Classical Congruences for Parameters in Binary Quadratic Forms
نویسنده
چکیده
Let Q(J!k) be an imaginary quadratic "eld with discriminant !k and class number h, with kO3, 4, or 8. Let p be a prime such that (~k p )"1. There are integers C, D, unique up to sign, such that 4ph"C2#kD2, p PC. Stickelberger gave a congruence for C modulo p which extends congruences of Gauss, Jacobi, and Eisenstein. Stickelberger also gave a simultaneous congruence for C modulo k, but only for prime k. We prove an extension of his result that holds for all k, giving along the way an exposition of his work. ( 2000 Academic Press
منابع مشابه
Some New Series for 1/π and Related Congruences
In this paper we prove some new series for 1/π as well as related congruences. We also raise several new kinds of series for 1/π and present some related conjectural congruences involving representations of primes by binary quadratic forms.
متن کاملPositive Definite Quadratic Forms, Elliptic Curves and Cubic Congruences
Let F (x, y) = ax + bxy + cy be a positive definite binary quadratic form with discriminant Δ whose base points lie on the line x = −1/m for an integer m ≥ 2, let p be a prime number and let Fp be a finite field. Let EF : y = ax + bx + cx be an elliptic curve over Fp and let CF : ax + bx + cx ≡ 0(mod p) be the cubic congruence corresponding to F . In this work we consider some properties of pos...
متن کاملOn Sums Related to Central Binomial and Trinomial Coefficients
A generalized central trinomial coefficient Tn(b, c) is the coefficient of x in the expansion of (x+bx+c) with b, c ∈ Z. In this paper we investigate congruences and series for sums of terms related to central binomial coefficients and generalized central trinomial coefficients. The paper contains many conjectures on congruences related to representations of primes by certain binary quadratic f...
متن کاملA ug 2 01 1 arXiv : 0911 . 5665 OPEN CONJECTURES ON CONGRUENCES
Abstract. We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while conjectures in Part B have been recently confirmed. We hope that this material will interest number theorists and stimulate further research. Number theoris...
متن کاملCongruences between Selmer groups ∗
The study of congruences between arithmetically interesting numbers has a long history and plays important roles in several areas of number theory. Examples of such congruences include the Kummer congruences between Bernoulli numbers and congruences between coefficients of modular forms. Many of these congruences could be interpreted as congruences between special values of L-functions of arith...
متن کامل