Hyunsuk Moon

THE NON-EXISTENCE OF CERTAIN MOD p GALOIS REPRESENTATIONS

The non-existence is proved of continuous irreducible representations p : Gal(Q/Q) \longrightarrow GL(F) with Artin conductor N outside p for a few small values of p and N.N.

On the Mordell-Weil Groups of Jacobians of Hyperelliptic Curves over Certain Elementary Abelian 2-extensions

Abstract. Let J be the Jacobian variety of a hyperelliptic curve over Q. Let M be thefield generated by all square roots of rational integers over a finite number field K. Thenwe prove that the Mordell-Weil group J(M) is the direct sum of a finite torsion groupand a free Z-module of infinite rank. In particular, J(M) is not a divisible group. On theother hand, if Mf is an extension of M which contains all the torsion points of J over Q,then J(Mf sol )/J(Mf ) tors is a divisible group of infinite rank, where Mf sol is the maximalsolvable extension of Mf. 1. IntroductionLet K be a number field. Let A be a nonzero abelian variety defined over K.For an extension M over K, we denote the group of M-rational points by A(M)and its torsion subgroup by A(M) tors . We call A(M) is the Mordell-Weil group ofA over M. In [1], Frey and Jarden have asked whether the Mordell-Weil group ofevery nonzero abelian variety A defined over K has infinite Mordell-Weil rank overthe maximal abelian extension K ab of K. They proved that for elliptic curves Edefined over Q, the Mordell-Weil group E(Q

On the finiteness and non‐existence of certain mod 2 Galois representations of quadratic fields

Some finiteness and non‐existence results are proved of 2‐dimensional mod 2 Galois representations of quadratic fields unramified outside 2.

ON MOD 3 GALOIS REPRESENTATIONS WITH CONDUCTOR 4

Let GQ be the absolute Galois group Gal(Q=Q) of Q. Let Fp be an algebraic closure of the flnite fleldFp of p elements. In this paper, we prove the non-existence of certain mod 3 Galois representation: Theorem 1. There exist no irreducible representations ‰ : GQ ! GL2(F3) with N(‰) dividing 4. Here, N(‰) = Q p-3 p n p(‰) is the Artin conductor of ‰ outside 3 ([6], x1.2; the deflnition of the exponent np(‰) will be recalled below). This proves a special case of Serre’s conjecture ([6]). Indeed, the conjecture predicts that such a representation, up to twist by a power of the mod 3 cyclotomic character, come from a cuspidal eigenform of level 4 and weight • 4, but there are no such forms. Such a result may serve as the flrst step of an inductive proof of Serre’s conjecture for N(‰) = 4 if Khare’s proof in the case of N(‰) = 1 ([3]) can be extended. Serre’s conjecture is known to be true if the image Im(‰) of ‰ is solvable ([4], Thm. 4). So, it remains for us to prove the Theorem 1 in the following two cases: (i) Im(‰) is non-solvable, (ii) ‰ is even and Im(‰) is solvable.

ON FOUR-DIMENSIONAL MOD 2 GALOIS REPRESENTATIONS AND A CONJECTURE OF ASH ET AL.

The non-existence is proved of four-dimensional mod 2 semisimple representations of Gal(Q=Q) which are unramified outside 2.

On the Non-Existence of Certain Galois Extensions

In this article, we give a survey of my results on the non-existence and finiteness of certain Galois extensions of the rational number field ℚ with prescribed ramification. The detail has been (will be) published in [8], [9], [10], [11], [12].