Phase IV — Arithmetic Geometry

Weeks 65–82 · ~117 hrs · Silverman + Milne

Goal: Add the number-theoretic dimension. Elliptic curves defined over \(\mathbb{Q}\), \(\mathbb{F}_q\), and \(\mathbb{Z}\); the Mordell-Weil theorem; the Hasse bound; and the Weil conjectures as a grand synthesis. By the end, you should understand the statement of BSD and why it’s hard, and be comfortable with how algebraic geometry over non-algebraically-closed fields works.

Primary text: Silverman, The Arithmetic of Elliptic Curves (AEC) Number theory supplement: Ireland-Rosen, A Classical Introduction to Modern Number Theory, Ch 8–11 Weil conjectures: Milne, Lectures on Étale Cohomology (free), Ch 1–2

Phase Bridge: From Geometry to Arithmetic

Phase III closed with the full scheme-theoretic treatment of curves over an algebraically closed field. Phase IV removes that assumption: the base field is now \(\mathbb{Q}\), \(\mathbb{F}_q\), or a number field. The key new phenomenon is that the Galois group \(G_k = \text{Gal}(\bar{k}/k)\) acts everywhere.

Weeks 65–67 — Elliptic Curves over Arbitrary Fields

Concepts to understand:

• Weierstrass equations and the discriminant: \(\Delta = -16(4a^3 + 27b^2) \neq 0\) for smoothness
• Short Weierstrass form (\(\text{char} \neq 2, 3\)): \(y^2 = x^3 + ax + b\)
• The group law over any field \(k\): the chord-and-tangent construction works over any field
• The group \(E(k)\): \(k\)-rational points form an abelian group
• Isogenies over non-algebraically-closed fields: a finite morphism \(\phi: E_1 \to E_2\) preserving the identity

Reading:

• Silverman AEC, Ch 1 (§1–4), Ch 2 (§1–4) (~5 hrs)

Problems:

• Silverman AEC: 1.1, 1.2, 1.3, 1.4, 2.1, 2.2, 2.5, 2.6, 2.8, 2.11

Weeks 68–69 — Isogenies and the Tate Module

Concepts to understand:

• Isogeny: nonzero morphism \(\phi: E_1 \to E_2\) of elliptic curves (automatically surjective over \(\bar{k}\))
• Degree of an isogeny: degree as a finite morphism; \(\deg([n]) = n^2\)
• The \(n\)-torsion \(E[n](\bar{k}) \cong (\mathbb{Z}/n)^2\) for \(\text{char}(k) \nmid n\)
• Tate module: \(T_\ell(E) = \varprojlim E[\ell^n] \cong \mathbb{Z}_\ell^2\) (as abelian groups over \(\bar{k}\))
• Galois acts on \(T_\ell(E)\): the \(\ell\)-adic representation \(\rho_\ell: G_k \to \text{GL}_2(\mathbb{Z}_\ell)\)

Reading:

• Silverman AEC, Ch 3 (§1–7) (~4 hrs)

Problems:

• Silverman AEC: 3.1, 3.2, 3.5, 3.6, 3.7, 3.10

Weeks 70–72 — Mordell-Weil Theorem

Concepts to understand:

• Mordell-Weil theorem: \(E(\mathbb{Q}) \cong \mathbb{Z}^r \oplus E(\mathbb{Q})_{\text{tors}}\) (finitely generated abelian group)
• The rank \(r\): the number of independent generators of infinite order; can be 0, 1, 2, …
• Proof strategy: (1) \(E(\mathbb{Q})/2E(\mathbb{Q})\) is finite (weak Mordell-Weil); (2) infinite descent via heights
• Weil height function \(h: E(\mathbb{Q}) \to \mathbb{R}\): measures arithmetic complexity of a rational point
• Canonical (Néron-Tate) height \(\hat{h}\): the unique quadratic form approximating \(h\); positive definite on \(E(\mathbb{Q}) \otimes \mathbb{R}\)

Reading:

• Silverman AEC, Ch 4 (§1–5: heights), Ch 8 (§1–2: Mordell-Weil) (~6 hrs)

Problems:

• Silverman AEC: 4.1, 4.3, 4.4, 8.1, 8.2, 8.5
• For \(E: y^2 = x^3 - x\), show \(E(\mathbb{Q})_{\text{tors}} \cong \mathbb{Z}/2 \times \mathbb{Z}/2\) and find a generator of infinite order if \(r = 1\)

Milestone: State and sketch the proof of Mordell-Weil from scratch. Identify where each of the two main steps (weak M-W and descent) uses the height function.

