Generalised Mertens and Brauer–Siegel theorems

Abstract

1. Introduction. In this article, we prove a generalisation of the Mertens theorem for prime numbers to number fields and algebraic varieties over finite fields, paying attention to the genus of the field (or the Betti numbers of the variety), in order to make it tend to infinity and thus to point out the link between it and the famous Brauer-Siegel theorem. Using this we deduce an explicit version of the generalised Brauer-Siegel theorem under GRH, and a unified proof of this theorem for asymptotically exact families of almost normal number fields.The classical Brauer-Siegel theorem describes the asymptotic behaviour of the quantity hR (the product of the class number and the regulator) in a family of number fields with growing genus under the conditions that the genus grows much faster than the degree and assuming some additional properties like normality or the Generalised Riemann Hypothesis (GRH) to deal with the Siegel zeroes. These two hypotheses are of different nature: omitting the first changes the final result, while the second is a technical hypothesis. Tsfasman and Vlȃduţ [9] were able to remove the first hypothesis, which led to the so called generalised Brauer-Siegel theorem, and Zykin [10] was able to replace "normality" by "almost normality" in the second hypothesis by using results of Stark and Louboutin. He also managed to generalise the Brauer-Siegel theorem to the case of smooth absolutely irreducible projective varieties over finite fields.As for the Mertens theorem, proven by Mertens in the case of Q, and much later generalised by Rosen [5] to both number and function fields, it can be regarded as the Brauer-Siegel theorem in the global field or variety constituting the family. An explicit Mertens theorem leads therefore to an explicit formulation of the generalised Brauer-Siegel theorem. We first recall the formulations of the (generalised) Brauer-Siegel theorem and Mertens theorem, then we prove their explicit versions for number fields and smooth

Lower bounds on the class number of algebraic function fields defined over any finite field

“…Then, Lebacque [9,Theorem 7] obtains an explicit version of the Generalized Brauer-Siegel theorem valid in the case of smooth absolutly irreducible abelian varieties defined over a finite field and for the number fields under GRH. Specialized to the case of smooth absolutly irreducible curves defined over a finite field, this theorem leads to the following result: Theorem 1.6.…”

On the number of rational points of Jacobians over finite fields

“…Step 2 5 2 7 7 30 12 3 13 2 14 10453 42898 26 4 33 2 27 343 733 443 618 1 543 267 494 985 74 6 We notice that our bound is better than the other ones except for the case of H 2 /F 4 where we can not beat the h AHL bound. The situation changes, however, if we use some additional information on the places of H 2 /F 4 .…”

“…Our starting point is the Mertens theorem ( [6]) for curves and its relation to the generalized Brauer-Siegel theorem. Our exposition differs slightly from [6]: we take the opportunity to sharpen (and sometimes correct) the corresponding bounds. Let us recall Serre's explicit formulae from [8].…”

“…Theorem 2.5. (Mertens theorem, [6]) Let X be a smooth projective absolutely irreducible curve of genus g defined over F q . Then…”

The Bateman–Horn conjecture: Heuristic, history, and applications

“…Thus, the asymptotic distribution of prime ideals (by norm) in a number field mirrors that of the prime numbers in the integers. Therefore, it is not surprising to find an analogue of Mertens' theorem (Theorem 5.3.2) that holds for prime ideals [69,Lemma 2.4] or [54,Prop. 2].…”