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By Joseph H. Silverman

In The mathematics of Elliptic Curves, the writer offered the fundamental conception culminating in basic international effects, the Mordell-Weil theorem at the finite iteration of the gang of rational issues and Siegel's theorem at the finiteness of the set of imperative issues. This e-book keeps the research of elliptic curves via offering six very important, yet a little bit extra really good issues: I. Elliptic and modular capabilities for the entire modular staff. II. Elliptic curves with complicated multiplication. III. Elliptic surfaces and specialization theorems. IV. Néron types, Kodaira-N ron category of specified fibres, Tate's set of rules, and Ogg's conductor-discriminant formulation. V. Tate's conception of q-curves over p-adic fields. VI. Néron's conception of canonical neighborhood top capabilities.

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Es sei nun to = co'. Wir konnen dann rt = co1 = 1 und folglich f(2) = f(,) + (f<2> - I'1') T setzen. + , (A; f «>, T) + .. = 0. Die Verbindungslinie von cr und r moge nun die Hyperebene xx = 0 in einem Punkt p schneiden, der bei der relationstreuen Spezialisierung (A, £(1), f<2>, T, cr) -> (y«,77, 77, w, co) in n hineinriicke, wobei wir o2 = n2 = 1 annehmen konnen. Wir haben dann (2) T = a + (T2 — 02) 5. -L. Chow. ), das bei der relationstreuen Spezialisierung (A, £(1>, | - (/*, T;, 77,

666 Die Aullosung der Singularitaten. Bisher haben wir nur die allgemeinen Punkte der Kurve C betrachtet. Jetzt wenden wir uns den speziellen oder nulldimensionalen Punkten zu. Die Verhaltnisse sind hier wesentlich komplizierter, da die speziellen Punkte, im Gegensatz zu den allgemeinen, nicht mehr alle miteinander algebraisch aquivalent sind. Dabei ergibt sich als besonders wichtig fiir algebraischgeometrische Untersuchungen der Begriff der Singularitaten, den wir jetzt definieren werden. Bin Punkt f = (p 0 , .

Mathematische Annalen. in. -L. Chow, Algebraisohe Funktionen fiber vollkommenen Korpern. Hasse, H. (2) Uber die Kongruenzzetafunktionen. -B. PreuB. Akad. d. , Phys.. Math. Kl. 1934. XVII. Severi, F. (3) Trattato di geometria algebrica. Vol. I, Parte I (1926). Schmidt, F. K. (4) Analytische Zahlentheorie in K6rpern der Charakteristik p. Math. Zeitschr. 33 (1931). van der Waerden, B. L. (5) Der Multiplizitatsbegriff der algebraischen Geometrie. Math. Annalen 97 (1927). (6) Moderne Algebra; Bd. I I (1931).

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