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 1982, Dr. Alan Stone raised a central dilemma in ethics for forensic psychiatry that has prompted significant and important discussion of the concerns about twisting justice, prostituting the profession, and operating without adequate ethics guidelines in the course of our work. In presidential addresses to the membership of the American Academy of Psychiatry and the Law (AAPL), Dr. Paul App...

The purpose of this chapter is to describe recent progress in the study of Teichmüller geometry. We focus entirely on the Teichmüller metric. A survey of the very important Weil-Petersson metric can be found in [W]. The study of the Teichmüller metric has different aspects. One major theme in the subject is to what extent Teichmüller space with the Teichmüller metric resembles a metric of negat...

We describe connections between the Fourier coefficients of derivatives of Eisenstein series and invariants from the arithmetic geometry of the Shimura varieties M associated to rational quadratic forms (V,Q) of signature (n, 2). In the case n = 1, we define generating series φ̂1(τ ) for 1-cycles (resp. φ̂2(τ ) for 0-cycles) on the arithmetic surface M associated to a Shimura curve over Q. These ...

This article is an introduction to the early life and work of Robert P. Langlands, creator and founder of the Langlands program. The story is, to a large extent, told by Langlands himself, in his own words. Our focus is on two of Langlands’ major discoveries: automorphic L-functions and the principle of functoriality. It was Langlands’ desire to communicate his excitement about his newly discov...

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...

the mordell-weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎in our previous paper, h‎. ‎daghigh‎, ‎and s‎. ‎didari‎, on the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎bull‎. ‎iranian math‎. ‎soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using selmer groups‎, ‎we have shown that for a prime $p$...