Algebraic Number Fields

By V. Harizanov, Julia F. Knight, C. Maher. 2006

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Noncommutative Geometry, Quantum Fields and Motives (2007, unknown): “For more details see [6], [187].”
Class Field Theory (2013, background): “/, viewed as a character ofWK , corresponds under 'K to the central character of ; (b) the map n preserves L-factors and "-factors of pairs of ’s (as defined by Jacquet, Piatetskii-Shapiro, and Shalika on the automorphic side, and by Langlands and Deligne on the Galois side); 15Note the similarity t”
The primes contain arbitrarily long polynomial progressions (2006, background): “) More generally, the Artin reciprocity law [23] relates Artin characters to certain local factors.”
Families of canonically polarized varieties over surfaces (2005, background): “Shafarevich conjectured [18] that the set of non-isotrivial families”
Hilbert's Tenth Problem (2006, methodology): “6, Chapter IV of [37], we haveδ(WK) = δ(W̄K) and the theorem holds for sufficiently large p.”