for when the margin is too small
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The abc conjecture is deceptively easy to state considering the deep implications it has for number theory. This E2 entry covers a couple of formulations of the conjecture, examines the connection to the ABC theorem for polynomials, and briefly discusses some of the problems it impacts upon.
As I continue to try and get a grasp of heights on elliptic curves, I’ve been looking into the various techniques available for computing them. Since the literal definition as a limit is unworkable, it is common to follow the approach of Silverman and consider the archimedean and nonarchimedean heights in turn. This isn’t too difficult to implement (it’s given in Cremona’s book as well as Silverman’s paper1), but requires a switching process to avoid certain technical wrinkles. An unpublished paper by my supervisor gives a variation on the theme which avoids the switching; I managed to get both working - and agreeing - in Maple fairly easily.
However, there is a radically different approach possible using Elliptic Divisibility Sequences. These a priori needn’t have anything to do with elliptic curves: they’re simply a type of sequence governed by a not-too-unpleasant rule for iteration: plenty of information on them is available from Graham Everest’s homepage. What’s remarkable, however, is that given an elliptic curve a sequence can be defined which allows you to recover the height of points on that curve. There are numerous advantages to this: it doesn’t matter whether you’re working over the rationals or some other number field; you needn’t have a minimal equation for the curve; and computation is essentially mechanical, as the algebraic geometry is hidden from view. For instance, to follow Silverman’s approach (or variations) it is necessary to factorise the discrinant of the curve to establish primes of bad reduction- this may take up the lions’ share of computation time, but isn’t needed when working via EDS.
There are however sacrifices to be made regarding efficiency or accuracy. Searching for points of very small height clearly only requires a few decimal places of accuracy to indicate whether you’re on the right track; but working purely from the definition of an EDS even this can be cripplingly slow to obtain. Thus, guided by Rachel Shipsey’s PhD thesis2, I’ve been trying to create a useful implementation in Maple. The fruits of my labour can be obtained here; they’re still not perfect, as a couple of the examples given by Everest and Ward still cause things to grind to a halt.
Nonetheless, they chew through the others quite happily, and are a lot simpler to use than they were to write! To get started, simply run eds_init with three arguments- the second, third and fourth terms of your sequence. Terms 0 and 1 are assumed to be 0 and 1, for a ‘proper’ EDS. Then, a call to termk(n) will give you the nth term of the sequence. You’ll find that globally there’s a vector h storing all known terms, and another h_known indicating just which those are: typically (and preferably) not all of them, since with Shipsey’s techniques determining the six neighbours of a term hn allows for a jump to the terms h2n (or h2n+1) and its neighbours. In this way, a double and add approach can be employed to obtain a desired term efficiently. There’s still room for improvement here, since this is only the case k=1 from Shipsey’s algorithm: knowing larger factors k allows yet greater jumps through the sequence (to h2kn from hk and hn etc). I’m also tempted to try coding them in PARI instead; I think Maple is stumbling more on performing calculations such as finding norms and gcds than determining the terms themselves, and PARI may be better suited to these.
To get an estimate of the height (as described by Everest and Ward3) , use height_everest(n,d,acc,x,y): n is term of the sequence you wish to work with (the higher the better but slower); d should be the degree of the field etension over the rationals; acc is the level of accuracy (in decimal places) passed to Maple’s eval function during the approximation of the logarithm; whilst x and y are the coordinates of the point of interest.
References
1: Silverman- Computing heights on elliptic curves. (MathSciNet entry)
2: Shipsey- Elliptic Divisibility Sequences (Download)
3: Everest, Ward- The canonical height of an algebraic point on an elliptic curve. (MathSciNet entry) (Download [PS])
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.
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,
\leq%20c(\vareps)\left(\prod_{p|abc}p\right)^{1+\vareps})
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 ΔE/K is the minimal discriminant and fE/K the conductor. Then given ε>0 there exist only finitely many elliptic curves E/K such that σE/K≥6+ε
Thus, in particular, σE/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 pointsatisfies
%20\geq%20{c_1}^{-[K:\mathbb{Q}]}%20\cdot%20{c_2}^{-\sigma_{E/K}}\cdot%20\log|N_{\mathbb{Q}}^K%20\Delta_{E/K}|)
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…
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.
Let E/Q be an elliptic curve. Then the torsion subgroup Etors(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 Etors(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 Etors(Q) is Z/7Z. Better still, there are results which aid in finding such points:
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.
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(Fp). 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.
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.
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, 3786613, 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.
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 ai 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.
I managed to arrange a talk by Nina Snaith for the final event by the undergraduate maths club I helped set up this year. As at Durham, she spoke on her work connecting quantum chaos to the moments of the riemann zeta function, via random matrix theory. Turnout was good, including some postgrads and staff, and the advantage of hearing the talk here is that I benefit from any feedback or insights they have.
To calculate the moments requires knowledge of a particularly elusive coefficient gk, about which very little is known: g0 is trivially 1, Hardy and Littlewood established that g1=1 in the early part of the 20th century, Ingham proved that g2=2 in 1926. No progress was made until 1995, when Conrey and Ghosh conjectured that g3 was, of all things, 42. At a conference in Vienna, Conrey and Gonek planned to present a conjecture for g4; yet the random matrix theorists had a conjecture for all values of k. Following some frantic checking at a blackboard, it was confirmed that the two conjectures agreed for g4=24024: and thus that quantum physics really could offer predictions about number theory!
One of my lecturers plugged the coefficients 1,1,2,42,24024 into the OEIS, and just one result comes up, sequence A039622. Curiously, this is about Young diagrams, which are linked to irreducible representations of the symmetric group. Young Tableau themselves describe a fairly elegant number theory puzzle. I’ve been studying representation theory this semester, and I’m always amazed by how different strands of mathematics can pull together like this- from quantum theory to distribution of primes to symmetry, somehow they’re all woven together.
For an overview of the current status of work on the Riemann Hypothesis, including Random Matrix Theory, see this article by Conrey. Marcus Du Sautoy’s the music of the primes also mentions Nina’s work, and Riemann’s own study of problems in Physics alongside the zeta function.
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A number-theoretic measure of complexity.
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A non-obvious polynomial identity. Writeup includes exercises from MA30172: conjecture and proof.
Graeme Taylor
PhD Student,
Number Theory
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