Modulo Errors

The Szpiro conjecture asserts that only finitely many elliptic curves have a Szpiro ratio greater than six, so I thought it’d be interesting to see if I could find any. Moreover, the result of Hindry and Silverman ensures that the greater the Szpiro ratio, the smaller the lower bound for the height of points on a curve; so there was a natural place to start looking- Elkies’ list of the nontorsion points of low height. Armed with PARI, and learning how to use it as I went, I set about computing the ratios for these curves of interest, alongside a couple of thousand test cases for comparison.

The largest ratio uncovered was around 5.6 (curve 3822bg1), but more importantly the lowest was around 2.96 (curve 1110m1), a not particularly inspiring value compared to the other data set. Further, there wasn’t a particularly strong correlation between the rankings according to ratio and height- the four largest Szpiro ratios arising from entries 6, 33, 52 and 38 on Elkies list by height. It seemed, therefore, that points could have impressively small height despite coming from curves of moderate ratio; whilst a high ratio was no guarantee of small height values.

To be thorough, I then set PARI and its elldata package loose on the curves of conductor 2000 or less, of which there are about eleven thousand, to see what kind of ratios could be reached. However, I quickly noticed that rank zero curves could generate high values, but are only relevant to the Szpiro conjecture itself; they have no nontorsion rational points to bound the height of! Below the cut I’ve therefore given the top 10 ratios observed (the best being 8.9), and then all the curves (23) of non-zero rank that attain a ratio above 6.

These I had hoped would therefore be good candidates for small heights, but as the tabulated values show this is very much not the case. Indeed, it seems that the ratio-dependent bounds given by Hindry & Silverman’s result are many orders of magnitude smaller than any heights known. Since no known points are anywhere near close enough to these bounds, ratio-based variations won’t create meaningful extra breathing space; and thus deliberately crafting curves with high Szpiro ratio is unlikely to be a sensible way to seek small heights. Which is a disappointing result, but one probably worth knowing; and there may still be interesting things to say about the distribution of values of the ratio. From my computations on 11308 curves: The lower bound is 1; the average is about 2.82; and the proportion of curves clearing the magical 6 mark is 1.19%, which must of course tend to zero as the sample size increases if the conjecture is to hold.

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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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I’ve added a couple of extra toys to my Maple procedures for elliptic curves. The major change is that it now supports calculation over some finite fields; that is, the integers modulo some prime. To activate this, set workModM to true and specify a modulus M. Then the usual commands ella, ellm, ncopies and mnadd will compute answers mod M instead.

This also makes it much more likely that you’ll be interested in the order of a point, so a procedure modgetorder is included to calculate this by brute force- that is, repeated addition until the zero element is reached.

This makes questions of the type I faced in MA40188: Algebraic Curves much easier. For instance, consider the curve

Over the field with 37 elements, and with a suitable dehomogenisation, the point P: (x,y)=(0,23) is easily verified as an element of E. Then we may easily determine the point Q=-2P, the third intersection of E with the tangent to E at P:

>read “gla.mpl”;

> a_1:=0;a_2:=0;a_4:=-9;a_3:=0;a_6:=11;

>workModM:=true;

>

>M:=37;

>Q:=ncopies(-2,0,23);

1,22

So Q=(1,22). Further, Q is an inflexion point: that is, the tangent to E at Q meets E three times at Q. In terms of the group law, this means -2Q=Q, or equivalently 3Q=0. We can verify this in a couple of ways:

> ncopies(3,Q);

zero

> modgetorder(Q);

3

Since Q=-2P and 3Q=0, it should follow that 6P=0. Which, fortunately, it does:

> modgetorder(0,23);

6

Here are some applications of the procedures developed in the previous post.

Example 1

We consider example 2.4/problem 3.4 from Silverman;

By inspection we can identify some integer points, such as P₁=(-2,3) and P₃=(2,5). A brute force search for x in the range -1000 to 1000 generates the following results-

> #naive point search;

> for k from -1000 to 1000 do

> if(type(simplify((k^3+17)^(1/2)),integer)) then print(k,simplify((k^3+17)^(1/2))); end;

> end;

-2, 3

-1, 4

2, 5

4, 9

8, 23

43, 282

52, 375

Silverman tells us there is a further point, P₈=(5234,378661), plus we have missed all the inverses of our points (since only the positive square root was computed). Brute force on the range -6000 to 6000 of course uncovers P₈, but this computation takes 70.6 seconds, 10.24mb of memory and produces alarming sounds from my new office computer. Silverman observes that (due to a result of Nagell) the rational points are generated by integer combinations of P₁, P₃, so we can proceed by testing some of these instead:

