Heights via $$p$$-Adic Points

Abstract In a paper published in 1980, the author gave an adelic Tamagawa number interpretation for Birch and Swinnerton-Dyer conjecture divisors on abelian varieties. Some years later, joint work with K. Kato, more general volume zeta values of motives weights $${<}\,{-1}$$ was proposed. at hand, is generalized to deal weight $$-1$$ . As points varieties are replaced by cohomology coefficients. Further, we introduce tori over groups mimic Néron–Severi 1980 paper.