Abstract. Let K be a number field, K an algebraic closure of K and E/K an elliptic curve defined over K. In this paper, we prove that if E/K has a K-rational point P such that 2P = O and 3P = O, then for each σ ∈ Gal(K/K), the Mordell-Weil group E(K σ ) of E over the fixed subfield of K under σ has infinite rank.

We study Chebyshev's bias in a finite, possibly nonabelian, Galois extension of global function fields. We show that, when the extension is geometric and satisfies a certain property, called, Linear Independence (LI), the less square elements a conjugacy class of the Galois group has, the more primes there are whose Frobenius conjugacy classes are equal to the conjugacy class. Our results are in line with the previous work of Rubinstein and Sarnak in the number field case and that of the first-named author in the case of polynomial rings over finite fields. We also prove, under LI, the necessary and sufficient conditions for a certain limiting distribution to be symmetric, following the method of Rubinstein and Sarnak. Examples are provided where LI is proved to hold true and is violated. Also, we study the case when the Galois extension is a scalar field extension and describe the complete result of the prime number race in that case.

We give conditions when the fixed points by the partial Atkin-Lehner involutions on X0(N ) are Weierstrass points as an extension of the result by Lehner and Newman [16]. Furthermore, we complete their result by determining whether the fixed points by the full Atkin-Lehner involutions on X0(N ) are Weierstrass points or not.

If the system of two diophantine equations X 2 + mY 2 = Z 2 and X 2 + nY 2 = W 2 has infinitely many integer solutions (X, Y, Z, W ) with gcd(X, Y ) = 1, equivalently, the elliptic curve E m,n : y 2 = x(x + m)(x + n) has positive rank over Q, then (m, n) is called a strongly concordant pair. We prove that for a given positive integer M and an integer k, the number of strongly concordant pairs (m, n) with m, n ∈ [1, N] and m, n ≡ k is at least O(N ), and we give a parametrization of them.

We give local conditions at ∞ ensuring that the intersection of n quadrics in P N , N n, satisfies weak approximation.Let K be a number field, V a finite-dimensional vector space over K, and W ⊂ Sym 2 V * a K-vector space of quadratic forms on V . We can regard W as a linear system of quadrics in PV , and we denote the intersection of all quadrics in the system X W . If S is a finite set of places of K, we can ask whether X W satisfies weak approximation with respect to S, i.e., whether the diagonal maphas dense image. Assuming that X W (K s ) = ∅ for all s ∈ S, this is a refinement of the basic question of whether X W (K) is non-empty.