STNB2023(36th edition)

On reduction steps for Leopoldt's conjecture


Fabio Ferri


Let p be a rational prime and let L/K be a Galois extension of number fields with Galois group G. Under some hypotheses, we show that Leopoldt's conjecture at p for certain proper intermediate fields of L/K implies Leopoldt's conjecture at p for L; a crucial tool will be the theory of idempotent relations in Q[G]. We also consider relations between the Leopoldt defects at p for intermediate extensions of L/K. We finally show that our results combined with some techniques introduced by Buchmann and Sands allow us to find infinite families of nonabelian totally real Galois extension of Q satisfying Leopoldt's conjecture for certain primes. This is joint work with Henri Johnston.


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