Gras's freeness conjecture for maximal pro-p extensions unramified outside p

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Let KK be a number field, let SpS_p be the set of pp-adic primes of KK, and let ΓSp=Gal⁡(KSp/K)\Gamma_{S_p}=\operatorname{Gal}(K_{S_p}/K), where KSpK_{S_p} is the maximal pro-pp extension of KK unramified outside SpS_p. Let r2r_2 be defined by the condition that 2r22r_2 is the number of complex embeddings of KK. Gras's conjecture. For p≫0p\gg 0, the pro-pp group ΓSp\Gamma_{S_p} is free on r2+1r_2+1 generators. This conjecture predicts the eventual freeness and rank of the relevant Galois groups; the source attributes it to Gras, but the supplied text gives no evidence resolving its status.

References

Primary source

Christian Maire, Ján Mináč, Ravi Ramakrishna and Nguyen Duy Tan, “On the strong Massey property for number fields”, arXiv:2409.01028 (2024).

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