We introduce the weak Brauer equivalence relation on the set of rational points on homogeneous spaces under connected reductive groups and consider its connections with Brauer and R-equivalence relations on such spaces over local fields and give some formulas for computing the set of equivalence classes in some important cases. Especially, for R-homogeneous spaces X under connected affine R-groups with connected R-stabilizers, any two elements x; y E X(R) are directly R-equivalent. As an application, we give a partial affirmative answer to a question raised by Colliot-Thele`ne and Kunyavskii.
Ryavec and Bombieri considered a certain linear functional defined on the Nyman-Beurling space which is closely related to the Riemann hypothesis. They showed that the Riemann hypothesis is true if and only if the linear functional is unbounded on the Nyman-Beurling space. In this paper, we study a linear functional defined on a proper subspace of the Nyman-Beurling space and give a exact operator norm. We show that each linear functional is bounded on the proper subspaces without the assumption of the Riemann hypothesis. Moreover we give a quantitative result to Ryavec and Bombieri’s theorem.
Let E be an elliptic curve defined over Q and (sic)E-p denote the reduction of E modulo a prime p. Due to work of Serre, Katz, and Mazur, it is known that the integer gcd(p subset of S) |(sic)E-p (F-p)| is an element of {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 12; 16} where S is a set of primes of density one containing primes of good reduction of E. In this note, we consider the integer gcd(p subset of S) |(sic)E-p (F-p)| when S is taken to be the set of all rational primes. We show that gcdpES Ep(Fp) <4. We also show that there are elliptic curves for which gcd(p subset of S) |(sic)E-p (F-p)| is either 2, 3 or 4.
We determine all pairs (p, n), where p is a prime and n a positive integer, such that there exists a reduced fraction u/v > 1 with u+ v= n and u/v has a nonterminating Schneider's p-adic continued fraction expansion. We also prove a bound on the length of the preperiod in the p-adic continued fraction of u/v when max{|u|,|v|} < p.
An imaginary biquadratic field K is a number field of the form Q(root-m(1,)root-m(2), where m(1) and m(2) are distinct positive squarefree integers. In this paper, we present an algorithm for determining all the imaginary biquadratic fields with small odd class numbers. We use this algorithm to determine all imaginary biquadratic fields K with odd class number h(K) satisfying 1 <= h(K) <50. Moreover, we note that many of these fields are not found in the L-functions and modular forms database (LMFDB) as of writing.
