[This lecture is also doubling as this week's "open problem of the week", as it discusses the Birch and SwinnertonDyer conjecture and the effective Mordell conjecture.]
Like many other maths departments, UCLA has a distinguished lecture series for eminent mathematicians to present recent developments in a field of mathematics, both to a broad audience and to specialists. Unlike most departments, though, our lecture series goes by the descriptive (but unimaginative) name of “Distinguished Lecture Series“, supported by the Gill Foundation. This week the lecture series is given by Shouwu Zhang from Columbia, and revolves around the topic of rational points on curves, a key subject of interest in arithmetic geometry and number theory. The first of three talks, which was on Tuesday, was a very accessible and enjoyable overview talk, which I am reproducing here (to use this opportunity to learn this stuff myself, and also to continue the diversification of subject matter here on this blog). As before, I do not vouch for 100% accuracy, and all errors are my responsibility rather than Shouwu’s.
Shouwu’s chosen topic involves a major and venerable component of number theory, namely the problem of solving Diophantine equations; this field is variously known as “Diophantine geometry” or “Diophantine analysis”, and indeed a major theme of Shouwu’s talk was that this field, which for most of its history was viewed primarily as a collection of “puzzles”, has now matured (particularly in the last century) to become a field rich in analytic, algebraic, and geometric structure and technique, and highly connected with other areas of mathematics. The definition of what a Diophantine equation is, and what it means to “solve” it, various from context to context, but for this talk it meant the study of solutions to polynomial equations where F is a polynomial with integer coefficients, and lie either in the integers or the rationals (and most of the talk eventually focused on the rationals). In the modern (geometric) approach to the subject, one views F as describing a hypersurface (or a scheme) on which one seeks to find integer points or rational points (or more generally, points over other commutative rings or fields). The three fundamental questions, for any given such Diophantine equation, are
 Existence. Does there exist at least one integer point or rational point?
 Structure. What structures (e.g. group structure, or other algebraic structure) does the space of all such points have?
 Effectiveness. Are there effective bounds on the number of points, or the size (height) of points? Are there effective algorithms to locate these points?
These questions are, in general, very hard. For instance, the famous theorem of Matiyasevich answers one version of Hilbert’s tenth problem by demonstrating (among other things) the existence of a Diophantine equation of several variables, whose solvability over the integers is undecidable. (It is still open whether a similar result holds true over the rationals.) However, for specific types of Diophantine equations, much more is known. For instance, two very classical (and “genus 0″) examples, both known to Euclid, include
 The line , which contains integer points precisely when c is divisible by the greatest common divisor of a and b (and whose solutions can be enumerated effectively via the Euclidean algorithm; and
 The unit circle , whose rational points are essentially equivalent to reduced Pythagorean triples , and can be enumerated explicitly as where m, n are integers that are not both zero.
Moving on to some more difficult “higher genus” or “higher dimensional” examples, we have
 The Fermat curve , which was famously proven by Wiles to have no nontrivial rational points for any ; and
 Euler’s hypersurface (an example, by the way, of a CalabiYau manifold), which was conjectured by Euler to have no nontrivial rational points for any , but which was disproven for n=5 by Lander and Parkin and n=4 by Elkies. While these counterexamples are completely explicit, they were found by geometric means; for instance, Elkies’ example was found by first locating Heegner points of an elliptic curve on the Euler surface, which turns out to be a K3 surface.
Given that these problems are so hard in general, it has been more profitable to focus on special types of hypersurfaces, such as
 Curves , with F irreducible for simplicity;
 Abelian varieties (connected to curves via motivic cohomology); and
 Moduli spaces (closely related to Shimura varieties).
Shouwu’s lectures are focused on curves, and specifically on finding rational (rather than integer) points on such curves. Such a curve C can be viewed in many ways:
 as an affine variety C;
 as a projective variety;
 as an abstract ring (the coordinate ring of C);
 as a Riemann surface (after compactifying and blowing up all singularities) of some genus g;
 as a set of points over a finite field (working projectively, and dealing with primes of bad reduction in standard ways);
 as a set of points over the rationals;
 etc.
Thanks to the great work of Grothendieck, we know that all of these different viewpoints are best interpreted through the unified perspective of schemes. For instance, if one is interested in rational points, one takes the associated ring of integers (in this case, just ), and views the spectrum as a new dimension in which to vary the curve C, thus converting the one dimensional object C to what is basically a twodimensional “arithmetic surface”, the scheme associated to C. In this specific case, the spectrum consists just of the primes (though in practice one should compactify and throw in as well), and the scheme sits over the spectrum as a kind of bundle, with the fibre over p just being (essentially) the finite curve , and the fibre over being (essentially) the Riemann surface . A rational point in can then be viewed as a special type of section of this bundle, whose value over p is just the mod p projection to and whose value over is formed simply by interpreting as complex numbers rather than integers. (If p divides the denominator of or then the section might have a pole at p, but these are easily dealt with by working projectively, compactifying, and blowing up singularities.) With this viewpoint, the problem of finding rational points in curves transforms now to one of finding special sections of bundles, which brings the subject much closer to many other “mainstream” areas of mathematics, and in particular to modern analysis, geometry, algebra, and topology.
