A cyclotomic polynomial n (x) is said to be ternary if n = pqr , with p, q and r distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behavior of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family pqr (x) with p < q < r , p and q fixed and r a free prime.
Abstract. In this paper we give new explicit formulas for Faltings' δ-invariant in terms of integrals of theta functions, and we deduce an explicit lower bound for δ only in terms of the genus and an explicit upper bound for the Arakelov-Green function in terms of δ. Furthermore, we give a canonical extension of δ and the Zhang-Kawazumi invariant ϕ to the moduli space of indecomposable principally polarised complex abelian varieties.
We give a close formula for the Néron-Tate height of tautological integral cycles on Jacobians of curves over number fields as well as a new lower bound for the arithmetic self-intersection number ω2 of the dualizing sheaf of a curve in terms of Zhang's invariant ϕ. As an application, we obtain an effective Bogomolov-type result for the tautological cycles. We deduce these results from a more general combinatorial computation of arithmetic intersection numbers of adelic line bundles on higher self-products of curves, which are linear combinations of pullbacks of line bundles on the curve and the diagonal bundle.
We study the additivity of Newton-Okounkov bodies. Our main result states that on two dimensional subcones of the ample cone the Newton-Okounkov body associated to an appropriate flag acts additively. We prove this by induction relying on the slice formula for Newton-Okounkov bodies. Moreover, we discuss a necessary condition for the additivity showing that our result is optimal in general situations. As an application, we deduce an inequality between intersection numbers of nef line bundles.
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