Number Theory
Arithmetic of Elliptic Curves and BSD
The rational points on an elliptic curve form a finitely generated abelian group - but computing its rank is one of the hardest open problems in mathematics. The BSD conjecture ($1M prize) says the answer is encoded in the order of vanishing of an $L$-function. Understanding this connection requires the full arithmetic theory of elliptic curves.
- Wiles' proof of Fermat's Last Theorem (1995) rests entirely on the modularity of elliptic curves - that every elliptic curve over $\mathbb{Q}$ corresponds to a modular form with matching $L$-function.
- Gross-Zagier formula (1986): when $L'(E,1) \neq 0$, it explicitly constructs a rational point (Heegner point) of infinite order - the only known method for proving rank $\geq 1$ analytically.
- Elliptic curve primality proving (ECPP): the fastest known primality test for large numbers (used by Wolfram Alpha and PARI/GP) relies on the structure of $E(\mathbb{F}_p)$.
Цели урока
- State and apply the Mordell-Weil theorem and Mazur's torsion theorem
- Describe the method of 2-descent for computing the Selmer group
- Explain the Gross-Zagier formula and its role in proving cases of BSD
Предварительные знания
- Elliptic curves: group law and reduction modulo p
- Zeta functions of varieties and the BSD conjecture
- Galois cohomology and class field theory (overview)
Mordell-Weil theorem and torsion
The Mordell-Weil theorem states that $E(\mathbb{Q}) \cong \mathbb{Z}^r \oplus E(\mathbb{Q})_{\mathrm{tors}}$ for some rank $r \geq 0$ and finite torsion subgroup. Mazur's torsion theorem (1977): $E(\mathbb{Q})_{\mathrm{tors}}$ is one of exactly 15 groups - $\mathbb{Z}/n\mathbb{Z}$ for $n \leq 10$ or $n=12$, or $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2n\mathbb{Z}$ for $n \leq 4$. This classification required years of work and uses the theory of modular curves.
2-descent and the Selmer group
The method of 2-descent bounds the rank via the Selmer group $\mathrm{Sel}^{(2)}(E/\mathbb{Q})$: $0 \to E(\mathbb{Q})/2E(\mathbb{Q}) \to \mathrm{Sel}^{(2)} \to \mathrm{Sha}(E/\mathbb{Q})[2] \to 0$. Here $\mathrm{Sha}$ is the Tate-Shafarevich group - its finiteness is conjectured but not proved in general. The rank satisfies $r \leq \dim_{\mathbb{F}_2} \mathrm{Sel}^{(2)} - \dim_{\mathbb{F}_2} E(\mathbb{Q})[2]$. Computing the Selmer group is algorithmic and implemented in Sage and Magma.
Heegner points: for an imaginary quadratic field $K = \mathbb{Q}(\sqrt{-d})$ satisfying the Heegner hypothesis, there is a canonical rational point $y_K \in E(K)$ constructed via the theory of complex multiplication. The Gross-Zagier formula (1986): $L'(E, 1) = c \cdot \hat{h}(y_K)$, where $\hat{h}$ is the canonical height. So $L'(E,1) \neq 0$ iff $y_K$ has infinite order, proving rank $= 1$ in this case.
The Tate-Shafarevich group $\mathrm{Sha}(E/\mathbb{Q})$ is the obstruction to the Hasse principle: it consists of homogeneous spaces for $E$ that have points over every $\mathbb{Q}_p$ but no rational point. Its finiteness is required by BSD but remains unproved in general. For curves of rank 0 and 1, finiteness follows from Kolyvagin's work (1990).
From Mordell (1922) to Wiles (1995): seven decades to FLT
Louis Mordell proved in 1922 that $E(\mathbb{Q})$ is finitely generated, generalizing a result of Poincare. Weil extended this to abelian varieties in 1928 (his doctoral thesis). Mazur's torsion theorem (1977) completed a program started by Nagell and Lutz in the 1930s. Wiles' modularity theorem (1995) built on three decades of work by Shimura, Taniyama, Serre, Ribet, Langlands, and many others.
Mordell-Weil Theorem and Group Structure
Louis Mordell proved in 1922 that the group E(Q) of rational points on an elliptic curve is finitely generated. Andre Weil generalized this to number fields in 1928. For y^2 = x^3 - x the rank is 0; for y^2 = x^3 - 2 the rank is 1 with generator (3, 5).
The current record rank is at least 29, due to Noam Elkies (2006). Whether ranks are unbounded remains open - a central question in the BSD program.
The Mordell-Weil theorem states E(Q) ≅ E(Q)_tors ⊕ Z^r. What does the rank r measure, and how does the regulator relate to it?
The rank r counts the number of independent infinite-order rational points (generators of the free part Z^r). The Neron-Tate height gives a positive semi-definite quadratic form on E(Q)/tors, and the regulator R = det(⟨Pᵢ, Pⱼ⟩) is the volume of a fundamental domain in this lattice. Large R means generators have large height and are hard to find. The regulator appears in the BSD formula alongside L'(E,1) and the period.
