On the spectrum of the Zhang-Zagier height

Christophe Doche*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


From recent work of Zhang and of Zagier, we know that their height Heng hooktop sign(α) is bounded away from 1 for every algebraic number α different from 0, 1, 1/2 ± √-3/2. The study of the related spectrum is especially interesting, for it is linked to Lehmer's problem and to a conjecture of Bogomolov. After recalling some definitions, we show an improvement of the so-called Zhang-Zagier inequality. To achieve this, we need some algebraic numbers of small height. So, in the third section, we describe an algorithm able to find them, and we give an algebraic number with height 1.2875274. . . discovered in this way. This search up to degree 64 suggests that the spectrum of Heng hooktop sign(α) may have a limit point less than 1.292. We prove this fact in the fourth part.

Original languageEnglish
Pages (from-to)419-430
Number of pages12
JournalMathematics of Computation
Issue number233
Publication statusPublished - Jan 2001
Externally publishedYes


  • Conjecture of Bogomolov
  • Mahler measure


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