> read “gla.mpl”;

> a_1:=0;a_3:=0;a_2:=0;a_4:=0;a_6:=17; #setting
up the curve;

>#smarter search

> for i from -5 to 5 do

> for j from -5 to 5 do

> if(type(mnadd(i,j,-2,3,-1,4)[1],integer) and type(mnadd(i,j,-2,3,-1,4)[2],integer)) then print(i,j,mnadd(i,j,-2,3,-1,4));

> end if;

> end:

> end:

-3, -2, 43, -282

-2, -3, 5234, -378661

-2, -1, 2, -5

-2, 0, 8, 23

-1, -1, 4, 9

-1, 0, -2, -3

-1, 1, 52, -375

0, -1, -1, -4

0, 1, -1, 4

1, -1, 52, 375

1, 0, -2, 3

1, 1, 4, -9

2, 0, 8, -23

2, 1, 2, 5

2, 3, 5234, 378661

3, 2, 43, 282

All sixteen points are recovered in 0.02 seconds, consuming merely 0.31mb of memory!

Of course, for this example I’m cheating somewhat because I know where I’d like to get to in that I know this list is complete; although a priori there’s no indication of how large the arguments m,n needed to be to generate points such as P₇ or P₈. Nonetheless, this indicates that the procedures allow for more rapid exploration of points on the curve, even if they don’t prove anything (besides existence) by themselves.

Example 2

Maple’s own algcurves package can also be useful to tackle problems given in projective terms. For instance, we can rapidly demonstrate the first result claimed in Exercise 3.3b. Here we are concerned with the curve

which homogenizing away from Z=0 gives

However, this is not of Weierstrass form; but we can retrieve this from Maple:

> with(algcurves):

> f:=x^3+y^3-A

> Weierstrassform(f,x,y,x0,y0)

This yields

But this is not quite of the Weierstrass form as used in Silverman; we substitute -x₀ for x₀ to arrive at

That is, we have that the curve coefficients a_i are all zero except a₆=27A²/4; we also have an isomorphism φ between E and its Weierstrass form given coordinatewise. We can verify with the procedure j_invariant that these are indeed the same curve (it turns out to have j invariant zero, too). Moreover, we can show the desired result, that

For this, let P=(x_0,y_0) a point on the curve in Weierstrass form. Then we compute -P:

> a_1:=0;a_3:=0;a_2:=0;a_4:=0,a_6=-27*A^2/4:

> read “gla.mpl”:

> ellm(x_0,y_0);

x_0, -y_0

Then, identifying P with a projective point via the isomorphism, we find

Which is the desired result.

Update: These procedures have been replaced with a more general (and efficient) set: see this post!

Overview

These maple procedures implement the group law algorithm for an elliptic curve as given in Chapter III Section 2.3 of Silverman’s The Arithmetic of Elliptic Curves. In particular, they can handle the group identity symbolically as it arises during calculations.

Loading the procedures

The procedures can be downloaded as the maple file gla.mpl, which should be placed in whatever directory Maple expects to find it. To be more helfpul, this is probably your home directory on Unix based systems; on Windows it could be the application directory, although if you invoke maple by opening a worksheet, it’ll be the directory that sheet resides in. If you’re unsure, entering the following:

> x:=5;

> save x, “test.m”;

Will create a file test.m which you can then search for to determine the appropriate directory.

Having established where the file goes, you then need to read it into Maple:

> read “gla.mpl”;

Which after some shameless self-promotion gives you the procedures. The assumption is that you have an Elliptic curve given by a Weierstrass equation determined by coefficients a₁,…,a₆ as in Silverman:

You can of course work in full generality without defining these coefficients. The point at infinity is referred to as zero, whilst a point P=(x,y) can be specified as x,y (using (x,y) will likely give errors).

The procedures

Looking at the source you’ll find various procedures, some of which are only needed for the internal workings- in particular, elladd cannot handle zero and should not be used directly. The operations available are:

Elliptic addition (ella)

Addition with the group law is achieved by a call to the ella procedure; a typical call is ella(x1,y1,x2,y2) to compute x1,y1+x2,y2=P₁⊕P₂; however, you may substitute zero for either or both points (for instance, ella(zero,x,y) is valid). In accordance with 2.3(b) this either returns zero or x(P₁+P₂),y(P₁+P₂).

Inverse of a point (ellm)

Given a point P=(x,y), ellm(x,y) returns the group inverse, i.e., the point -P. zero is understood and is its own inverse.