The schemetheoretic perspective can be unintuitive at first glance, especially to those mathematicians (such as myself) who are more used to analysis than to algebra. To explain this viewpoint better, Shouwu described the function field model, in which the “arithmetic” ring of integers are replaced by the “analytic” ring of complex polynomials of one variable t, and the field of rational numbers then replaced by the field of rational functions . The spectrum here is identifiable with the complex line , since every complex number z gives rise to a prime ideal , and these are all the prime ideals. (Conversely, is the function field of its spectrum , viewed as an algebraic variety). If one is interested in, say, finding all “rational” points to the Fermat curve (i.e. all rational functions such that is identically equal to 1), one can replace the curve C by the surface , which one should think of as a bundle of curves over the spectrum, i.e. the complex line. (In this case, the curve turns out to be independent of the base point z; in more general examples of schemes, this is unlikely to be true.) A rational point on the Fermat curve then corresponds to a rational (in particular, meromorphic) section of the surface X. Thus we see that by introducing the spectral parameter z we have introduced some more geometry into the problem; in this case, some complex geometry.
To illustrate how we can use this additional geometric structure, Shouwu recalled the classic proof of “Fermat’s last theorem for polynomials”, namely that when , the only rational points on the Fermat curve are the constant points (where are independent of t). If for contradiction we had a nontrivial rational point, which we write in reduced fraction form as , with f(t), g(t), h(t) coprime, then by evaluating at the spectral parameter z we have
for all complex z. Now the complex structure implies in particular that both sides here are complex differentiable in z, so we can differentiate and obtain
;
combining this with the previous equation and using some algebra, one eventually obtains
.
Since f and g are coprime, we thus see that divides and divides , and by considering the degrees of f,g,h we quickly obtain a contradiction for unless f,g,h are all constant.
Returning to the ordinary rationals and integers, Shouwu noted that the “arithmetic geometry” of the primes does not have the type of differential geometric structure that the function field model had; nobody has succeeded, for instance, in proving Fermat’s last theorem by taking an integer solution modulo each prime p, and then “differentiating with respect to p” in analogy to the above argument. Nevertheless there are other analytic methods one can use to exploit the arithmetic geometry in this situation. Shouwu then listed the two major analytic methods in the subject:

The method of infinite descent, which was first used to great effect in this subject by Fermat (though arguably the Euclidean algorithm already implicitly appeals to this method); in the modern theory, this method has developed into the theory of heights (which measure the complexity of equations and their solutions), and thence to Arakelov theory, which Shouwu interpreted as the intersection theory of certain types of Chern classes, which can be viewed as invariants of the arithmetic surface ; and

The method of Lfunctions, which creates a single and very rich analytic invariant – the Lfunction – associated to the arithmetic surface – rather than studying a lot of “smaller” invariants.
Despite looking completely different, in recent years it has been realised that these two methods are connected in several ways (to be discussed in later lectures). For this lecture, Shouwu focused on the Lfunction approach. The philosophy here is that the arithmetic surface is easily studied over any given prime p (where it is just an algebraic curve); to quote Max Noether, “algebraic curves are created by God, but algebraic surfaces are created by the Devil”. To implement this philosophy, one collects all of this “local” information and place it into a convenient analytic object, the HasseWeil Lfunction
where is a multiplicative sequence that measures a kind of correction term for the behaviour of the curve at the place n and are the Frobenius eigenvalues in ; the individual factors in the product over primes are the local zetafunctions of the curve, and relate to the number of points over for integer k. In particular, the number of points at p is given by the formula (this identity apparently relates to the Lefschetz principle, though I didn’t understand this point); this identity is also a special case of the GrothendieckLefschetz fixed point formula. Weil (building on earlier work of Hasse) famously proved the Riemann hypothesis for function fields, which among other things demonstrated that all the eigenvalues all have magnitude at most , which thus implies that . (An equivalent formulation of this fact is that all the local zetafunctions have all the zeroes on the critical line, thus explaining the terminology “Riemann hypothesis”.) In particular we see that is nonzero for all sufficiently large p.
