Modulo Errors

As the recent technical posts possibly indicate, I’ve become interested in computational aspects of elliptic curves. Most recently, I’ve been thinking about points of small height. Since a defining property of a height function is that there be only finitely many points of height less than any given value, it follows that for a fixed curve there is a lower bound on the height of non-torsion rational points (the torsion points have height 0, so there are only finitely many of those, too, but Mazur’s theorem ensures that and more anyway). It’s natural to ask, therefore, just how close to zero these heights can be. This is very similar to the very first question I discussed with my supervisor during my interview: the Lehmer conjecture, which asks for similar constraints on the Mahler measure of a polynomial.

Of course, we’re not the first to ponder this. There is a conjecture of Lang which looks very strong indeed:

Dem’Janeko, Lang Conjecture
There exist absolute constants c₁, c₂> 0 st for all elliptic curves E/Q and all nontorsion points P∈ E(Q)

It’d take to long to unravel the definition of the minimal discriminant here, but suffice to say it gives an indication of the complexity of an elliptic curve- so the more complicated the curve, the greater the bound on the heights.

Is there evidence for such an ambitious claim? The conjecture can be shown to hold for all elliptic curves with integral j invariant, for instance. But more impressively, it follows from a deceptively simple looking hypothesis, the abc conjecture:

abc Conjecture
For any ε>0 there exists a constant c depending only on ε such that, given integers a,b,c with a+b=c and gcd(a,b,c)=1,

This hypothesis (first advanced in the 1980s) is shaping up to be the new Fermat’s Last theorem- elementary to state, yet likely to require serious mathematical heavy-lifting to resolve. It also has far reaching implications for more demanding number theoretic issues, and can also be brought to bear on the problem at hand, since it would validate another conjecture:

Szpiro Conjecture (over a number field, ratio version)

Define the Szpiro ratio by

Where Δ_E/K is the minimal discriminant and f_E/K the conductor. Then given ε>0 there exist only finitely many elliptic curves E/K such that σ_E/K≥6+ε

Thus, in particular, σ_E/K is bounded above.

The Szpiro conjecture is not as strong as the abc conjecture- whilst abc implies Szpiro, Szpiro only implies a weaker formulation of abc with exponent of 6/5+ε rather than 1+ε. There is a stronger version, the modified Szpiro conjecture, which is equivalent to the abc conjecture: but this extra wiggle room is actually desirable as it holds out hope of bounding the heights even if abc turns out to be intractable. That’s because with Szpiro’s conjecture, the following theorem of Hindry and Silverman implies Lang’s conjecture:

Theorem (Hindry, Silverman)
There exist explicit constants c₁, c₂> 0 such that for all number fields K and for all elliptic curves E/K, any nontorsion point satisfies

Setting K=Q and using the (conjectured) boundedness of the Szpiro ratio therefore gives the desired result; so Lang’s conjecture is as easy as abc!

In fact, for the reasons noted above, it’s even easier than abc, although there’s still no obvious way in! Nonetheless, at this stage I’m really just trying to absorb as many diferent ideas as possible, and having an interesting problem like this to guide me through all the mathematics I need to study is helpful. If nothing else, I could probably generate a vast amount of experimental evidence; there’s a supercomputer around here somewhere…

• Much much more on the abc conjecture can be found at Abderrahmane Nitaj’s page.
• Notes in a variety of formats from a seminar talk, Elliptic curves, the abc conjecture and points of small canonical height, which covers the interplay between the various conjectures in somewhat greater depth than above (including modified Szpiro and proofs of equivalence).
• An introduction to height functions by Joseph Silverman (PDF)
• Hindry, Silverman The canonical height and integral points on elliptic curves Invent. Math. 93 (1988) (MathSciNet entry)

One of the central results in the study of Elliptic curves is the Mordell-Weil theorem, which asserts that the group E(K) is finitely generated. Thus it consists of a finite part- the torsion subgroup - and a free abelian part, the rank of which is notoriously difficult to compute. However, the torsion subgroup is relatively accessible, and this is something I’ve been playing with for a while. It covers a range of techniques and ideas and attempting a concrete implementation in Maple has helped considerably in my understanding of those, even if it is effectively reinventing the wheel given the existence of John Cremona’s Algorithms for elliptic curves. The procedures themselves and worked examples are after the cut; first, some theory.

Mazur’s Theorem

Let E/Q be an elliptic curve. Then the torsion subgroup E_tors(Q) is one of the following fifteen groups:

Z/nZ for 1≤n≤10 or n=12;

Z/2Z X Z/2nZ 1≤n≤4.

Further, each of these groups does occur as an E_tors(Q).

This result is particularly handy as it allows for an experimental approach to be taken, gathering enough computational evidence to determine which form the torsion subgroup takes; knowledge of the order of points being especially useful. For instance, the presence of an order 7 element instantly shows that E_tors(Q) is Z/7Z. Better still, there are results which aid in finding such points:

Nagell-Lutz Theorem

Let E/Q be an elliptic curve of the form

(that is, with the usual labelling of coefficients, a_1=a_3=0) with a,b,c integers. If P an element of E(Q) has finite order then x(P), y(P) are also integers.

Further, For such a point either y(P)=0 or y(P) divides

Hence for such curves it is sufficient to look for integer points; and only finitely many such points are suitable candidates for being torsion points.

Good and Bad Reduction

What of Elliptic curves not in the above form? It is possible to bound the number of torsion elements (and generate candidates) by working over finite fields (which I’ve coincidentally considered before). Save for finitely many primes of bad reduction - those which divide the discriminant of the elliptic curve - it transpires that the torsion subgroup maps injectively to E(F_p). For small primes, this is readily found without anything more sophisticated than brute force. Testing a number of primes can give an upper bound whilst naive searches for integer points can provide a lower bound: appealing again to Mazur’s theorem then usually settles the question.

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