In number theory, an eigencurve is a rigid analytic curve that parametrizes certain p-adic families of modular forms, and an eigenvariety is a higher-dimensional generalization of this. Eigencurves were introduced by Coleman and Mazur (1998), and the term "eigenvariety" seems to have been introduced around 2001 by Kevin Buzzard (2007).
References Buzzard, Kevin (2007), "Eigenvarieties" (PDF), in Burns, David; Buzzard, Kevin; Nekovář, Jan (eds.), L-functions and Galois representations, London Math. Soc. Lecture Note Ser., vol. 320, Cambridge University Press, pp. 59–120, doi:10.1017/CBO9780511721267.004, ISBN 978-0-521-69415-5, MR 2392353 Coleman, R.; Mazur, Barry (1998), "The eigencurve", Galois representations in arithmetic algebraic geometry (Durham, 1996), London Math. Soc. Lecture Note Ser., vol. 254, Cambridge University Press, pp. 1–113, doi:10.1017/CBO9780511662010.003, ISBN 9780511662010, MR 1696469