Mazur's Torsion Theorem
Barry Mazur proved in 1977 that E(Q)_tors is one of exactly 15 groups: Z/nZ for n = 1,...,10,12 or Z/2Z x Z/2nZ for n = 1,...,4. This sharp classification uses modular curves X_1(N) and the theory of formal groups.
Torsion computation: y² + y = x³ − x
The curve y^2 + y = x^3 - x has E(Q)_tors = Z/5Z. Checking: the point (0, 0) satisfies 5*(0,0) = O (the point at infinity). Modulo p = 7: #E(F_7) = 5, confirming the torsion order.
Mazur's theorem classifies E(Q)_tors as one of exactly 15 groups. Which groups are allowed?
Mazur's 1977 theorem: E(Q)_tors is one of exactly 15 groups - cyclic groups Z/nZ for n ∈ {1,2,...,10,12} (note: n=11 is excluded) and products Z/2Z × Z/2nZ for n ∈ {1,2,3,4}. The proof uses modular curves X_1(N): for large N these have genus ≥ 2, so by Faltings only finitely many rational points exist, ruling out large torsion orders.
2-Descent and Selmer Groups
The 2-descent algorithm computes an upper bound on rank via the 2-Selmer group. The difference between the Selmer rank and the true rank is measured by the 2-part of the Tate-Shafarevich group. BSD predicts this group is finite.
Sage and Magma implement 2-descent. For most curves of conductor < 500,000 the rank is known exactly. The Cremona database lists minimal models, ranks and generators for curves with conductor up to 500,000.
The exact sequence 0 → E(Q)/2E(Q) → Sel^(2)(E/Q) → Sha(E/Q)[2] → 0 relates three groups. What does this imply for the rank?
From the exact sequence: dim E(Q)/2E(Q) = rank E(Q) + dim E(Q)_tors[2] ≤ dim Sel^(2). The gap is dim Sha[2]. So rank ≤ dim Sel^(2) − dim E(Q)_tors[2]. Computing Sel^(2) (which is finite and algorithmically accessible via local conditions) gives an upper bound on rank. To pin down the exact rank one must bound Sha - Kolyvagin's theorem does this when the analytic rank is ≤ 1.
Heegner Points and Gross-Zagier
Kurt Heegner's 1952 paper constructed rational points on elliptic curves via CM theory, though it was largely ignored until Bryan Birch rediscovered the method in the 1960s. The landmark Gross-Zagier theorem (1986) connected the height of Heegner points to the derivative L'(E,1).
Heegner point for conductor-37 curve
For E: y^2 + y = x^3 - x^2 - 10x - 10 (conductor 37, rank 1) and K = Q(sqrt(-7)): the Heegner point y_K = (0, 0) in E(Q) after tracing. Since L'(E,1) is nonzero, Gross-Zagier confirms rank = 1 and Sha is finite.
Zhang Shou-Wu (2001) and Yuan-Zhang-Zhang (2013) extended Gross-Zagier to Shimura curves over totally real fields. This is essential for BSD over number fields beyond Q.
The Gross-Zagier formula states ĥ(y_K) = c·L'(E/K, 1). What does this imply for the rank of E(Q) when L'(E/K, 1) ≠ 0?
Gross-Zagier: ĥ(y_K) = c·L'(E/K,1). If L(E/K,1) = 0 and L'(E/K,1) ≠ 0, then ĥ(y_K) > 0, so y_K has infinite order. Kolyvagin's theorem (1990) then shows: if the Heegner point y_K has infinite order, then rank E(Q) = 1 exactly and Sha(E/Q) is finite. This proves the BSD conjecture for analytic rank ≤ 1 - the most important known case.
Computing torsion and finding generators
For $E: y^2 = x^3 - x$: the torsion subgroup is $\{O, (0,0), (1,0), (-1,0)\} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ (the three 2-torsion points). The rank is 0: the only rational points are these four. Compare $E: y^2 = x^3 - x + 1$: torsion is trivial but rank is 1, generated by $(1, 1)$. The point $(1,1)$ has infinite order: $(1,1), P+P, 3P, \ldots$ are all distinct rational points.
Итоги
- The Mordell-Weil theorem gives $E(\mathbb{Q}) \cong \mathbb{Z}^r \oplus T$ with $T$ classified by Mazur's theorem.
- The rank $r$ is controlled by the Selmer group via 2-descent.
- The Gross-Zagier formula connects $L'(E,1)$ to the height of a Heegner point, proving BSD for rank 1 in many cases.
Connections to other topics
The BSD conjecture (nt-27-en) is the arithmetic statement; Mordell-Weil and 2-descent are the algebraic tools for bounding the rank. The Weil conjectures (proved by Deligne) and the modularity theorem (Wiles) together make $L$-functions of elliptic curves a central object in the Langlands program.
- Nt 27 En — related
Вопросы для размышления
- The Gross-Zagier formula $L'(E,1) = c \cdot \hat{h}(y_K)$ equates an analytic quantity (derivative of an $L$-function) with a geometric one (height of a point). Why does this formula prove that if the analytic rank is exactly 1, then the geometric rank is at least 1?
- What would it take to prove the converse?