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The problem of quantum unique ergodicity (QUE) of weight Eisenstein series for leads to the study of certain double Dirichlet series involving automorphic forms and Dirichlet characters. We study the analytic properties of this family of double Dirichlet series (analytic continuation, convexity estimate) and prove that a subconvex estimate implies the QUE result.
We strengthen the compatibility between local and global Langlands correspondences for when is even and . Let be a CM field and a cuspidal automorphic representation of which is conjugate self-dual and regular algebraic. In this case, there is an -adic Galois representation associated to , which is known to be compatible with local Langlands in almost all cases when by recent work of Barnet-Lamb, Gee, Geraghty and Taylor. The compatibility was proved only up to semisimplification unless has Shin-regular weight. We extend the compatibility to Frobenius semisimplification in all cases by identifying the monodromy operator on the global side. To achieve this, we derive a generalization of Mokrane’s weight spectral sequence for log crystalline cohomology.
We prove that some skew group algebras have Noetherian cohomology rings, a property inherited from their component parts. The proof is an adaptation of Evens’ proof of finite generation of group cohomology. We apply the result to a series of examples of finite-dimensional Hopf algebras in positive characteristic.
Let be a Hopf algebra and let be a Koszul -module algebra. We provide necessary and sufficient conditions for a filtered algebra to be a Poincaré–Birkhoff–Witt (PBW) deformation of the smash product algebra . Many examples of these deformations are given.
Chebyshev observed in a letter to Fuss that there tends to be more primes of the form than of the form . The general phenomenon, which is referred to as Chebyshev’s bias, is that primes tend to be biased in their distribution among the different residue classes . It is known that this phenomenon has a strong relation with the low-lying zeros of the associated -functions, that is, if these -functions have zeros close to the real line, then it will result in a lower bias. According to this principle one might believe that the most biased prime number race we will ever find is the versus race, since the Riemann zeta function is the -function of rank one having the highest first zero. This race has density , and we study the question of whether this is the highest possible density. We will show that it is not the case; in fact, there exist prime number races whose density can be arbitrarily close to . An example of a race whose density exceeds the above number is the race between quadratic residues and nonresidues modulo , for which the density is . We also give fairly general criteria to decide whether a prime number race is highly biased or not. Our main result depends on the generalized Riemann hypothesis and a hypothesis on the multiplicity of the zeros of a certain Dedekind zeta function. We also derive more precise results under a linear independence hypothesis.
Fix an integer that is not a perfect square. In 1927, Artin conjectured that there are infinitely many primes for which is a primitive root. Forty years later, Hooley showed that Artin’s conjecture follows from the generalized Riemann hypothesis (GRH). We inject Hooley’s analysis into the Maynard–Tao work on bounded gaps between primes. This leads to the following GRH-conditional result: Fix an integer. Ifis the sequence ofprimes possessing asa primitive root, then , where is a finiteconstant that depends on but not on . We also show that the primes in this result may be taken to be consecutive.