Hasse-Weil L-function Research Articles

Novel solutions for the heat equations arising in the elliptic curves over the field of rational numbers

In this article we consider the solutions of the heat equations with use of the elliptic curves over the field of rational numbers. We propose the entire functions associated with the Hasse-Weil L-function. We show the conjectures that new functions have only real zeros in the entire function plane. The obtained results are proposed as new tool to describe the complex behaviors of the heat problems as well as number theory.

Tamagawa number divisibility of centralL-values of twists of the Fermat elliptic curve

Given any integer N>1 prime to 3, we denote by C N the elliptic curve x 3 +y 3 =N. We first study the 3-adic valuation of the algebraic part of the value of the Hasse–Weil L-function L(C N ,s) of C N over ℚ at s=1, and we exhibit a relation between the 3-part of its Tate–Shafarevich group and the number of distinct prime divisors of N which are inert in the imaginary quadratic field K=ℚ(-3). In the case where L(C N ,1)≠0 and N is a product of split primes in K, we show that the order of the Tate–Shafarevich group as predicted by the conjecture of Birch and Swinnerton-Dyer is a perfect square.

Class numbers of real quadratic fields

Let d>0 be a fundamental discriminant of a real quadratic field. Let h(d) be the class number and εd the fundamental unit of the real quadratic field Q(d). In this paper, we prove that if there is an elliptic curve E over Q whose Hasse-Weil L-function LE/Q(s) has a zero of order g at s=1, then there is an effectively computable constant κ>0 satisfyingh(d)log⁡εd>1κ(log⁡d)g−3∏p|d,p≠d(1−⌊2p⌋p+1).

Varieties via their L-functions

We describe a procedure for determining the existence, or non-existence, of an algebraic variety of a given conductor via an analytic calculation involving L-functions. The procedure assumes that the Hasse–Weil L-function of the variety satisfies its conjectured functional equation, with no assumption of an associated automorphic object or Galois representation. We demonstrate the method by finding the Hasse–Weil L-functions of all hyperelliptic curves of conductor less than 500.

Analytic properties of the Hasse–Weil L-function

We prove that the Hasse–Weil L -function associated to an elliptic curve E over is of growth order one and belongs to the Selberg class with degree two. Moreover, we show that has an infinite product representation associated with the conductor .

Geometric Kac–Moody modularity

It is shown how the arithmetic structure of algebraic curves encoded in the Hasse–Weil L-function can be related to affine Kac–Moody algebras. This result is useful in relating the arithmetic geometry of Calabi–Yau varieties to the underlying exactly solvable theory. In the case of the genus three Fermat curve we identify the Hasse–Weil L-function with the Mellin transform of the twist of a number theoretic modular form derived from the string function of a non-twisted affine Lie algebra. The twist character is associated to the number field of quantum dimensions of the conformal field theory.

The Shimura–Taniyama conjecture and conformal field theory

The Shimura–Taniyama conjecture states that the Mellin transform of the Hasse–Weil L-function of any elliptic curve defined over the rational numbers is a modular form. Recent work of Wiles, Taylor–Wiles and Breuil–Conrad–Diamond–Taylor has provided a proof of this longstanding conjecture. Elliptic curves provide the simplest framework for a class of Calabi–Yau manifolds which have been conjectured to be exactly solvable. It is shown that the Hasse–Weil modular form determined by the arithmetic structure of the Fermat type elliptic curve is related in a natural way to a modular form arising from the character of a conformal field theory derived from an affine Kac–Moody algebra.

Arithmetic of Calabi–Yau varieties and rational conformal field theory

It is proposed that certain techniques from arithmetic algebraic geometry provide a framework which is useful to formulate a direct and intrinsic link between the geometry of Calabi–Yau manifolds and the underlying conformal field theory. Specifically, it is pointed out how the algebraic number field determined by the fusion rules of the conformal field theory can be derived from the number theoretic structure of the cohomological Hasse–Weil L-function determined by Artin’s congruent zeta function of the algebraic variety. In this context, a natural number theoretic characterization arises for the quantum dimensions in this geometrically determined algebraic number field.

Selmer groups of quadratic twists of elliptic curves

(1.1) E : y + a1xy + a3y = x 3 + a2x 2 + a4x+ a6 where a1, a2, a3, a4, a6 ∈ Z. Let N(E) denote the conductor of E, j(E) the j-invariant of E, and L(E, s) = ∑∞ n=1 a(n)n −s the Hasse-Weil L-function of E. If E is modular, then let FE(z) = ∑∞ n=1 aE(n)q n ∈ S2(N(E), χ1) be the associated weight 2 cusp form. Here χ1 denotes the trivial Dirichlet character. Throughout, D will denote a square-free integer, and χD shall denote the Kronecker character for the field Q( √ D). Let h(D) denote the order of Cl(D), the ideal class group of Q( √ D). If l is prime, then define h(D)l by

On the 3-part of the Birch and Swinnerton-Dyer conjecture

Let K=ℚ(√−3). Assume E is an elliptic curve defined over K, with complex multiplication by OK, the ring of integers of K. Ω∈[Copf ]X is a minimal period of E. For each prime v of K, let cv denote the Tamagawa number [E(Kv)[ratio ] E0(Kv)]. Let L(E/K, s) denote the Hasse–Weil L-function of E/K. [lfloor ][mid ][rfloor ](E/K)= ker{H1(K, E)→ [oplus ]vH1(Kv, E)} denotes the Tate-Shafarevic group of E/K.

Eta-quotients and elliptic curves

In this paper we list all the weight 2 newforms f(r) that are products and quotients of the Dedekind eta-function 71(r) = 1/24 17 (1 qn) n=1 where q := e2iT,. There are twelve such f (r), and we give a model for the strong Weil curve E whose Hasse-Weil L-function is the Mellin transform for each of them. Five of the f(T) have complex multiplication, and we give elementary formulae for their Fourier coefficients which are sums of Hecke Gr6ssencharacter values. These formulae follow easily from well known q-series infinite product identities. In light of the proof of Fermat's Last Theorem by A. Wiles and R. Taylor, there have been many expository articles describing the nature of the ShimuraTaniyama conjecture, the conjecture which asserts that every elliptic curve E over Q is modular. This implies that the Hasse-Weil L-function of an elliptic curve E with conductor N over Q,

On Waring's problem for cubes and biquadrates. II

In discussing the consequences of a conditional estimate for the sixth moment of cubic Weyl sums, Hooley [4] established asymptotic formulae for the number ν(n) of representations of n as the sum of a square and five cubes, and for ν(n), defined similarly with six cubes and two biquadrates. The condition here is the truth of the Riemann Hypothesis for a certain Hasse–Weil L-function. Recently Vaughan [8] has shown unconditionally , a lower bound of the size suggested by the conditional asymptotic formula. In the corresponding problem for ν(n) the author [1] was able to deduce ν(n) > 0, as a by-product of the result that almost all numbers can be expressed as the sum of three cubes and one biquadrate. As promised in the first paper of this series we return to the problem of bounding ν(n) from below.