研究路径Number课程已上线

Arithmetic geometry and elliptic curves

Optional research direction: solving polynomial equations in integers and rationals with geometry. Elliptic curves and their group law, the Mordell–Weil theorem, curves over finite fields and the Hasse bound, the Weil conjectures, modularity and Fermat's Last Theorem, and the Birch and Swinnerton-Dyer conjecture.

math.number.arithmetic-geometry

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前置知识

Algebraic number theory Algebraic geometry

53 项必需的前置知识（一直追溯到底）

你将学会

Define elliptic curves, construct the chord-and-tangent group law and compute torsion points.
State the Mordell–Weil theorem and the Hasse bound, and count points over small finite fields.
Explain the statements of the Weil conjectures, the modularity theorem and the Birch and Swinnerton-Dyer conjecture, distinguishing theorems from conjectures.

定义

An elliptic curve over a field K (of characteristic not 2 or 3) is a curve y² = x³ + ax + b with 4a³ + 27b² ≠ 0, together with a point at infinity.
The rank of an elliptic curve over ℚ is the rank of the finitely generated abelian group E(ℚ).
An elliptic curve over ℚ is modular if its L-function coincides with the L-function of a weight-2 modular newform.

定理

Mordell–Weil: E(K) is a finitely generated abelian group for every number field K.
Hasse: |#E(𝔽_q) − (q + 1)| ≤ 2√q.
Modularity theorem (Wiles; Taylor–Wiles; Breuil, Conrad, Diamond and Taylor): every elliptic curve over ℚ is modular; this implies Fermat's Last Theorem.
The Birch and Swinnerton-Dyer conjecture, relating the rank to the order of vanishing of the L-function at s = 1, is open in general.

常见误解

Thinking elliptic curves are ellipses; the name comes from elliptic integrals, which compute arc length on ellipses.
Believing there is a known algorithm guaranteed to compute the rank of every elliptic curve over ℚ; no such algorithm is known unconditionally.

可选发现

Cryptography Elliptic-curve cryptography uses the group of points of an elliptic curve over a finite field.
The Langlands programme: an orientation The modularity theorem — every elliptic curve over ℚ is modular — is an instance of Langlands reciprocity, and implies Fermat's Last Theorem.
← Riemann surfaces
← Complex and Kähler geometry; Hodge theory
← Stable homotopy theory and generalised cohomology

学习路径

Algebra and number theory