to try the actual checking a few times until it runs through."  (2010-07-06: Stoll just retried the computation in the absolutely latest version of Magma and the crashes are now fixed.)

Conjecture (Stein): $J_1(p)(\QQ)_{\rm tor}$ is generated by differences of rational cusps.

See extensive data about this conjecture in Conrad-Edixhoven-Stein.  Proof of this for certain $p$ is a key input to the above theorem. 

Function fields project in Sage: I really want to implement enough of Hess's algorithm(s) so that the above computation can be done entirely with Sage.  See trac 9054. This is challenging, since nobody has every implemented Hess's algorithm, except Hess. 

Future Work

1. Implement Hess's algorithm, to be able to deal with $p\leq 31$ using Sage.
2. Determine if $J_1(p)(\QQ)_{\rm tor}$ is cuspidal for $p=29$. (I proved this for $p<29$ and $p=31$.)
3. Make the algorithm for showing that $\max(S(4)) \leq 31$ much more efficient, then repeat my calculations, but for $d=5$ and hope to replace the Oesterle bound of $\max(S(5)) \leq 271$ by $$\max(S(5)) \leq 43 \quad\text{  (or something close)}.$$
4. Pie in the sky -- Isogeny degrees? -- still an open problem even over quadratic fields!  
   • Cremona: "I'm also very interested in the corresponding question for $X_0(\ell)$, so we know what the possible prime degrees of isogenies are for a given field (or degree). I had some interesting correspondence about this with Parent about 6 months ago; he says that is still wide open for quadratic fields!    My student Kimi is implementing isogenies of degree 11, 17, 19 (the genus 1 cases) in Sage (work in progress). But to have a genuine isogeny_class() function over any non-Q number fields we need a bound."   
   • Mazur (email): "It would be also interesting if you could, say, rule out a few  primes $p$ occurring as $p$-isogenies over such fields (for non CM curves)?"