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Stark units and Kolyvagin’s “Euler systems”. (English) Zbl 0752.11045
In his important work on the conjecture of Birch and Swinnerton-Dyer, V. Kolyvagin [The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 435-483 (1990; Zbl 0742.14017)] introduced the concept of an Euler system. Loosely speaking, an Euler system is a set of points on an algebraic group satisfying two conditions: a norm property and a congruence property. In the present paper, the author shows that the norm property implies a weak form of the congruence property that is sufficient for most applications. The proof is fairly formal, relying on the introduction of a “universal Euler system”.
The main importance of the result is that the units predicted to exist by Stark’s conjecture satisfy the norm condition, hence the congruence condition, so they conjecturally yield a new family of Euler systems, which can be used to study the structure of ideal class groups as in the work of F. Thaine [Ann. Math., II. Ser. 128, 1-18 (1988; Zbl 0665.12003)], the author [Invent. Math. 89, 511-526 (1987; Zbl 0628.12007)], and V. Kolyvagin.

MSC:
11R27 Units and factorization
11G16 Elliptic and modular units
11R29 Class numbers, class groups, discriminants
14G05 Rational points
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