The Arithmetic of Elliptic Curves Theory Conjectures

Computations

Applications Applications

John Cremona University of Exeter

Example 1 Zagier, Dudeney, …

Find m, n ∈ ℕ satisfying

1^{2} + 2^{2} + 3^{2} + … + n^{2} = m^{2}.  (1)

Two solutions: (m, n) = (1, 1), (70, 24). Substituting X = 12n + 6, Y = 72m leads to

E : Y^{2} = X^{3} − 36X (2)

which is an elliptic curve. There are infinitely many rational solutions (X, Y) to (2), but only a finite number of integral solutions.

E(ℚ) = ⟨P, T_{1}, T_{2}⟩ ≅ ℤ × ℤ/2ℤ × ℤ/2ℤ

where

P = (−2, 8), T_{1} = (0, 0), T_{2} = (6, 0)

and

Q = −2P − T_{1} = (294, 5040) ↦ (m, n) = (70, 24).

Elliptic Curves: definition

An elliptic curve (defined over any field K) is a plane cubic curve with equation

E :  Y^{2} = X^{3} + aX + b  a, b ∈ K

where Δ = −4a^{3} − 27b^{2} ≠ 0.

Fields of interest: K = ℂ: elliptic functions, classical complex analysis. K = 𝔽_{q}: applications in computational number theory and computer science (factorization, primality testing and proving, cryptography, coding theory). K = ℚ (or a number field): Diophantine equations, Arithmetical Algebraic Geometry. Also: K = ℝ, K = ℚ_{p}, etc.

Two types of question to ask:

GENERAL: classify all elliptic curves over K (up to isomorphism, or isogeny, say)

• easy over ℂ (via j-invariant)

• well understood over 𝔽_{q}

• much harder over ℚ: the Weil-Taniyama conjecture states that all elliptic curves E defined over ℚ are modular, i.e. attached to modular forms.

SPECIFIC: study individual curves over K

In particular, describe the group of “rational points”:

E(K) = {(x, y) ∈ K^{2} | y^{2} = x^{3} + ax + b} ∪ {∞}.

The group law is

P + Q + R = 0 ⟺ P, Q, R are collinear.

(The “tangent-chord” process). This turns E(K) into an Abelian group (with ∞ = 0).

Group structure

E(ℂ) ≅ ℂ/Λ ≅ (ℝ/ℤ)^{2} where Λ = ℤω_{1} + ℤω_{2}.

E(𝔽_{q}) is finite: |#E(𝔽_{q}) − (q + 1)| ≤ 2\sqrt{q} (“Riemann Hypothesis” for elliptic curves over 𝔽_{q}). E(𝔽_{q}) is either cyclic or is a product of two cyclic factors (since nP = 0 has ≤ n^{2} solutions ∀n ≥ 1).

If K is a number field, E(K) is called the Mordell-Weil group. Mordell-Weil Theorem: E(K) is finitely generated: so

E(K) ≅ T × ℤ^{r}

where T is the (finite) torsion subgroup and r is the rank. Questions:

• Which groups T arise?

• How to find T for a given curve?

• Which ranks r occur?

• How to find r for a given curve?

• How to find a basis of r independent points?

TWO APPLICATIONS

I. Fermat’s Last Theorem

Frey, Ribet: Counterexample to FLT ⇒ ∃ non-modular (semi-stable) elliptic curve over ℚ

⇒ Weil-Taniyama fails. Wiles, Taylor: Weil–Taniyama holds for semi-stable curves over ℚ. Corollary: FLT holds! N.B. Proof theoretical, not computational, but now all semi-stable elliptic curves over ℚ may be explicitly constructed (in principle) from modular forms.

II. Gauss’s Class Number Problem

Goldfeld (1975) proved a good bound on those d > 0 for which the field ℚ(\sqrt{−d}) has class number h. Proof needs an L-function with certain analytic properties, supplied (Gross, Zagier 1985) by L(E, s), where

E : Y^{2} + Y = X^{3} − 7X + 6

which is modular and has rank 3. These two facts were established by computational means (the former originally by Mestre).

Rational newforms for Γ_{0}(N)

N ≤ 5077

Range of N Total r = 0 r = 1 r = 2 r = 3
1–1000 2463 1321 1124  18  0
1001–2000 3391 1575 1737  79  0
2001–3000 3661 1663 1852 146  0
3001–4000 3837 1665 2006 166  0
4001–5000 3962 1690 2092 180  0
5001–5077  284 121 148  14  1
1–5077 17598 8035 8959 603  1

These are also the numbers of (isogeny classes of) modular elliptic curves of conductor N. The ranks r here are analytic ranks (order of vanishing of Lseries at s = 1), but also agree with the algebraic ranks of the Mordell–Weil groups of the corresponding curves in all cases (by a combination of Theorems and explicit computation).

N = 5077 is the first case with r = 3.

MORDELL-WEIL GROUPS

E(K) ≅ T × ℤ^{r}

I. Torsion Over ℚ, possible torsion groups classified by Mazur c.1970:

T ≅ \{ \begin{matrix}C_{n}, & (1 ≤ n ≤ 10, n = 12); \\ C_{2} × C_{n}, & (1 ≤ n ≤ 4). \end{matrix}

Also, (x, y) ∈ T ⇒ x, y ∈ ℤ and either y = 0 or y^{2} | Δ (assuming a, b ∈ ℤ), so T is easy to find for a given curve.

