We formulate a Serre-type conjecture for n-dimensional Galois representations that are tamely ramified at p. The weights are predicted using a representation-theoretic recipe. For n = 3 some of these weights were not predicted by the previous conjecture of Ash, Doud, Pollack, and Sinnott. Computational evidence for these extra weights is provided by calculations of Doud and Pollack. We obtain t...

We present a framework for encoding information about objects from higher arithmetic geometry. This framework is built around a new kind of data type called a Tannakian symbol. The arithmetic objects we have in mind include modular forms (and more general automorphic representations), elliptic curves (and more general schemes, motives and algebraic stacks), finite graphs, group representations,...

We show that if the L-function of an irreducible 2-dimensional complex Galois representation over Q is not automorphic then it has infinitely many poles. In particular, the Artin conjecture for a single representation implies the corresponding strong Artin conjecture. Introduction Let ρ : Gal(Q/Q) → GLn(C) be an irreducible continuous representation of the absolute Galois group of Q. Brauer [2]...

A loop is (right) automorphic if all its (right) inner mappings are automorphisms. Using the classification of primitive groups of small degrees, we show that there is no non-associative simple commutative automorphic loop of order less than 2, and no non-associative simple automorphic loop of order less than 2500. We obtain numerous examples of non-associative simple right automorphic loops. W...

A fundamental problem in number theory is to estimate the values of L-functions at the center of the critical strip. The Langlands program predicts that all L-functions arise from automorphic representations of GL(N) over a number field, and moreover that such L-functions can be decomposed as a product of primitive L-functions arising from irreducible cuspidal representations of GL(n) over Q. T...

Abstract We prove an asymptotic expansion of the second moment central values $\mathrm {GL}(n)\times \mathrm {GL}(n)$ Rankin–Selberg L -functions $L(1/2,\pi \otimes \pi _0)$ for a fixed cuspidal automorphic representation $\pi _0$ over family $ with analytic conductors bounded by quantity that is tending to infinity. Our proof uses integral representations -functions, period regularised Eisenst...