## Logic of Elliptic Curves

In my 1997 doctoral dissertation, the main idea came as I was lying in bed one Sunday morning. I write about this now, because an email friend expressed curiosity. I am also testing the assertion of a mathematician friend:

You may revisit some corner [of mathematics] after being away for 30 years, and discover that everything there is the same as it was when you left it.

What was a correct mathematical argument, when I completed my dissertation, is correct now. This has to do with the universality of mathematics, which I described, though not by that term, in “Some Say Poetry.”

Our right to decide what is mathematically correct is inseparable from our responsibility to find agreement with interested others. Here then is an antithesis. What I thought right in the 1990s, I am likely to find right still, because my work was checked, and in some cases corrected, by me, an advisor, an examining committee, and a referee, in the manner also discussed in “Antitheses.”

Nonetheless, as Aristotle says in the second sentence of the Physics,

πέφυκε δὲ ἐκ τῶν γνωριμωτέρων ἡμῖν ἡ ὁδὸς καὶ σαφεστέρων
ἐπὶ τὰ σαφέστερα τῇ φύσει καὶ γνωριμώτερα
The natural way is from what is clearest and most knowable to us
to what is clearest and most knowable by nature.

This is consonant with Alexandre Borovik’s observation in another recent blog post:

This is the Catch-22 of learning mathematics: only at the next stage of learning it becomes possible to tell whether the learner mastered the previous stage.

What we have known may still become clearer when we look back at it.

Art and physics are academic fields; but unlike mathematics, they produce work that anybody can enjoy. With art this is obvious; with physics, there are products like the photographs of nebulae that Christopher Hitchens commends to our attention, in the quotation from God Is Not Great that I made at the end of my last post, on Gödel’s Incompleteness Theorem.

Mathematics can give you beautiful images, as of fractals like the Mandelbrot set. However, these are not what most of mathematics is about. I appreciate how Timothy Gowers declines to share such images in A Very Short Introduction to Mathematics (2002):

Very little prior knowledge is needed to read this book—a British GCSE course or its equivalent should be enough—but I do presuppose some interest on the part of the reader rather than trying to drum it up myself. For this reason I have done without anecdotes, cartoons, exclamation marks, jokey chapter titles, or pictures of the Mandelbrot set.

Mathematics originates in geometry. Etymologically speaking, this is surveying. Each of the propositions in Euclid’s Elements has a diagram, which anybody can look at. You can treat the diagram as a map of something on the ground. However, the fundamental concept of Greek geometry is the ratio, and this cannot be drawn in a diagram. A ratio is not really an object, but a relation between two objects, namely magnitudes. We can draw two magnitudes, and then draw two more magnitudes that have the same ratio; but we cannot point to what is the same.

Euclid points to things with letters. Letters can label points, or line segments, or angles, or regions. As far as I have seen, letters do not label ratios, except in the last proposition of Book VII of the Elements. Therefore I suspect that this proposition was tacked on later, when a more modern, symbolic conception of mathematics started to develop.

Euclid gives two different accounts of when two ratios are the same. Before these, historically, there was a third account. I talked somewhat about these matters in “The Geometry of Numbers in Euclid.” Thus my doctoral work took up an ancient theme: when two things are the same, how can you tell? For me, those things were not ratios, but elliptic curves.

A curve is the solution-set of a polynomial equation in two variables. If we allow any number of variables, and more than one equation, the solution-set is a variety. A variety has a function-field, namely the field of polynomial functions that can be defined on the variety. The function-field is indeed a field, because any two members have a sum, a difference, and a product, and members other than zero have reciprocals.

In papers of 1986 and 1992 in the Journal of Symbolic Logic, Jean-Louis Duret took up the question of whether you can distinguish two varieties by the first-order theories of their function-fields. This means looking only at what can be said about sums and products of individual elements of the function-fields. Duret showed that you can make the desired distinction, unless one of the varieties is an elliptic curve with complex multiplication. I ended up showing that Duret’s theorem still holds, even for some elliptic curves with complex multiplication. By a theorem of algebraic number theory, there are only thirteen of these curves.

