# Universal Norms and (Phi,Gamma)-Modules

## Introductory Workshop: Algebraic Cycles, L-Values, and Euler Systems January 23, 2023 - January 27, 2023

**Speaker(s):**Laurent Berger (École Normale Supérieure de Lyon)

**Location:**SLMath: Online/Virtual

**Primary Mathematics Subject Classification**No Primary AMS MSC

**Secondary Mathematics Subject Classification**No Secondary AMS MSC

#### Universal Norms And (Phi,Gamma)-Modules

Let K_n=Q_p(mu_{p^n}) for n>0, and let K_infty be the cyclotomic extension of Q_p, namely the union of the K_n. Let Gamma=Gal(K_infty/Q_p). If E is an elliptic curve, or a formal group, the module of universal norms is the projective limit for the trace maps of the E(K_n). What can we say about these universal norms, as a module over the Iwasawa algebra of Gamma? This question has been studied by Mazur, Hazewinkel, Schneider, and others. I will describe a far reaching generalization of this question, that was answered by Perrin-Riou using her big exponential map. One can reinterpret Perrin-Riou's proof in terms of (phi,Gamma)-modules for the cyclotomic extension of Q_p.

#### Universal Norms And (Phi,Gamma)-Modules

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