On Relative Computability for Curves

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On Relative Computability for Curves Asia Pacific Mathematics Newsletter 1 OnOn Relative Relative Computability Computability for for Curves Curves MinhyongMinhyong KimKim The following essay is based on a lecture delivered conjecture in question probes this relationship at the Isaac Newton Institute in the summer of more deeply. 2005. Conjecture 1 (Matiyasevich). The finiteness prob- In the fall of 2004, I visited the excellent mathe- lem for integral points is undecidable relative to the matical logic group at the University of Illinois, existence problem. and greatly appreciated the hospitality of Carl Jockusch, who was gracious enough to share his In other words, even given an “existence ora- office with me for the duration of my stay. In cle,” i.e. a decision oracle for the existence prob- due course, I had the pleasure of conversing with lem (or equivalently, an oracle for the halting him over lunch. It was then that I learnt of the problem), the finiteness problem should be un- term “relative computability,” a subject to which decidable. I am sure one can make this conjecture I am told Carl has made profound contributions. more precise or generalise it in many ways using Furthermore, Carl was kind enough to inform the sophisticated machinery of recursion theory, me of a remarkable conjecture that provides a of which I am woefully ignorant. Also, for a comfortable framework for the topic I am set to naive number-theorist, the subtleties of relative discuss. So let us start with that. computability are often hard to comprehend in a situation where the oracle whose existence we Here, the objects of interest are integral need to assume is known not to exist. This is of Diophantine equations, that is, equations of the course because we are more obsessed with solving form problems than classifying them. In any case, when f (x , x , ... , x ) = 0 1 2 n I heard this conjecture, it seemed natural enough to ask this question in a context where a relative where f (x1, ... , xn) is a polynomial with integral computability result still has a chance of leading coefficients. I am sure you know about the exis- to an actual reduction of the problems of interest. tence problem related to this equation, that is, the That is, we can shift our attention (seemingly problem of determining the existence of integral slightly) to rational solutions rather than just the solutions, as well as the undecidability result of integral ones. Having done that, one finds that Matiyasevich [1]. The subject of the conjecture, this conjecture relates rather well to established however, goes beyond existence. That is, it con- programs in Diophantine geometry, and a precise siders simultaneously the problem of determining articulation of this relationship becomes quite de- the finiteness of the solution set. It is easy to see, sirable. I will not attempt to carry this out today, by adjoining a dummy variable, that the finiteness out of pure laziness. However, I do wish to give problem is undecidable as well: f (x1, x2, ... , xn) = 0 some sense of the issues that come up, and maybe has a solution if and only if put forth a suggestion or two as to the kind of g(x1, x2, ... , xn, xn+1) = f (x1, x2, ..., xn) = 0 phenomena one should expect. For example, here is something of a guess. has infinitely many solutions. However, even with the general result, you probably know that some Guess 2. For rational solutions, the finiteness problem of the most celebrated theorems of arithmetic are is decidable relative to the existence problem. about finiteness for specific sorts of equations. In fact, many of them state finiteness in total That is, as far as rational solutions are con- ignorance of existence. And then, sometimes you cerned, my expectation goes counter to the con- know existence and nothing about finiteness. But jecture for integral solutions. That this could be so as far as the decision problem is concerned, the is not too surprising since, I believe, experienced 16 April 2013, Volume 3 No 2 Asia Pacific Mathematics Newsletter2 recursion theorists and number theorists do find — If g(f ) = 0 then the solution set is empty or the nature of rational and integral solutions to infinite. be very different. In particular, I do not think — If g(f ) = 1 then the solution set can be empty, too many people expect Z to be definable in Q. non-empty finite, or infinite. Barry Mazur has pointed out that the explicit — Finally, if g(f ) ≥ 2, then the solution set is family of equations constructed by Matiyasevich empty, or non-empty finite. all have rational solutions for trivial reasons (I A fact that emerges from this classification is have not verified this myself). As another illustra- that if we restrict our attention to equations with tion, consider the case of elliptic curves (curves of g(f ) ≥ 2, then my guess is trivially correct (in so genus one equipped with a rational point) where far as the stated classification is trivial). However, the finiteness of integral solutions is well-known. what seems not entirely trivial is that even more For rational solutions, by contrast, the decision is true, in some sense. To flesh out this cryptic problem for finiteness forms an important part of comment, we start by recalling the situation in the Birch and Swinnerton-Dyer (BSD) conjecture. genus zero. Here, after some change of variables, In other words, the rational case is notoriously one essentially reduces to equations of the form difficult. More on this point later. I wish to post- 2 + 2 = pone to the end of the lecture an explanation ax by c, of the intuition behind my guess which needs where a, b, c are non-zero. In this classical case, to be vague anyway, since otherwise, I would there is the fact involving the Hilbert symbol, that have done the work necessary to elevate the a solution exists if and only if guess to a conjecture. Instead, I will concentrate = for now on the case of curves. That is, to say, (c, −1)v(c, a)v(c, b)v(a, b)v 1 we are interested in Diophantine equations in for v = ∞ and v = p for all prime factors p of abc. If two-variables, this criterion tells us a solution exists, one can just = f (x, y) 0, search until one is found. Afterwards, I am sure where f is again a polynomial with integral coef- you are familiar with the method of sweeping ficients, but assumed now to be irreducible over lines, whereby all the solutions can be constructed the complex numbers. [A note for geometers: In from just one. In short, the existence oracle (which this essay, although I cannot help lapsing into is available) already provides us with a method geometric terminology, the emphasis really is on for “constructing” all solutions. A few years ago, I the equations themselves. That is, the precise pre- was happy to discover that a similar phenomenon sentation under discussion, as input for machines, occurs when the genus is at least two. That is to is the focal point.] As stated, we will be interested say, in the rational solutions, which I will mostly refer to merely as solutions, for brevity. A rough Observation 3. For equations of genus at least two, classification of the solution set, representing the relative to the existence problem, the full solution set main achievements of 20th century number the- is computable as a function of f. ory, depends on the genus, g(f ), of the equation (or the polynomial). That is, one considers the Incidentally, the computability of the solution field set for curves of higher genus is one of the two most important questions regarding the arith- C(x)[y]/(f (x, y)), metic of curves, the other being the BSD conjec- which can be realised as the field of meromor- ture. Usually, number-theorists like to consider phic functions on a unique Riemann surface. The computability in a specific form, like a specific genus of this surface is the genus of f . In most bound on the size of the (numerators and de- cases, it can be computed readily from f using nominators of the) solutions in terms of some the formula simple algebraic invariants of f . This is the sub- ject of the effective Mordell conjecture, or Vojta’s g(f ) = (d − 1)(d − 2)/2 conjecture, or the ABC conjecture, and so on. where d is the degree of f . And then one Our theorem proceeds in a different direction, knows: and merely constructs an algorithm relative to the April 2013, Volume 3 No 2 17 Asia Pacific Mathematics Newsletter 3 existence oracle. In particular, no bounds on size, lies on at most finitely many secants to X and even relative ones, are obvious in this approach. then, actually find those secants and check if any My motivation for pointing out this relative com- of them are rational. All of this can be achieved putability in fact was to emphasise a different using standard computational algebra programs. perspective from the usual arithmetic-geometric In this manner, searching exhaustively, one locates one, since an algorithm can be quite different the desired c. Now one applies the oracle to h = 0 from a special kind of formula. An additional and proceeds. The point of this discussion is that remark is that if a more powerful decision oracle for curves of genus different from 1, the existence is given, for example, allowing equations and oracle is indeed very powerful. Not only does inequations, then the theorem is trivial.
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