Notions relatives de régulateurs et de hauteurs

Berge, Anna; Martinet, Jean

doi:10.4064/aa-54-2-155-170

Cited by 14 publications

(6 citation statements)

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“…[F k : I l/k ] Reg(k) Reg E l/k = Reg(l), which was established in [10, Theorem 1]. A slightly different definition for a relative regulator was considered by Bergé and Martinet in [2], [3], and [4]. We have used the definition proposed by Costa and Friedman in [10] and [11], as it leads more naturally to the inequality (3.6).…”

Section: Relative Regulatorsmentioning

confidence: 99%

Heights, regulators and Schinzel’s determinant inequality

Akhtari

Vaaler

2016

Acta Arith.

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We prove inequalities that compare the size of an S-regulator with a product of heights of multiplicatively independent S-units. Our upper bound for the S-regulator follows from a general upper bound for the determinant of a real matrix proved by Schinzel. The lower bound for the S-regulator follows from Minkowski's theorem on successive minima and a volume formula proved by Meyer and Pajor. We establish similar upper bounds for the relative regulator of an extension l/k of number fields.2000 Mathematics Subject Classification. 11J25, 11R04, 46B04.

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Section: Relative Regulatorsmentioning

confidence: 99%

Heights, regulators and Schinzel’s determinant inequality

Akhtari

Vaaler

2016

Acta Arith.

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“…In the spirit of [1], we consider the relative extension K/k . Let T>K¡k be the relative discriminant for the extension K/k .…”

Section: Fundamental Unitsmentioning

confidence: 99%

A family of cyclic quartic fields arising from modular curves

Washington¹

1991

Math. Comp.

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Abstract. We study a family of cyclic quartic fields arising from the covering of modular curves -Y,(16) -» ^(16).An integral basis and a fundamental system of units are found. It is shown that a root of the quartic polynomial we construct is a translate of a cyclotomic period by an integer of the quadratic subfield of the quartic field.Recently, O. Lecacheux [9,10] and H. Darmon [4] showed how to use coverings of modular curves to obtain cyclic extensions of Q. In particular, they were able to give a geometric construction of a family of cyclic quintic fields discovered by E. Lehmer [11]. The covering XX(N) -» XQ(N) (for N > 2) has degree 4>(N)/2 and group (Z//VZ)X/{±1}. For the quintic case, they took N = 25 , which gave a cyclic covering of degree 10, then took the subcovering of degree 5. An important ingredient in the construction was the fact that X0(25) has genus 0. This also occurs for N = 1, ... , 10, 12, 13, 16, 18. These all give trivial or quadratic coverings except for N = 7, 9, 13, 16, 18 . The values N -1, 9, IS yield cubic extensions and can be shown to yield the family of polynomials X3 -aX2 -(a + 3)X -1, namely the "simplest cubic fields" [17]. (However, it should be remarked that every cyclic cubic extension of Q comes from a polynomial of this form if a is allowed to be rational. Similarly, we are guaranteed that the quadratic extensions obtained from the coverings mentioned above correspond to polynomials of the form X -aX -1 with a rational.) The case N = 13 is treated by Lecacheux [9]. It might be suspected that the sextic fields she obtains are the same as the "simplest sextics" constructed by M.-N. Gras [6]. However, these latter fields were found by taking the fixed field of an element of order 6 in PGL2(Q) = Aut(Q(X)). Therefore, they come from a covering of curves of genus 0. But ^(13) has genus 2. Alternatively, these sextic fields must be different because the discriminants of the quadratic, and cubic, subfields are different.In the present paper, we study the case N -16. As above, it might be hoped that this case would give a geometric construction of the quartic fields studied

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“…As Silverman suggested (cf. [BM2,p. 156]), E K can be replaced by any multiplicative group G contained in the algebraic closure Q of the rational field Q.…”

Section: Introductionmentioning

confidence: 99%

“…Despite the rather ad hoc appearance of the above definitions, Bergé and Martinet [BM1,BM2] found an explicit formula for H(θ ; E K ) which is reminiscent of a product expression for the classical height. Recall [La2,p.…”

Section: Introductionmentioning

confidence: 99%