Sato-Tate equidistribution of certain families of Artin \(L\)-functions. (English) Zbl 1469.11443
This paper streched out over 62 pages, has to do with several examples and arithmetic statistics of families of \(L\)-function dealing with the Katz-Sarnak heuristics. The list of References counts for 79 papers and books. Out of those, of older and recent times, important developments are combined, treated and extended. In particular, reference [in: Families of automorphic forms and the trace formula. Proceedings of the Simons symposium, Puerto Rico, 2014. Cham: Springer. 531–578 (2016; Zbl 1417.11078)] is used as a framework; it concerns the contribution by P. Sarnak et al. The Introduction of the paper under review consists of nine pages, whereas the next sections 2, 3, 4 preface in detail for results to be given in section 5, so-called \(S_n\)-families, in particular the parametric family of monogenized degree-\(n\) numbers and the proof of their Theorem 1.1 concerning Sato-Tate equidistribution. In section 6, one finds geometric families in which the fields yield \(L\)-functions in the families where the fields do not all have the same Galois group. Skipping section 7 here, one sees in section 8 the appearance of one-parameter families of quaternionic fields.
The paper can be regarded as informative and as a worked-out compendium of the following papers in the References (to name but a few): work by M. Bhargava [Ann. Math. (2) 162, No. 2, 1031–1063 (2005; Zbl 1159.11045); among others]; work by D. Fiorilli et al. [Math. Proc. Camb. Philos. Soc. 160, No. 2, 315–351 (2016; Zbl 1371.14026)]; N. M. Katz [Enseign. Math. (2) 59, No. 3–4, 359–377 (2013; Zbl 1320.14042)], N. M. Katz and P. Sarnak [Bull. Am. Math. Soc., New Ser. 36, No. 1, 1–26 (1999; Zbl 0921.11047)]; Sarnak et al. [loc. cit. ]; M. M. Wood [Algebra Number Theory 2, No. 4, 391–405 (2008; Zbl 1176.11063); among others]; etc., etc.
A very rich paper, indeed! It took me a few hours in order and organize the totality of ideas, theorems, connections, and the like. But it was worth to do so. The contents of the paper give a good impression of results in this area, starting (say) in modern times, \(\pm\) 1990, up to now, thereby strategically involving results from the past.
The paper can be regarded as informative and as a worked-out compendium of the following papers in the References (to name but a few): work by M. Bhargava [Ann. Math. (2) 162, No. 2, 1031–1063 (2005; Zbl 1159.11045); among others]; work by D. Fiorilli et al. [Math. Proc. Camb. Philos. Soc. 160, No. 2, 315–351 (2016; Zbl 1371.14026)]; N. M. Katz [Enseign. Math. (2) 59, No. 3–4, 359–377 (2013; Zbl 1320.14042)], N. M. Katz and P. Sarnak [Bull. Am. Math. Soc., New Ser. 36, No. 1, 1–26 (1999; Zbl 0921.11047)]; Sarnak et al. [loc. cit. ]; M. M. Wood [Algebra Number Theory 2, No. 4, 391–405 (2008; Zbl 1176.11063); among others]; etc., etc.
A very rich paper, indeed! It took me a few hours in order and organize the totality of ideas, theorems, connections, and the like. But it was worth to do so. The contents of the paper give a good impression of results in this area, starting (say) in modern times, \(\pm\) 1990, up to now, thereby strategically involving results from the past.
Reviewer: Robert W. van der Waall (Amsterdam)
MSC:
| 11R42 | Zeta functions and \(L\)-functions of number fields |
| 11M41 | Other Dirichlet series and zeta functions |
| 11M50 | Relations with random matrices |
Keywords:
Katz-Sarnak heuristics; Sato-Tate equidistribution; Artin \(L\)-functions; Dedekind \(\zeta\)-function; number fields; quadratic and bilinear forms; binary forms; quaternionic fieldsCitations:
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