Masterseminar on number theory_SS 2026

In this seminar we will study Iwasawa theory through the arithmetic of cyclotomic fields, one of the central meeting points of algebraic number theory and the theory of zeta functions. One of the most important results in Iwasawa theory is Iwasawa Main Conjecture, which relates p-adic L-functions to ideal class groups of p-power cyclotomic fields. We will follow Coates and Sujatha's book and use Euler systems to give an elementary and short proof of this important and beautiful result. Along the way, we will study various arithmetic properties of cyclotomic fields, Iwasawa algebras, p-adic Mahler transforms, etc, which are fundamental concepts in modern algebraic number theory.

Tentative program:

  1. [CS] §2.1-2.3. Cover the proof of Theorem 2.1.2. If time allows, prove the Weierstrass preparation theorem 2.1.3.
  2. [CS] §2.4. Cover the proof of Theorem 2.4.6.
  3. [CS] §2.5-2.6. Cover the proof of Theorem 2.5.2 and Proposition 2.6.3 and Theorem 2.6.4.
  4. [CS] §3.1-3.3. Define p-adic integral pseudo-measures and intregration on a profinite abelian groups. Define and study the Mahler transform.
  5. [CS] §3.4-3.6. Cover the proof of Theorem 3.5.1, Proposition 3.5.2.
  6. [CS] §4.1-4.4. Cover the proof of Theorem 4.2.6, Theorem 4.4.1.
  7. [CS] Appendix.
  8. [CS] §4.5-4.6. Cover the proof of Theorem 4.5.2. Proposition 4.5.5, Proposition 4.5.7, Theorem 4.6.3. State Theorem 4.5.6.
  9. [CS] §4.7-4.8. Cover the proof of the Theorems in §4.7. Cover Theorems 4.8.1 and 4.8.2 with a level of details depending on time. If time allows, Proposition 4.7.2.
  10. [CS] §5.1-5.3. Cover the definition and first properties of an Euler systems in §5.2, and the proof of Theorem 5.3.1.
  11. [CS] §5.4-5.5. Cover the proof of Theorem 5.4.9 and Theorem 5.5.1.
  12. [CS] §5.4-5.5. Reach the proof of the main conjecture: Theorem 6.3.1.
  13. [G] Iwasawa Main Conjecture for elliptic curves.

Time and Room: Friday 10h15-11h45 in WSC-S-U-3.02.

Reference:

  1. [CS] Coates,Sujatha, Cyclotomic fields and zeta values (online accessible through our university library).
  2. [G] Greenberg, Iwasawa theory for elliptic curves (https://arxiv.org/abs/math/9809206).
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