Let E be an elliptic curve defined over Q and P ∈ E(Q) a rational point of infinite order. Suppose that E has complex multiplication by an order in the quadratic imaginary field k. Denote by ME,P the set of rational primes ` such that ` splits in k, E has good reduction at `, and P is a primitive point modulo `. Under the generalized Riemann hypothesis, we can determine the positivity of the de...

systems, each of which violates the like-numbered postulate and satisfies all the other postulates of the set. These examples are all arithmetic systems, the elements being the numbers 0 and 1. The symbol/(a , b) (mod 2) in an arithmetic system denotes the least positive residue modulo 2 obtained from /(a , b) by rejecting multiples of 2. The operations + and X are to be interpreted as the oper...

In this paper, we completely characterize the quadratic polynomials modulo 3 with the largest (hence “optimal”) correlation with parity. This result is obtained by analysis of the exponential sum