A positive density of fundamental discriminants with large regulator

Abstract

We prove that there is a positive density of positive fundamental discriminants D such that the fundamental unit ε(D) of the ring of integers of the field ‫(ޑ‬ √ D) is essentially greater than D 3 .

On the sup-norm of Maass cusp forms of large level. III

“…If l 1 > 1 then we write the solutions of the equation in terms of the fundamental unit. The fundamental unit is always greater than 1+ √ 5 2 = 1.618 · · · , which is bounded away from 1 (for better estimates, see [4] and the references herein). We deduce that the number of pairs (a + d, l 2 ) of solutions of (2.12) is ≪ Λ o(1) 1.…”

Fundamental solutions to Pell equation with prescribed size

“…By construction, we have U (2) = T (1) U (1) . Since ε D (1) is a root of the Pell equation with even parameter D (1) , we deduce that T (1) is odd and the same applies to U (2) . We continue the induction process by constructing D (3) , D (4) , .…”

“…Lemma 6 implies the equality ε 2 D (1) = ε 4D (1) . Write D (2) := 4D (1) and ε D (2) = T (2) + U (2) √ D (2) . By construction, we have U (2) = T (1) U (1) .…”

“…In this work, we intend to pursue the study initiated by Hooley [5], and followed by the authors [1][2][3], of the statistical properties of the size of ε D . To do so, we consider for x ≥ 2 and 0 ≤ α 1 < α 2 the following set: and let S f (x; α 1 , α 2 ) be its cardinality.…”

Height of rational points on quadratic twists of a given elliptic curve

“…The first step consists in restricting our attention to squarefree integers which have a large prime factor. We note that a similar construction was exploited by Fouvry and Jouve [FJ13] in their investigation of the size of the fundamental solution of the Pell equation. Then, we observe that it is possible to take advantage of this property by parametrizing rational points using a complete 2-descent process as in the previous work of the author [LB16].…”