Weeks 73–74 — Torsion Subgroups over \(\mathbb{Q}\)

Concepts to understand:

• Nagell-Lutz theorem: if \(E: y^2 = x^3 + ax + b\) with \(a, b \in \mathbb{Z}\), then \(E(\mathbb{Q})_{\text{tors}}\) has coordinates in \(\mathbb{Z}\) and \(y = 0\) or \(y^2 \mid \Delta\)
• Using Nagell-Lutz to find all torsion points: a finite computation
• Mazur’s torsion theorem (statement): \(E(\mathbb{Q})_{\text{tors}}\) is one of exactly 15 groups:
  • \(\mathbb{Z}/n\mathbb{Z}\) for \(n = 1, 2, \ldots, 10, 12\)
  • \(\mathbb{Z}/2 \times \mathbb{Z}/2n\) for \(n = 1, 2, 3, 4\)
• Reduction mod \(p\): for good primes \(p\), torsion injects into \(\tilde{E}(\mathbb{F}_p)\)

Reading:

• Silverman AEC, Ch 7 (§1–3: torsion) (~3 hrs)

Problems:

• Silverman AEC: 7.1, 7.2, 7.3, 7.5, 7.7
• Find all rational torsion points of \(E: y^2 = x^3 - x\) using Nagell-Lutz

Weeks 75–76 — Elliptic Curves over Finite Fields

Concepts to understand:

• \(E(\mathbb{F}_q)\): the group of \(\mathbb{F}_q\)-rational points; a finite abelian group
• Hasse’s theorem: \(|E(\mathbb{F}_q)| = q + 1 - t\) where \(|t| \leq 2\sqrt{q}\) (the “trace of Frobenius”)
• The Frobenius endomorphism \(\phi_q: (x,y) \mapsto (x^q, y^q)\): an isogeny of degree \(q\)
• Characteristic polynomial of Frobenius: \(T^2 - tT + q\); roots \(\alpha, \bar{\alpha}\) with \(|\alpha| = \sqrt{q}\)
• \(|E(\mathbb{F}_{q^n})| = q^n + 1 - \alpha^n - \bar{\alpha}^n\) for all \(n\)

Reading:

• Silverman AEC, Ch 5 (§1–4: finite fields) (~4 hrs)
• Ireland-Rosen, Ch 8 §1–3 (~2 hrs)

Problems:

• Silverman AEC: 5.1, 5.2, 5.3, 5.4, 5.7, 5.9
• For \(E: y^2 = x^3 + x\) over \(\mathbb{F}_p\), compute \(|E(\mathbb{F}_p)|\) for \(p = 5, 7, 11, 13\)

Weeks 77–78 — Zeta Functions of Curves

Concepts to understand:

• Zeta function of a curve \(X/\mathbb{F}_q\): \(Z(X, T) = \exp\left(\sum_{n=1}^\infty |X(\mathbb{F}_{q^n})| \frac{T^n}{n}\right)\)
• For a smooth projective curve of genus \(g\) over \(\mathbb{F}_q\):
  • \(Z(X, T) = \frac{P(T)}{(1-T)(1-qT)}\) where \(P(T) = \prod_{i=1}^{2g}(1 - \alpha_i T)\) is a polynomial of degree \(2g\)
  • Rationality: \(Z(X,T) \in \mathbb{Q}(T)\)
  • Functional equation: \(Z(X, \frac{1}{qT}) = q^{1-g} T^{2-2g} Z(X, T)\)
  • Riemann hypothesis: \(|\alpha_i| = \sqrt{q}\) for all \(i\)
• For elliptic curves: \(P(T) = 1 - tT + qT^2\) where \(t\) is the trace of Frobenius

Reading:

• Ireland-Rosen, Ch 11 §1–4 (~3 hrs)
• Milne, Lectures on Étale Cohomology, Ch 1 §1.1–1.3 (introduction) (~2 hrs)

Problems:

• Compute \(Z(E, T)\) for \(E: y^2 = x^3 + x\) over \(\mathbb{F}_5\) by counting points for \(n = 1, 2, 3\)
• Verify the functional equation for this example

Weeks 79–80 — The Weil Conjectures

Concepts to understand:

