Solving Diophantine equation is one of the main problem in number theory for a long time. It is very difficult but wonderful. For example, it took over 300 years to see that Xn + Y n = Zn has no nontrivial integers solution when n ≥ 3. We would like to consider an easier problem: solving the Diophantine equation modulo p, where p is a prime number. We expect that this problem is easy enough to ...

Hecke generalized this equivalence, showing that an integral form has an associated Dirichlet series which can be analytically continued to C and satisfies a functional equation. Conversely, Weil showed that, if a Dirichlet series satisfies certain functional equations, then it must be associated to some integral form. Our goal in this paper is to describe this work. In the first three sections...

The character associated to a quadratic extension field K of Q, χ : Z −→ C, χ(n) = (disc(K)/n) (Jacobi symbol), is in fact a Dirichlet character; specifically its conductor is |disc(K)|. This fact encodes basic quadratic reciprocity from elementary number theory, phrasing it in terms that presage class field theory. This writeup discusses Hilbert quadratic reciprocity in the same spirit. Let k ...

• Y 2 −X is reducible, • Y 2 − αX is irreducible, but not absolutely irreducible, • Y 2 −X + 1 is absolutely irreducible. To see the last item, note that any factorization of H(X,Y ) = Y −X+1 in F[X,Y ] is also a factorization of H(X,Y ) in K[Y ], where K is the field F(X), and thus the factorization must be of the form (Y −a(X))(Y + a(X)), where a(X) ∈ K satisfies a(X) = X − 1. But this cannot...

The goal of this paper is to define an analogue the Weil-pairing for Drinfeld modules using explicit formulas and deduce its main properties from these formulas. Our result generalizes formula given rank 2 by van der Heiden works as a more explicit, elementary proof Weil-pairing’s existence Heiden.

The Weil conjecture is a delightful theorem for algebraic varieties on finite fields and an important model for dynamical zeta functions. In this paper, we prove a functional equation of Lefschetz zeta functions for infinite cyclic coverings which is analogous to the Weil conjecture. Applying this functional equation to knot theory, we obtain a new view point on the reciprocity of the Alexander...

By the Mordell-Weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎There is no known algorithm for finding the rank of this group‎. ‎This paper computes the rank of the family $ E_p:y^2=x^3-3px $ of elliptic curves‎, ‎where p is a prime‎.

We study the double cosets of a Lie group by a compact Lie subgroup. We show that a Weil formula holds for double coset Lie hypergroups and show that certain representations of the Lie group lift to representations of the double coset Lie hypergroup. We characterize smooth (analytic) vectors of these lifted representations.

Artykuł jest próbą porównania, w jaki sposób poetyka paradoksu realizuje się pisarstwie filozoficzno-teologicznym Simone Weil oraz języku muzycznym fińskiej kompozytorki Kaiji Saariaho jej oratorium La Passion de Simone, poświęconym życiu i twórczości Weil. Przewodnikiem na drodze analizy komparatystycznej etymologia paradoksu, począwszy od jego znaczeń związanych z przeciwstawianiem doksie, ro...