0.1. Growth of Elliptic Curves. My Ph.D. thesis is about Iwasawa theory of elliptic curves. An elliptic curve E over ℚ is a subset of the (projective) plane given by a cubic polynomial with coefficients in ℚ, for example

E_{0} : y^{2} + xy = x^{3} − 4764702x − 4003545844

For any number field K/ℚ, the set E(K) of points on E with coordinates in K has a geometrically defined addition law, under which E(K) is a finitely generated Abelian group (the Mordell-Weil theorem). For instance, we find that my curve E_{0} has E_{0}(ℚ) = ℤ. It is natural to ask: how does the ℤ-rank of E(K) vary as K ranges over all number fields?

Two issues conspire to make this question intractable as stated. First, the lattice of all finite extensions of ℚ is too unwieldly for a ‘parameter space’. To deal with this, Iwasawa had the idea of focusing on a ‘1-parameter’ rank variation problem by choosing a rational prime p and letting the field K range through a tower of extensions K_{∞} ⊃ ⋯ ⊃ K_{n} ⊃ ⋯ ⊃ K_{0} = ℚ such that Gal(K_{∞}/ℚ) ≅ ℤ_{p} (there is only one such K_{∞} for each p, and it’s contained with finite index in the field generated by the pth-power roots of unity).

The second difficulty is that, even for a fixed K, E(K) is difficult to compute: the algorithm underlying the proof of Mordell-Weil theorem is not known to terminate for every elliptic curve. It is therefore convenient, for p chosen above, to relate E(K) to a finite-dimensional 𝔽_{p}-vector space called the p-Selmer group and denoted Sel_{p}(E_{/_{K}}). Its big advantage over E(K) is that its dimension is effectively computable and gives an upper bound for rank_{ℤ}E(K). The main question of Iwasawa theory of elliptic curves now becomes: how does dim_{𝔽_{p}} Sel_{p}(E_{/_{Kn}}) grow as K_{n} ranges over the subfields of the ℤ_{p}-tower K_{∞}/ℚ?

0.2. The μ-invariant. We distinguish between two possible growth behaviors for Sel_{p}(E_{/_{Kn}}): if dim_{𝔽_{p}} Sel_{p}(E_{/_{Kn}}) is bounded as n → ∞, we say that μ(E)_{p} = 0; if not, we say that μ(E)_{p} > 0. This somewhat unnatural lingo is the elementary shadow of the general Iwasawa theory which in most cases (when E is ‘ordinary at p’), defines μ(E)_{p} as an actual non-negative integer. This number is called ‘the μ-invariant of E’ and measures the rate of exponential growth, as n → ∞, of a more general relative of Sel_{p}(E_{/_{Kn}}).

It is possible to define another μ invariant of an elliptic curve via padic analysis. The conjecture that this analytic μ-invariant is equal to the algebraic one defined in terms of growth of Sel_{p}(E_{/_{Kn}}) is part of what is rightly called the Main Conjecture of Iwasawa theory of elliptic curves.

There has been a recent upsurge of interest of computing μ-invariants defined in other contexts. Hida [?], for instance, proves that μ = 0 for the anticyclotomic Katz p-adic L-functions, using a deatiled study of q-expansions of Eisenstein series and deformation spaces of Hilbert-Blumenthal abelian varieties. Vatsal [?] uses his results on the equidistribution of Heegner points to conclude μ = 0 for most p-adic L-function associated to elliptic curves over quadratic imaginary fields. Both these results concern analytically defined μ-invariants. By contrast, my results on the vanishing of μ(E)_{p} for curves with reducible torsion are squarely on the algebraic side of the Main Conjecture.

For any given E_{/_{ℚ}}, the algebraic μ-invariant is in principle hard to compute. For example, I can prove μ(E_{0})_{3} = 2 for the example curve E_{0} above; no previous method, to the best of my knowledge, even gives a provable upper bound on μ(E)_{p} in this and similar rank one examples. Despite the computational difficulty, Greenberg conjectures that for E defined over ℚ, μ(E)_{p} is ‘explained’ by the presence of a special finite subgroup of E(ℚ̄):

Conjecture 1. (Greenberg) Let M ⊂ E(ℚ̄) be the maximal Gal(ℚ̄/ℚ) stable-subgroup which is cyclic p-primary, and such that the action of Gal(ℚ̄/ℚ) is odd (conjugation acts non-trivially) and ramified at p.

