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 coefficie...

Abstract A classical theorem of Balcar, Pelant, and Simon says that there is a base matrix height ${\mathfrak h}$ , where the distributivity number ${\cal P} (\omega ) / {\mathrm {fin}}$ . We show if continuum c}$ regular, then are matrices any regular uncountable $\leq {\mathfrak in Cohen random models. This answers questions Fischer, Koelbing, Wohofsky.

There are two fundamental problems motivated by Silverman’s conversations over the years concerning nature of exact values canonical heights $$f(z)\in \bar{\mathbb {Q}}(z)$$ with $$d:=\deg (f)\ge 2$$ . The first problem is conjecture that $$\hat{h}_f(a)$$ either 0 or transcendental for every $$a\in \mathbb {P}^1(\bar{\mathbb {Q}})$$ ; this holds when f linearly conjugate to $$z^d$$ $$\pm C_d(z)...

We consider a family of heights defined by the $L_p$ norms polynomials with respect to equilibrium measure lemniscate for $0 \le p \infty$, where $p=0$ corresponds geometric mean (the generalized Mahler measure) and $p=\infty$ standard supremum norm. This special choice allows find an explicit form polynomial, estimate it via certain resultant. For lemniscates satisfying appropriate hypotheses,...