Abstract
We study various families of Artin$L$-functions attached to geometric parametrizations of number fields. In each case we find the Sato–Tate measure of the family and determine the symmetry type of the distribution of the low-lying zeros.
Highlights
The Katz–Sarnak heuristics [43] concern the arithmetic statistics of a family F of L-functions
We verify the heuristics for certain families arising from number fields
We recall that in [61] the authors distinguish two ways of forming a family: harmonic families, which can be studied with the trace formula; and geometric families arising from algebraic varieties defined over the rationals
Summary
The Katz–Sarnak heuristics [43] concern the arithmetic statistics of a family F of L-functions. The family parametrizing monogenized degree-n number fields ordered by height has Sato–Tate group Sn ⊂ GLn−1(C), rank zero, and Symplectic symmetry type. It satisfies Sato–Tate equidistribution in the sense of [61, Conjecutre 1], and the Katz–Sarnak heuristics for one-level density. It is always the case that the rank is zero for any geometric family of number fields because the construction involves the H 0 of the zerodimensional fibers This is consistent with the belief that Artin L-functions never vanish at the central point except when forced by the root number being −1. It should be possible to improve the remainder terms and support for one-level density using for example large sieve inequalities and Fourier transforms of orbital measures
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