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Non-solvable Galois number fields ramified at 2, 3 and 5 only

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In the mid 90s, Dick Gross made the following conjecture.

Conjecture: For every prime p, there exists a non-solvable Galois number field K ramified at p only.

For p >= 11 this conjecture follows from results of Serre and Swinnerton-Dyer using mod p Galois representations attached to classical modular forms. However, it a consequence of the Serre conjecture, now a theorem thanks to Khare and Wintenberger, et al, that classical modular forms cannot yield the case p <= 7. In this talk, we show that the conjecture is true for p = 2, 3 and 5 using Galois representations attached to Hilbert modular forms. We will also explain the limitations of this technique for the prime p = 7, and outline an alternative strategy using the unitary group U(3) attached to the extension Q(zeta_7)/Q(zeta_7)^+.

This talk is part of the Number Theory Seminar series.

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