Abstract In this paper we consider two piecewise Riemannian metrics defined on the Culler–Vogtmann outer space which call entropy metric and pressure . As a result of work McMullen, these can be seen as analogs Weil–Petersson Teichmüller closed surface. We show that while geometric analysis is similar to metric, from point view group theory, behave very differently than metric. Specifically, wh...

Suppose E is an elliptic curve over Q and χ a Dirichlet character. We use statistical properties of modular symbols to estimate heuristically the probability that L(E,χ,1)=0. Via Birch Swinnerton-Dyer conjecture, this gives heuristic Mordell–Weil rank grows in abelian extensions Q. Using heuristic, we find large class infinite F where expect E(F) be finitely generated. Our work was inspired by ...

In this paper, we investigate the relationship between the squared Weil/Tate pairing and the plain Weil/Tate pairing. Along these lines, we first show that the squared pairing for arbitrary chosen point can be transformed into a plain pairing for the trace zero point which has a special form to compute them more efficiently. This transformation requires only a cost of some Frobenius actions. Ad...

In this chapter we will relate the topology of smooth projective varieties over the complex numbers with counting points over finite fields, via the Weil conjectures. If X is a variety defined over a finite field Fq, one can count its points over the various finite extensions of Fq; denote Nm = |X(Fmq )| (for instance, if X ⊂ AnFq is affine, given by equations f1, . . . , fk, then Nm = |{x ∈ Fm...

Abstract We identify a geometric relation between the Laplace–Beltrami spectra and eigenfunctions on compact Riemannian symmetric spaces Borel–Weil theory using ideas from symplectic geometry quantization. This is done by associating with each space, via Marsden–Weinstein reduction, generalized flag manifold that covers space parametrizing all of its maximal totally geodesic tori. In process, w...

A central problem in Diophantine geometry is to uniformly bound the number of $K$-rational points on a smooth curve $X/K$ terms $K$ and its genus $g$. recent paper by Stoll proved uniform bounds for hyperelliptic $X$ provided that rank Jacobian at most $g - 3$. Katz, Rabinoff Zureick-Brown generalized his result arbitrary curves satisfying same condition. In this paper, we prove conditional r...