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Book description: From the reviews of Vol. IV: "This is the fourth volume of J-P. Serre's Collected Papers covering the period 1985-1998. Items, numbered 133-173, contain "the essence'' of his work from that period and are devoted to number theory, algebraic geometry, and group theory. Half of them are articles and another half are summaries of his courses in those years and letters. Most courses have never been previously published, nor proofs of the announced results. The letters reproduced, however (in particular to K. Ribet and M.-F. Vignéras), provide indications of some of those proofs. Also included is an interview with J-P. Serre from 1986, revealing his views on mathematics (with the stress upon its integrity) and his own mathematical activity. The volume ends with Notes which complete the text by reporting recent progress and occasionally correct it. The volume is an indispensable addition to previous ones and the whole set presents a fine piece of modern mathematics." Zentralblatt MATH In this softcover edition of volume IV, two recently published articles have been added, one on the life and works of Andrè Weil, the other one on Finite Subgroups of Lie Groups.