The Galois groupoid of G-spectra is equivalent to the étale fundamental groupoid of the Burnside ring of G.
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For primes N and p with N ≡ 1 mod p, the rank r of Mazur's Eisenstein Hecke algebra equals one plus the vanishing order of a mod-p zeta element interpolating L-values at -1 when r is 2 or 3, with a uniform extension to level N² and partial results for higher ranks.
For n greater than or equal to 3 and sufficiently generic weights, the universal supersingular representation of GL_n(k) is non-admissible and of infinite length.
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The Galois theory of $G$-spectra and the Burnside ring
The Galois groupoid of G-spectra is equivalent to the étale fundamental groupoid of the Burnside ring of G.