Then log_{p} |M| = μ(E)_{p}.

The structure of the p-torsion E[p] seems to determine how accessible the conjecture is.

1. When E[p] is irreducible, the conjecture predicts μ(E)_{p} = 0 and seems intractable at present. This is too bad, since this is the generic case. The best we can do is produce some examples using a formula of Greenberg for the constant term of a characteristic polynomial closely related to μ.

2. When E[p] is reducible, i.e. when it lives in a short exact sequence of Gal(ℚ̄/ℚ) modules

(1)  0 → Φ → E[p] → Ψ → 0,

we are much closer to proving Greenberg’s conjecture. This case further splits into two:

2a. When Φ is even and unramified at p, or odd and ramified at p, Greenberg and Vatsal [?] prove that μ = 0 (as expected). This is relatively easy, essentially because we can bound dim_{𝔽_{p}} Sel_{p}(E_{/_{Kn}}) in terms of contributions from Φ and Ψ. These contributions are easy to compute since they depend only on Φ and Ψ themselves, and not on the structure of E[p].

2b. For the remaining cases with reducible torsion, Greenberg’s conjecture can be reduced to its special case:

Conjecture 2. If Φ is even and unramified at p, and if the sequence (1) is non-split, then μ(E)_{p} = 0.

It is this form of Greenberg’s conjecture that I tackle in my thesis in the special case when Φ ≅ ℤ/pℤ, i.e. when E has a rational p-torsion point.

0.3. My Thesis. Unlike in case 2a, we have to take into account the structure of the extension 0 → Φ → E[p] → Ψ → 0, as hinted by the nonsplit assumption in Conjecture 2. From the structure of E[p], I derive a sequence of geometrically defined bilinear pairings (⟨·, ·⟩_{n}), and show that dim_{𝔽_{p}} Sel_{p}(E_{/_{Kn}}) is bounded as n → ∞ if and only if the pairings are nonzero for n ≫ 0. Roughly speaking, we can imagine ⟨·, ·⟩_{n} as a pairing O_{n}^{×}/O_{n}^{×p} × O_{n}^{×}/O_{n}^{×p} → 𝔽_{p}, where O_{n}^{×} is the group of units in the ring of integers of K_{n}.

To show that the pairing ⟨·, ·⟩_{n} is non-zero I compute it on a large subset of its domain, using a subtle dévissage argument for spectral sequences, which may well be useful in other contexts. The formula for the pairing is a sum, over all primes in K_{n}, of perfectly explicit local factors. Interesting numerological questions about units in K_{n} arise in trying to show this sum is non-zero. At present, I can answer these questions for many examples, including some fundamentally new ones. For a precise statement of a general result encompassing all my examples, please see the attached technical summary.

0.4. Research Plans. There are several directions in which I could extend my thesis. First, the ‘numerology’ of my pairing formula deserves closer study, with an eye towards settling Conjecture 2 in the case Φ ≅ ℤ/pℤ. Second, I am currently adapting my method to the case of a general even and unramified-at-p Φ. Third, I want to attempt the proof of the generic case of Greenberg’s conjecture, i.e. when E[p] is irreducible. One possible line of a attack would be to extend my method to a number field K over which E[p] acquires a rational point, and then descend the result to ℚ. Fourth, I would like to explore p-adic analytic μ invariants and compare them with the algebraic computations in my thesis, in parallel with the results of Greenberg-Vatsal concerning the case 2a.

My general mathematical interest is centered on the nexus of p-adic Lfunctions, relations between Heegner points and L-functions, and constructions of rational points on elliptic curves, as in the recent work of Darmon, Vatsal, Cornut and others. Specifically, I would like to explore analogous constructions for a non-abelian (rather than imaginary quadratic extension). For example, let F/ℚ be a complex S_{3}-extension, and let F_{∞}/F be its maximal ‘anti-cyclotomic’ pro-p extension. Assuming Leopoldt’s conjecture, Γ = Gal(F_{∞}/F) ≅ ℤ_{p}^{3}. We can consider the contribution to L(E_{/_{F}} , s from characters induced from Γ. Under suitable assumptions we expect three lines of characters of Γ such that L(E/K, χ, s) has functional equation with sign −1 and therefore vanishes. I’d like to look for a construction of points predicted by this vanishing, possibly only a p-adic construction since a global one à la Heegner seems too hard at present.