A hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field. If at least one of the rational prime divisors is rational over the field of constants, then it also can be represented as an imaginary quadratic extension of the rational function field. The arithmetic in the divisor class group can be realized in the second case by Cantor’...

We begin with the same setup as before: Let x be a Heegner point on X0(N), let c be the divisor (x)− (∞), and let d be the divisor (x)− (0). We need to compute the Néron height pairing 〈c, Tmc〉 = 〈c, Tmd〉. We have seen that this global height pairing decomposes into a sum of local height pairings. Andrew computed the local height pairings at the archimedean places two weeks ago; our goal will b...

We obtain computational hardness results for f-vectors of polytopes by exhibiting reductions the problems DIVISOR and SEMI-PRIME TESTABILITY to on polytopes. Further, we show that corresponding simplicial are polytime solvable. The situation where prove this difference (conditioned standard conjectures density primes $$P \ne NP $$ ) is when dimension d tends infinity number facets linear in d.

Let $X$ be a generic determinantal affine variety over perfect field of characteristic $p \geq 0$ and $P \subset X$ standard prime divisor generator $\mathrm{Cl}(X) \cong \mathbb{Z}$. We prove that the pair $(X,P)$ is purely $F$-regular if $p>0$ so log terminal (PLT) $p=0$ $\mathbb{Q}$-Gorenstein. In general, using recent results Z. Zhuang S. Lyu, we show PLT-type, i.e. there $\mathbb{Q}$-divis...

In the present work, we prove that every class of divisor group a Krull monoid algebra contains infinitely many prime divisors. Several attempts to this result have been made in literature so far, unfortunately with open gaps. We complete proof fact.