Over a number field K of degree d: similar classification exists for d = 2 (Kamienny, Kenku, Momose). The Uniform Boundedness Conjecture states that ∀d∃B: |T | ≤ B for all curves E over all fields K of degree d. In particular, the set S_{d} of primes p such that there exists a curve E defined over a field K of degree d with a point of order p should be finite for each d. This was proved in 1993 by L. Merel: he showed p < d^{3d^{2}}. (This bound is likely to be improved.) More work is then needed to determine exactly which groups T can occur for each d. Examples: S_{1} = {2, 3, 5, 7}, S_{2} = {2, 3, 5, 7, 11, 13}.

MORDELL-WEIL GROUPS (contd.) II. Rank No bound on the rank r is known, or believed to exist, for any given number field K. Over ℚ, curves of rank up to 21 are known; including infinite families of rank up to 11. (Mestre, Nagao) Computing r: via descent. Same method as used to prove the rank is finite, made very explicit.

Descent Take K = ℚ for simplicity. Basic idea: cover E with a finite number of auxiliary curves C_{1}, C_{2}, …, C_{k} (all of genus 1 and isomorphic to E over ℂ).

• If C_{i}(ℚ) ≠ ϕ, then its image is a complete coset of mE(ℚ) in E(ℚ) for some m ≥ 2 (usually m = 2).

• Hence once hopes to find [E(ℚ) : mE(ℚ)], and hence the rank r, by deciding which C_{i} have rational points.

• Problem: no algorithm exists for deciding this! One can fairly easily eliminate those C_{i} for which either C_{i}(ℝ) = ϕ, or C_{i}(ℚ_{p}) = ϕ for some prime p. BUT some C_{i} may have points “everywhere locally but not globally”. The extent to which this can happen is measured by the Tate-Shafarevich group � (“Sha”).

Higher descents We can carry out “further descents”, covering each C_{i} by a finite set of curves C_{ij}, so that C_{i}(ℚ) ≠ ϕ ⟺ C_{ij}(ℚ) ≠ ϕ for some j. If we knew that � was finite, this process would stop after a finite number of steps. If not, we have a problem: but this procedure might still be practical.

� Conjecture: For all elliptic curves E defined over ℚ,

|� (E/ℚ)| < ∞.

In practice: we can always bound r by descent (using the local solubility conditions) and can often determine it exactly. There now exist general efficient programs for doing this over ℚ which are currently being extended to certain quadratic fields. But making these ideas explicit enough to implement as programs is very difficult. Current status of the � conjecture: Proved for modular elliptic curves of rank 0 or 1 by Rubin and Kolyvagin, as part of their work on the famous Birch Swinnerton-Dyer Conjecture.

The Birch Swinnerton-Dyer Conjecture

This relates the rank r of E(ℚ) to the behaviour of a certain analytic function L(E, s), called the Lfunction of E over ℚ, at s = 1:

(1) ord_{s=1}L(E, s) = r = rank(E(ℚ)); It also gives a formula for |� |, analogous to the analytic class-number formula for number fields: (2)

|� | = \frac{L^{(r)}(E, 1) · |T |^{2}}{r! · Ω · R · ∏_{p} c_{p}}

where : Ω is the real period of E; R is an r × r determinant of the “height pairing” for generating points; c_{p} is a positive integer (= 1 except for the finite set of “bad” primes).

The BSD conjecture has only been proved for certain modular elliptic curves of rank 0 or 1. Until this, the � Conjecture was not known to be true for any elliptic curve over ℚ!

INTEGRAL POINTS

E : Y^{2} = X^{3} + aX + b (a, b ∈ ℤ)

The set of integral points E(ℤ) = E(ℚ) ∩ ℤ^{2} is always finite, even when the group of rational points E(ℚ) is infinite. There are bounds on the x-coordinate of integral points in terms of a and b (due to Siegel, Baker) using transcendence theory, but they are too large to be practical.

Recent advances use bounds on linear forms in elliptic logarithms to make the determination of all integral points on a given curve a practical proposition, provided that a basis for the Mordell group E(ℚ) is known. If this basis is P_{1}, …, P_{r}, then the method can provide small bounds for the coefficients k_{i} of an integral point P = ∑ k_{i}P_{i}. These methods have been developed by DeWeger, Stroeker, Tzanakis, Smart (and others) implemented by these and Zimmer and Gebel. All these implementations rely on first having a basis for E(ℚ), so the demand for programs (such as mine, with improvements due to S. Siksek) which can find such a basis (in many cases) is growing.

Example 2

(J. P. Robinson, with help from JC and Smart)

Find all m, n ∈ ℕ satisfying

1 + 2 + 3 + … + m = 1^{2} + 2^{2} + 3^{2} + … + n^{2}.

Setting X = 6n + 3 and Y = 18m + 9, this becomes an elliptic curve:

E : Y^{2} = X^{3} − 9X + 81.

E(ℚ) has no torsion and rank 2. E is not proved to be modular (it is not semistable, conductor N = 8604 = 2^{2}3^{2}239). E(ℚ) is generated by P = (−3, 9) and Q = (0, 9). A search for integral points with X ≤ 6 × 10^{7} finds the following:

X = −5, −3, 0, 3, 7, 9, 24, 33, 39, 513, 1099, 5112.

Using the generators P and Q, Smart showed that all integral points are iP + jQ with |i|, |j| ≤ 13. Computing all these points we find just the ones above. To fit the original problem we require X = 6n + 3: only X = 9, 33, 39, 513 fit, giving four solution to the original problem:

(m, n) = (1, 1), (10, 5), (13, 6), (645, 85).