Origin of the name "Torelli group" The genus $g$ Torelli group $I_g$ is the kernel of the action of the mapping class group of a genus $g$ surface on the first homology group of the surface.
The first paper I am aware of that uses the name "Torelli group" for this group is 
Johnson, Dennis A survey of the Torelli group. Low-dimensional topology (San Francisco, Calif., 1981), 165–179, Contemp. Math., 20, Amer. Math. Soc., Providence, RI, 1983. (Reviewer: J. S. Birman) 57N05 (14H15 32G15 57-02)
It had previously been studied without this name in numerous papers (e.g. earlier papers of Johnson, work of Powell, and work of Birman-Craggs).  In Johnson's survey above, he says "The topologists had no name for it, but it has been known for a long time to the analysts, so I will use their label: the Torelli group".
Now, I understand the mathematical justification for calling it the Torelli group, namely Torelli's theorem.  My question is instead historical : Johnson's sentence above seems to imply that he did not make up this name, and thus that there are older papers (maybe by analysts or algebraic geometers?) that use it.  Is anyone aware of any?
 A: As abx noted, the real object that algebraic geometers and complex analysts considered was not the Torelli group per se, but rather Torelli space, which is the quotient of Teichmuller space by the Torelli group.  It's always hard to trace the origins of a name like this, but I believe that the terminology was introduced by Weil in his 1958 Séminaire Bourbaki lecture
A. Weil, Modules des surfaces de Riemann, in Seminaire Bourbaki; 10e annee: 1957/1958. Textes des conferences; Exposes 152 a 168; 2e ed.corrigee, Expose 168, 7 pp, Secretariat Math., Paris.
Notice that this predates the paper of Grothendieck that abx references.  As evidence that this is the correct origin, Rauch explicitly attributes the terminology to Weil on page 544 of
Rauch, H. E. Weierstrass points, branch points, and moduli of Riemann surfaces. Comm. Pure Appl. Math. 12 1959 543-560.
When searching for old papers on the subject, it's also worth pointing out that before Johnson, the Torelli group was often called the Torelli modular group; similarly, the mapping class group was often called the Teichmuller modular group and the symplectic group was often called the Siegel modular group.
EDIT : Just to point out something a little confusing -- I hadn't looked through these old papers in a while, but while doing so I found a few papers that use the phrase Torelli modular group for the symplectic group!  So make sure to read any of these older papers carefully to make sure you understand exactly what they are talking about...
A: I think the terminology was coined by Grothendieck in the Cartan seminar 1960-61. He introduces the term Torelli space for the quotient of the Teichmuller space by the Torelli group. The fact that he says that the corresponding functor "pourrait s'appeler" (could be called) Torelli functor seems to indicate clearly that he is inventing the name.
