Abstract
We utilize harmonic analytic tools to count the number of elements of the Galois cohomology group fin H^1(K,T) with discriminant-like invariant text {inv}(f)le X as Xrightarrow infty . Specifically, Poisson summation produces a canonical decomposition for the corresponding generating series as a sum of Euler products for a very general counting problem. This type of decomposition is exactly what is needed to compute asymptotic growth rates using a Tauberian theorem. These new techniques allow for the removal of certain obstructions to known results and answer some outstanding questions on the generalized version of Malle’s conjecture for the first Galois cohomology group.
Highlights
F /K p for some finite extension F /K . (We call the finitely many places where this is not satisfied irregular places.) the result applies if we impose a local condition Lp ⊆ H1(Kp, T ) at each place p, provided that these conditions are again
Harmonic analysis on adelic cohomology has been used by Frei–Loughran–Newton [5,6] as an alternate approach to counting abelian extensions of number fields with prescribed local conditions, and by the second author [10] to give a streamlined proof of the Ohno–Nakagawa reflection theorem and its generalizations
2 Fourier analysis on adelic cohomology Our goal is to study the average value of certain functions over the global cohomology classes H1(K, T )
Summary
T. It is natural to count such coclasses asymptotically by an invariant such as the norm of the discriminant or the product of the ramified primes. The first author [1] proved a result counting coclasses by discriminant-like invariants which are Frobenian; that is, they are determined at almost all places p of K by the Frobenius element. This result can be interpreted as giving the asymptotic size of an “infinite Selmer group,” where for any family of local conditions. (For example, let Lp = H 1(Kp, T ) for all p so that HL1 (K, T ) = H 1(K, T ).) We can still say something about the “size” by talking about how the 1-coclasses are distributed with respect to some ordering. Given an admissible ordering inv : H 1(K, T ) → IK valued in the group of fractional ideals (as in [1]), we can define
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