Caliber number of real quadratic fields

We obtain lower bound of caliber number of real quadratic field using splitting primes in K. We find all real quadratic fields of caliber number 1 and find all real quadratic fields of caliber number 2 if d is not 5 modulo 8. In both cases, we don't rely on the assumption on ζK(1/2).     

On fundamental units of real quadratic fields of class number 1

Archiv der Mathematik - In this paper, we give a nontrivial lower bound for the fundamental unit of norm $$-1$$ of a real quadratic field of class number 1.     

On the number of real quadratic fields with class number divisible by 3

We find a lower bound for the number of real quadratic fields whose class groups have an element of order . More precisely, we establish that the number of real quadratic fields whose absolute discriminant is and whose class group has an element of order is improving the existing best known bound of R. Murty.

     

Computing the rank of elliptic curves over real quadratic number fields of class number 1

In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and Swinnerton-Dyer for curves over . Several examples are included.

     

The conjugacy problem of the modular group and the class number of real quadratic number fields

We shall discuss the conjugacy problem of the modular group, and show how its solution, in conjunction with a theorem of Olga Taussky can be used to compute the class number of certain real quadratic number fields.     

On real quadratic function fields of Chowla type with ideal class number one

Let be the finite field with elements, (2), , where is a square-free polynomial in with and . In this paper several equivalent conditions for the ideal class number to be one are presented and all such quadratic function fields with are determined.

     

Imaginary quadratic fields of class number 2

A conjecture concerning linear forms in the logarithms of algebraic numbers is made. It is shown that this conjecture allows an effective determination of all imaginary quadratic fields of class number 2.     

Computations of class numbers of real quadratic fields

In this paper an unconditional probabilistic algorithm to compute the class number of a real quadratic field is presented, which computes the class number in expected time . The algorithm is a random version of Shanks' algorithm. One of the main steps in algorithms to compute the class number is the approximation of . Previous algorithms with the above running time , obtain an approximation for by assuming an appropriate extension of the Riemann Hypothesis. Our algorithm finds an appoximation for without assuming the Riemann Hypothesis, by using a new technique that we call the `Random Summation Technique'. As a result, we are able to compute the regulator deterministically in expected time . However, our estimate of on the running time of our algorithm to compute the class number is not effective.

     

Sums of Kloosterman sums for real quadratic number fields

We estimate sums of Kloosterman sums for a real quadratic number field F of the type

     

A note on the relative class number in function fields

Let be a finite field, , and . Let be the field extension of obtained by adjoining the -torsion on the Carlitz module. The class number of can be written as a product . The number is called the relative class number. In this paper a formula for is derived which is the analogue of the Maillet determinant formula for the relative class number of the cyclotomic field of -th roots of unity. Some consequences of this formula are also derived.

     

Computation of class numbers of quadratic number fields

We explain how one can dispense with the numerical computation of approximations to the transcendental integral functions involved when computing class numbers of quadratic number fields. We therefore end up with a simpler and faster method for computing class numbers of quadratic number fields. We also explain how to end up with a simpler and faster method for computing relative class numbers of imaginary abelian number fields.

     

Note on the class numbers of certain real quadratic fields

We prove that the class number of the real quadratic field is divisible byn forany integern ≥ 2 andany odd integera ≥ 3.