Now, in order for C to have any rational points at all, it is necessary for all the to be nonzero (otherwise there is a local obstruction at p, similar to what I discussed in my first Simons lecture). Hasse formed his famous localtoglobal principle, which is the heuristic that the converse should be true, namely if there are no local obstructions at any prime p then there should be at least one rational point. This principle turns out to not always be true, but nevertheless seems to be an excellent first approximation.
Shouwu closed by quickly sketching the state of the art for the theory of rational points on curves for various values of the genus g, which is the decisive parameter in the theory (and is roughly related to the degree of F via the RiemannRoch theorem):

For genus 0 curves (in particular, lines and conics, i.e. the case when F has degree at most 2), we have a satisfactory and effective theory. The HasseMinkowski theorem asserts that the localtoglobal principle is true in this case; a rational point exists if and only if a point exists over each padic field, which (in nonsingular cases) is equivalent to the existence of a point in . The latter can be computed easily using quadratic reciprocity. As for describing the structure of all rational points, there is an extremely classical observation of Diophantus himself, which in modern language states that once one rational point of a genus zero curve is found, the space of all rational points can be identified with the projective line over the rationals, essentially by stereographic projection from the given rational point.

For genus 1 curves (which happens when F has degree 3), we have a partially satisfactory theory. The localtoglobal principle does not quite work; for instance, the curve has a point over every , but has no rational points. Nevertheless the obstruction to this principle can be quantified as follows. If one marks a rational point P on a genus 1 curve C to serve as a group identity then the resulting pair (C,P) is an elliptic curve E. Given a genus 1 curve C, there is an abstract elliptic curve E (the Jacobian of C) one can canonically assign to C, by quotienting out the space of divisors of C (formal integer sums of points in C in which the weights sum to zero) by the principal divisors (which in this case are generated by whenever and are collinear triples of points in C; note that an elliptic curve intersects any line in three points (in an algebraically complete field, of course)). The fibre of curves C which give the same abstract elliptic curves E is the principal homogeneous space of E, and can be given the structure of a group: the TateShafarevich group of E. [Update, May 2: actually, the TateShafarevich group is a bit smaller than this; as we do not want to see local obstructions, we restrict attention to curves which have points over all completions of the underlying field, which in the case of is roughly like asking the curves to have a point over for every p.] This group controls the number of rational points on C (and whether they exist at all), after accounting for local obstructions, but it is not fully understood at present. As is well known, the rational points of C is itself an abelian group; a famous theorem of Mordell and Weil asserts that this is isomorphic to a finitely generated subgroup of the twotorus (which is in turn isomorphic to , from the classical work of Weierstrass). The dominant feature of this subgroup is the rank; the remaining bit, the torsion group, is known by a deep theorem of Mazur to be quite small, and more precisely to have order at most 16. Computing the rank of this group is in fact an extremely important problem in the subject; the famous Birch and SwinnertonDyer conjecture asserts (among other things) that the rank of is always equal to the order of vanishing of the Lfunction at s=1. (It also asserts that the TateShafarevich group is always finite.) This conjecture can be formulated without explicit reference to the Lfunction as the assertion that the product diverges like , which can be viewed as a more analytical version of the localtoglobal principle (relating the number (or more precisely rank) of rational points to the number of points over ). The BSD conjecture is supported by much numerical evidence and an array of partial results and special cases, but is still far from complete resolution. In addition to BSD, there are a number of other conjectures (including one of Lang) which assert that various characteristics of are controlled in terms of the height of the discriminant; these conjectures, if true, would enable one to describe the rational points fairly effectively.

For higher genus curves (which is the typical situation when F has degree 4 or higher), we are quite far from a satisfactory theory, in large part because there is little group structure (or grouplike structure) to exploit here. The celebrated proof by Faltings of the Mordell conjecture shows that the number of rational points is necessarily finite in this case, but as the proof proceeds by comparing rational points to each other, rather than to an external reference point, there is no effective bound known on the height of these points (though one can get a weak but effective bound on the number of points in terms of the rank of the Jacobian, which is in turn controlled by a “weak MordellWeil conjecture”). It would be a substantial breakthrough to get this effective bound. In particular there are the effective Mordell conjectures, one (rather naive) version of which can be stated as follows: given any curve C of genus , one should have for all rational points in C, where the height of the rational point is the maximum of the logarithms of the numerators and denominators of , and the height of the curve C is (say) 1 plus the logarithm of the largest coefficient in the defining polynomial F. This conjecture is already so strong that it implies the abc conjecture. It is in turn implied by an even stronger bound which was motivated from the function field model (i.e. complex surfaces rather than arithmetic ones) as an analogy. More precisely, since the wellknown BogomolovMiyaokaYau inequality for complex surfaces controls the selfintersection of the first Chern class of the canonical complex line bundle of X by the second Chern class; one can conjecture an arithmetic analogue of this inequality in which the selfintersection of a “relative canonical bundle” from Arakelov theory (a substitute for the canonical line bundle, which is not available due to the lack of differential structure in ) would be controlled by the logarithm of some sort of discriminant (plus some other correction terms), times a factor depending on the genus; I didn’t fully understand the statement here, but apparently it implies the effective Mordell conjecture.
