Heights via $p$-Adic Points

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