In mathematics, an Igusa curve is (roughly) a coarse moduli space of elliptic curves in characteristic p with a level p Igusa structure, where an Igusa structure on an elliptic curve E is roughly a point of order p of E(p) generating the kernel of V:E(p) → E. An Igusa variety is a higher-dimensional analogue of an Igusa curve. Igusa curves were studied by Igusa (1968) and Igusa varieties were introduced by Harris and Taylor (2001) with the motivation that they have application to studying the bad reduction of some PEL Shimura varieties, the ℓ-adic cohomology of Igusa varieties sheds some light on that of Shimura varieties.
References Harris, Michael; Taylor, Richard (2001), The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, ISBN 978-0-691-09090-0, MR 1876802 Igusa, Jun-ichi (1968), "On the algebraic theory of elliptic modular functions", Journal of the Mathematical Society of Japan, 20: 96–106, doi:10.2969/jmsj/02010096, ISSN 0025-5645, MR 0240103