[Update, May 2: Corrected the reference to the GrothendieckLefschetz fixed point formula, as pointed out by Emmanuel Kowalski, and corrected the definition of the TateShafarevich group, as pointed out by bb.]
10 comments
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2 May, 2007 at 1:52 pm
Emmanuel Kowalski
The “Lefschetz principle” is presumably a reference to the GrothendieckLefschetz
trace formula, which states that the number of points on an algebraic variety
defined over a finite field with elements is the alternating
sum of traces of the “global” Frobenius automorphism acting on some finitedimensional
vector spaces, the index ranging from 0 to , where is the dimension.
For a smooth irreducible projective curve, , the first trace () is (trivial action on a onedimensional space), the last trace () is (multiplication by on a
onedimensional space) and the middle (interesting) trace () is the trace on
a vector space of dimension (analogue of the classical first (co)homology group of a Riemann surface), written as sum of eigenvalues .
2 May, 2007 at 5:08 pm
bb
I’d just like to point out a tiny mistake in the definition of the TateShafarevich group (Sha, for short). What you’ve defined (the set of genus 1 curves over a field k with jacobian E, upto isomorphism) is actually the cohomology group H^1(k,E) (this can also be defined as the set of curves over k with a free and transitive Eaction defined over k). In order to get Sha, we must account for local obstructions as we’re trying to capture the failure of the Hasse principle. So over a number field k, we define Sha(E,k) to be the subgroup of H^1(k,E) consisting of curves which have points over all completions of k. This is the group that’s conjectured to be finite — the H^1(k,E) could very well be infinite.
Also, as perhaps an interesting aside, I believe that in the function field case the truth of BSD conjecture is now equivalent to the finiteness of Sha which, in turn, follows from the 2dimensional case of the remarkable conjecture of Artin that any smooth projective variety over a finite field (in fact, any proper scheme over Spec(Z) works!) has a finite Brauer group (Morita equivalence classes of central simple algebras).
2 May, 2007 at 5:45 pm
Terence Tao
Dear Emmanuel and bb,
Thanks very much for the corrections; they are especially helpful for me as I am trying to learn this subject properly. (I sort of absorbed bits and pieces of this through osmosis while a graduate student at Princeton – I was there when Wiles proved his theorem, in particular – but it’s only now that I think I’m getting a bit of a handle on things.)
3 May, 2007 at 9:24 am
Distinguished Lecture Series II: Shouwu Zhang, “GrossZagier formula and Birch and SwinnertonDyer conjecture ” « What’s new
[...] In the previous lecture, Shouwu defined an elliptic curve E as a genus 1 curve with a marked point P (which will eventually become the group identity element). It is traditional to move this point P to the point at infinity and view the curve affinely, in which case the curve can be placed via an affine transformation in the normal form [...]
4 May, 2007 at 8:52 am
Distinguished Lecture Series III: Shouwu Zhang, “Triple Lseries and effective Mordell conjecture” « What’s new
[...] discussed in the first lecture, one of the landmark achievements in the higher genus theory is Faltings’ theorem (proving [...]
5 May, 2007 at 12:23 pm
Not Even Wrong » Blog Archive » All Sorts of Stuff
[...] continues to come up with amazingly good blog entries. His latest is a series of three postings (here, here and here), reporting on my colleague Shouwu Zhang’s lectures at UCLA on the topic of [...]
18 May, 2007 at 7:49 pm
Ars Mathematica » Blog Archive » Tao on Zhang
[...] Overview of rational points on curves [...]
27 July, 2007 at 7:34 am
Dyadic models « What’s new
[...] factorisation of large numbers) are surprisingly easy to solve in the dyadic world (see e.g. Zhang’s talk for the dyadic version of FLT), but nobody knows how to convert the dyadic arguments to the [...]
17 May, 2008 at 8:04 pm
Anonymous
Very deep lecture.
20 September, 2010 at 12:41 pm
Sylvain JULIEN
Please let me add something on bb’s comment. What would be the exact consequences of the finiteness of Sha for an elliptic curve defined over the field of rational numbers? Would it be enough to prove BSD conjecture, or would one also need to build new connections with the function field case?