Abstract. Let E/Q be an elliptic curve defined over Q of conductor N and let Gal(Q/Q) be the absolute Galois group of an algebraic closure Q of Q. For an automorphism σ ∈ Gal(Q/Q), we let Q σ be the fixed subfield of Q under σ. We prove that for every σ ∈ Gal(Q/Q), the Mordell-Weil group of E over the maximal Galois extension of Q contained in Q σ has infinite rank, so the rank of E(Q σ ) is infinite. Our approach uses the modularity of E/Q and a collection of algebraic points on E -the so-called Heegner points -arising from the theory of complex multiplication. In particular, we show that for some integer r and for a prime p prime to rN , the rank of E over all the ring class fields of a conductor of the form rp n is unbounded, as n goes to infinity. This paper is motivated by the following conjecture of M. Larsen [8]:Conjecture. Let K be a number field and E/K an elliptic curve overIn [3] and [4], we have proved this conjecture in certain cases: For a number field K and an elliptic curve E/K over K,• if E/K has a K-rational point P such that 2P = O and 3P = O, or • if 2-torsion points of E/K are K-rational, then for every automorphism σ ∈ Gal(K/K), the rank of the Mordell-Weil groupRecall that a field L is said to be PAC (pseudo algebraically closed) if every absolutely irreducible nonempty variety defined over L has an L-rational point. We note that if K is a countable separably Hilbertian field, then M. Jarden has proven in [6, Theorem 2.7] that for almost all σ ∈ Gal(K/K) in the sense of Haar measure on Gal(K/K), the maximal Galois extension contained in K σ is a PAC field with the absolute Galois group isomorphic to the free profinite group on countably many generators, so the maximal Galois extension of K in K σ is smaller than K σ for almost all σ. In [3], we have shown that under the first assumption above, the Mordell-Weil group of E over the maximal Galois extension of K in K σ for every σ ∈ Gal(K/K) has infinite rank, and under the second assumption, we have shown a stronger result in [4] that the rank of E over the maximal abelian extension of K in K σ (hence, over the maximal Galois extension of K in K σ ) is infinite.

Abstract. Let K be a field of characteristic = 2 such that every finite separable extension of K is cyclic. Let A be an abelian variety over K. If K is infinite, then A(K) is Zariski-dense in A. If K is not locally finite, the rank of A over K is infinite.

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