• The Weil conjectures (1949) for a smooth projective variety \(X/\mathbb{F}_q\) of dimension \(n\):
  1. Rationality: \(Z(X, T) = \prod_{i=0}^{2n} P_i(T)^{(-1)^{i+1}}\), each \(P_i \in \mathbb{Z}[T]\)
  2. Functional equation: \(Z(X, \frac{1}{q^n T}) = \pm q^{n\chi/2} T^\chi Z(X, T)\) where \(\chi = \chi_{\text{top}}(X)\)
  3. Riemann hypothesis: roots of \(P_i\) have absolute value \(q^{-i/2}\)
  4. Betti numbers: \(\deg P_i = b_i\), the \(i\)-th Betti number of the “corresponding” complex variety
• Weil’s proof for curves (1948): using the Riemann-Roch theorem and a positivity argument
• Grothendieck’s proof strategy: \(\ell\)-adic cohomology \(H^i_{\text{ét}}(X_{\bar{\mathbb{F}}_q}, \mathbb{Q}_\ell)\) plays the role of \(H^i_{\text{sing}}\); Frobenius acts on it; the Lefschetz fixed-point formula gives the zeta function
• Deligne’s proof of the Riemann hypothesis (1974): the deepest result; uses weights and Hard Lefschetz

Reading:

• Milne, Lectures on Étale Cohomology, Ch 1–2 (for statement, intuition, and Weil’s proof for curves) (~4 hrs)
• Ireland-Rosen, Ch 11 §5–6 (Weil’s proof for curves) (~2 hrs)

Problems:

• Verify the Weil conjectures for \(\mathbb{P}^n_{\mathbb{F}_q}\) directly: compute \(Z(\mathbb{P}^n, T)\) and check all four properties
• Verify the Weil conjectures for a smooth plane conic over \(\mathbb{F}_q\) (genus 0, so \(\mathbb{P}^1\) after base change)
• Sketch Weil’s proof of the Riemann hypothesis for curves: where does Riemann-Roch enter?

Geometric insight: The Weil conjectures say that the zeta function of a variety over \(\mathbb{F}_q\) “behaves like” the zeta function of a compact complex manifold. This is the deep bridge between arithmetic and geometry: number theory over \(\mathbb{F}_q\) is secretly topology of the corresponding complex variety.

Week 81 — The BSD Conjecture

Concepts to understand:

• The \(L\)-function of an elliptic curve \(E/\mathbb{Q}\): \(L(E, s) = \prod_{p \nmid N} \frac{1}{1 - a_p p^{-s} + p^{1-2s}} \prod_{p \mid N} (\text{bad factors})\) where \(a_p = p + 1 - |E(\mathbb{F}_p)|\)
• Modularity theorem (Wiles-Taylor-Wiles, 1995): every elliptic curve over \(\mathbb{Q}\) is modular — \(L(E,s) = L(f, s)\) for a modular form \(f\)
• BSD conjecture: \(\text{ord}_{s=1} L(E, s) = \text{rank}(E(\mathbb{Q}))\)
• Birch-Swinnerton-Dyer: numerical evidence by computing \(|E(\mathbb{F}_p)|\) for many primes
• Why it is hard: the rank is “global” (points over \(\mathbb{Q}\)) while \(L(E,s)\) is “local” (product over all primes)

Reading:

• Silverman AEC, Appendix C (BSD) (~2 hrs)
• Silverman, The Arithmetic of Elliptic Curves, Ch 5 §5 (introduction to \(L\)-functions) (~2 hrs)

Problems:

• For \(E: y^2 = x^3 - x\) (rank 0), verify numerically that \(L(E, 1) \neq 0\)
• For \(E: y^2 = x^3 - x^2 - 2x\) (rank 1), observe numerically that \(L(E, s)\) has a simple zero at \(s = 1\)

Week 82 — Final Qual Preparation

Use this week for a full mock qualifying exam.

Mock exam checklist:

• Work the complete Ritvik Ramkumar transcript from memory (algebraic geometry section)
• Work the complete Will Fisher transcript from memory (algebraic geometry section)
• Work all Harvard qualifying problems — aim for confidence on \(> 80\%\)
• Write clean answers to:
  • State and prove Riemann-Roch from scratch
  • State Serre duality and deduce \(h^1(\mathcal{O}_E) = 1\) for an elliptic curve
  • Define Weil divisors, Cartier divisors, and \(\text{Pic}(X)\); give an example where Weil \(\neq\) Cartier
  • State Hurwitz’s theorem and apply it to classify unramified covers of elliptic curves
  • Define a scheme. Give an example that is not a variety. Give an example of a non-separated scheme.
  • State the Weil conjectures and verify them for \(\mathbb{P}^1_{\mathbb{F}_q}\)

Final Milestone: A successful qual preparation means you can respond to unexpected follow-up questions, not just recite definitions. The measure is: given any statement in the transcripts, can you reconstruct the why — the geometric picture — without having seen that specific question before?