I found my current job in Bordeaux thanks to this blog post of Minhyong Kim, and will now try to make someone else a similar favour.
Firstly, there are postdoc positions at INRIA advertised here. The subject area is computer science and applied mathematics, but I know that in the past this has included computational arithmetic geometry. If anyone is interested, I can put you in touch with someone at the maths department here in Bordeaux who has hired math postdocs via INRIA before (am hesitant to post his name and email address here without his permission). Just leave a comment or send me an email.
Secondly, there is some European Union money for a number of postdoc positions in the Aquitaine region. As far as I understand, the first step towards an application is to contact a potential host researcher, for example at the IMB.