Integer multiples (ncopies)

Repeated iteration of ella for a single point P=(x,y) is made available by ncopies(n,x,y), for n an integer. As before, zero may be (somewhat pointlessly) subsituted for x,y. Care is taken to ensure zero is appropriately handled at each stage, and thus may be returned as an answer (always, for n=0). Negative values of n are of course handled by returning n copies of -P, so this provides an alternative to ellm.

Addition of integer multiples (mnadd)

For convenience, two such integer multiples [m]P₁,[n]P₂ can be added using mnadd(m,n,x1,y1,y2); as usual zero can replace a pair of coordinates (or both).

So, I attended my first mathematics conference last week; two days of pure mathematics talks to lure us into postgraduate study. There are very few ‘pure’ topics I wouldn’t enjoy a lecture on, and I’ve been attending my own university’s staff/postgrad colloquia series this semester simply out of mathematical curiosity and enthusiasm. But beyond this entertainment value, the Durham lectures helped confirm/deny some opinions on potential research areas, so the event was certainly worthwhile.

Michael Drischel (Nottingham) gave the first talk, on Sums of Squares, which you can find online, so I won’t discuss the content too much.

Bill Jackson (Queen Mary London) presented a talk on Rigidity of Graphs concerning combinatorics and graph theory. The first section was presented using the geometry package Cinderella with which I was working for my summer research, demonstrating its many applications. This isn’t a field I’ve studied at all, but the ideas are both accessible and interesting so the talk was one of my favourites. There were even some connections to organic chemistry, which I haven’t thought about for a long time!

Patrick Dorey (Durham) gave a talk entitled Surprises in Quantum Mechanics; sadly I doubt I can ever get to grips with this topic (I can only remember abandoning two books partly read, and both were on Quantum Physics). However (ignoring a talk on funding) the next talk managed to overcome even my general dislike of Physics- Nina Snaith (Bristol)’s talk Every moment brings a treasure: how physicists came to the rescue of number theory. This was one of the more entertaining presentations anyway, but the central result was genuinely intriguing- how random matrix theory, a topic developed in the context of mathematical physics, was able to back up conjectures related to the Riemann zeta function arrived at by traditional number theoretic approaches. The method has turned out to have applications in other areas, and even features as a plot device in the film Proof!

The first day closed with a traditional talk-and-chalk on Geometry and integrability by David Calderbank. Due to a quirk of the MMath structure, I wasn’t allowed to take our differential geometry course. So this was a topic I knew very little about; the talk itself was interesting but I don’t think the field holds much appeal for me. Playing with surfaces is fun, but I prefer my analysis to be more topological rather than heavily connected to calculus.

Some of the ideas of the previous talk were picked up in the first of day two; Michael Singer (Edinburgh) giving an outline of a popular example of an integrable systems in a talk entitled The geometry of nonlinear waves. I’m hoping to track down the Maple worksheet for this one; you really have to see the graphs (or perform experiments with canals!) to appreciate what’s going on.

The most influential talk for me was John Cremona (Nottingham)’s Explicit methods in Number Theory. This was more accurately subtitled Rational points on curves and has cemented my interest in Algebraic Number Theory. For some time I’ve been deliberating between algebraic geometry and algebraic number theory; hindered by our lack of a number theory course at Bath! Based on this talk (and fortunate discussions with John at breakfast) it seems that the aspects of the algebraic geometry course I particularly liked more naturally fall within the remit of number theory; as do the bits of computer algebra that I enjoyed.

Norbert Peyerimhoff (Durham) spoke on averaging and equidistribution problems in geometry; I think this was another talk-and-chalk but I didn’t make notes because the content didn’t really appeal (didn’t help that it got very difficult very quickly!). Similarly Cobordism and groups of formal power series by Neil Strickland (Sheffield) confirmed that algebraic topology, whilst utterly fascinating, is really really difficult. Maple users can find the talk itself at this address, a non-interactive pdf version is also available if you can’t read Maple10 (it seems that, with Maple9.5, I can’t).

The final talk, given by Farid Tari (Durham) Singularities and the imagination offered more diffgeo, and an opportunity for me to demonstrate my complete lack of spatial awareness during the ‘practical’ component where we attempted to build some surface out of a piece of paper! Again, interesting as a talk but not as a career (although Farid was a very good speaker and well suited to holding our attention at the end of a demanding two days).

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Discussion of (affine) Varieties, ideals and examples of how they relate. Statement and proof of the Nullstellensatz.