8
$\begingroup$

In the end of Section 9.2 of Bosch's book Lectures on Formal and Rigid Geometry, a rigid $S$-space $E_Q$ is constructed, for a variable $Q$ replacing the classical parameter $q\in k$. (Here $k$ is a non-archimedean field and $S=\mathrm{Spf}\mathbb Z[[Q]]$). Then $E_Q$ may be viewed as the family of all Tate elliptic curves by looking at the morphism $Z[[Q]] \to k^\circ, Q\mapsto q$.

(1) It looks interesting but the reference is missing in Bosch's book. Which one of Raynaud's paper is about this construction?

(2) Could we in some sense further generalize it by considering more than one variables, i.e. $S'=\mathbb Z[[Q_1,\dots, Q_k]]$? Notice that $S'$ seems still satisfy the condition $(N)$ required by Bosch (see p 162)

Improve this question
$\endgroup$
1

1 Answer 1

Reset to default
3
$\begingroup$

Q1: Michel Raynaud, Géométrie analytique rigide d’après Tate, Kiehl..., Bull. Soc. Math. France, Mémoire 39-40, p. 319-327 (1974).

Improve this answer
$\endgroup$
2
  • 2
    $\begingroup$ As far as I know, the first place where the construction of the universal elliptic curve appears is in "Les schémas de modules de courbes elliptiques" by Deligne and Rapoport, in section VII.1., page 149, where they attribute it to Raynaud without any reference. This was in 1972, in the proceedings of the international summer school "Modular Functions of One Variable II". $\endgroup$
    – Xarles
    Commented Dec 9, 2019 at 8:55
  • $\begingroup$ I have the same question as the OP. It seems to me that the reference you give does not contain any mention of Tate universal elliptic curve. What Raynaud does there, is showing how to construct rigid spaces as generic fibers of formal schemes. Xarles comment seems to give the right reference, as long as there's nothing else by Raynaud to be found. $\endgroup$
    – Daniele Turchetti
    Commented Jan 31, 2020 at 15:45

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .