Abstract
In this chapter we describe how points of finite order on certain elliptic curves can be used to generate interesting extension fields of \(\mathbb{Q}\). Here we mean points of finite order with arbitrary complex coordinates, not just the ones with rational coordinates that we studied in Chapter 2 So we will need to use some basic theorems about extension fields and Galois groups, but nothing very fancy. We start by reminding you of most of the facts that we need, and you can look in any basic algebra text such as [14, 23, 26] for the proofs and additional background material.
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Notes
- 1.
This makes sense because \(\mathcal{O} = [0, 1, 0]\) in homogeneous coordinates, so \(\sigma (\mathcal{O}) = [\sigma (0),\sigma (1),\sigma (0)] = [0, 1, 0] = \mathcal{O}\).
- 2.
There are many possible choices for P 1 and P 2, just as a vector space has many different bases. It will not matter which basis we choose.
- 3.
The theory of group representations is an extremely powerful tool for studying groups, and it is used extensively in mathematics, physics, and chemistry. We do not need the general theory, but for those who are interested, a very nice introduction to the representation theory of finite groups is given in Serre [43].
- 4.
This may remind you of our proof of the Nagell–Lutz theorem in Section 2.4 There we proved that a certain rational number a∕d was an integer by checking, for each prime ℓ, that ℓ did not divide d. This idea of looking at one prime at a time, which in fancy language is called localization, is a powerful number theoretic tool. It is the algebraic equivalent of looking at a neighborhood of a point when you are studying real or complex analysis.
- 5.
A heuristic argument suggests that the number of primes p ≤ T that satisfy the congruence \(2^{p} \equiv 1\ (\text{mod}\ p^{2})\) should be roughly loglogT. As of 2015, the only primes known to have this property are p = 1093 and p = 3511.
- 6.
In its present form, the Frey equation may be quite badly singular when reduced modulo 2, but a change of variables takes care of the problem. For ease of exposition, we will mostly ignore the prime 2 in our discussion.
- 7.
For ease of exposition, we have, and will continue to, ignore a number of technical issues. First, we should take a “minimal” equation for E, which roughly means that there is no change of variables that makes the discriminant smaller while keeping integer coefficients. Second, we completely ignore the prime 2. Third, for primes p that divide D, we should take ε p ∈ {−1, 0, 1}, with the exact value depending on whether \(\tilde{E}\) has a node or cusp, and if a node, whether the tangent slopes are in \(\mathbb{F}_{p}\). Then the corresponding factor in the L-series is \((1 -\epsilon _{p}p^{-s})^{-1}\).
- 8.
You may have seen that the Riemann ζ-function similarly has a meromorphic continuation to \(\mathbb{C}\) and satisfies a functional equation. Setting \(\xi (s) = \frac{1} {2}\pi ^{-s/2}s(s - 1)\varGamma (s/2)\zeta (s)\), the functional equation for ζ(s) says that \(\xi (s) =\xi (1 - s)\).
- 9.
We have omitted a technical condition that f(z) vanish at every cusp. A more intrinsic way to describe the transformation formula, which may be less mysterious, is to say that the differential form f(z) dz is invariant under the transformation sending z to \((az + b)/(cz + d)\). More precisely, we want f(z) dz to be a well-defined differential form on (a smooth completion of) the quotient \(\mathfrak{H}/\varGamma _{0}(N)\). We also mention that our modular forms have weight 2, and that there also exist modular forms of other weights, where one replaces the (cz + d)2 with (cz + d)k for some other value of k.
- 10.
What we have stated is a consequence of Ribet’s theorem, whose full statement requires concepts and terminology that would take too long to develop here.
- 11.
We recall that the decomposition group of \(\mathfrak{P}\) is the set of \(\sigma \in \mathop{\mathrm{Gal}}\nolimits {\bigl ( \mathbb{Q}{\bigl (E[n]\bigr )}/\mathbb{Q}\bigr )}\) such that \(\sigma (\mathfrak{P}) = \mathfrak{P}\).
- 12.
In mathematical terminology, the extension generated by q 1∕ℓ is unramified at every prime p dividing D.
- 13.
The criterion of Néron–Ogg–Shafarevich [49, VII.7.1] says that if for every t ≥ 1, the representation \(\rho _{\ell^{t}}\) on \(\mathcal{D}_{\mathfrak{p}}\) is unramified at p, then \(\tilde{E}_{p}\) is non-singular. For the Frey curve, we know this property for t = 1, which is a start.
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Silverman, J.H., Tate, J.T. (2015). Complex Multiplication. In: Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-18588-0_6
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