We are working here with varieties whose defining polynomials have coefficients from an algebraically closed field, such as the field ℂ of complex numbers. This means every polynomial equation in one variable has a solution. The field ℂ has characteristic zero, by default, since it has no prime number as a characteristic. Duret considered only this case for his elliptic curves. However, for every prime number p, such as 2, 3, 5, 7, 11, and so on, there are fields, and therefore algebraically closed fields, having characteristic p. This means the pth multiple of unity is zero in the field. For example, in the two-element field, effectively studied by George Boole, 1 + 1 = 0, so the characteristic is 2. Only as a post-doc was I able to generalize, to arbitrary characteristic, Duret’s result concerning elliptic curves with no complex multiplication.

An elliptic curve is not an ellipse, but somehow arises from trying to compute lengths along an ellipse. I shall not go into this.

We can obtain an elliptic curve as the quotient of the additive group of complex numbers by a certain kind of discrete subgroup. In a simpler illustration of the process, we can obtain a circle as the quotient of the additive group of real numbers by a discrete subgroup. I shall spell this out below. Meanwhile, the desired discrete subgroups of ℂ are lattices. An example is the lattice ⟨1, i⟩ of Gaussian integers, namely the complex numbers k + mi, where k and m are integers. If L is a lattice, then, multiplying each point of the lattice by the same integer n, we obtain the lattice nL, which is a sub-lattice of L. In some cases, we can obtain a sub-lattice by multiplying by a complex number. For example, in the case of the Gaussian integers, the number i works. The corresponding elliptic curve is then said then to have complex multiplication.

If L and M are lattices, and α (alpha) is a nonzero complex number for which αL is a sub-lattice of M, then multiplication by α is an embedding of L in M. The index of the embedding is the size of the quotient ML. For example, multiplication by an integer n embeds a lattice in itself with index n2. The thought I had in bed one Sunday morning was that, if there are two different embeddings of L in M, and the indices of these embeddings are prime to one another, then the function fields of the corresponding elliptic curves are, to a certain extent, logically indistinguishable. If L is required to have complex multiplication, there are only the thirteen cases that I mentioned, where M would have to be isomorphic to L itself, in order for the embeddings of co-prime index to exist. Thus there are lots of examples of non-isomorphic elliptic curves that we do not know how to distinguish with first-order logic. This is actually an example of what makes logic interesting: that it cannot say everything.

To be precise, in the situation described, we cannot distinguish the curves using sentences of the form

x1 … ∀xny φ(x1, …, xn, y),

where φ is quantifier-free. We don’t know (or at least I don’t know) whether more complicated sentences would work.

I said I would elaborate on obtaining circles and elliptic curves. The range of the function f defined by

f(t) = (sin t, cos t)

from the real line ℝ to the Cartesian plane ℝ × ℝ is the circle defined by

x2 + y2 = 1.

The function f is periodic, with period 2π (two pi). Thus f is a one-to-one correspondence between

• the real numbers in the interval [0, 2π), and
• the points on the indicated circle.

The real numbers t in the interval correspond to the sets of real numbers of the form

t + 2kπt,

where k ranges over the set ℤ of integers. The sets

{t + 2kπt : k ∈ ℤ}

then make up the quotient denoted by

ℝ / 2πℤ.

In this quotient, addition and subtraction remain well-defined; thus the quotient is an additive group.

By means of the analytic functions of sine and cosine, we have obtained a geometric object, a circle. We may observe further that the cosine function is the derivative of the sine function.

Likewise, for every lattice L of ℂ, there is a function, the Weierstrass ℘ function, whose period is the given lattice. The function is the P function, but the P is given its own special form, even in Unicode, as here. The ordered pair of ℘ and its derivative ℘′ satisfy an equation

y2 = 4x3axb.

This is the equation that defines the elliptic curve corresponding to L. The equation being cubic, every straight line cuts the curve in three points. The sum of those three points is zero, in the addition in the curve that corresponds to addition in ℂ / L. This makes elliptic curves especially interesting.

For the record, when I was first writing the present account, here in Istanbul, I felt an earthquake, and so I went to report my experience.