Abstract. The notion of isomorphism of stable AF-C ∗-algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e., is associated with a single square primitive nonsingular incidence matrix. C ∗-isomorphism induces an equivalence relation on these matrices, called C ∗-equivalence. We show that the associated isomorphism equivalence problem is decidable, i.e., there is an algorithm that can be used to check in a finite number of steps whether two given primitive nonsingular matrices are C ∗-equivalent or not.

The purpose of this paper is to prove an absolute version of the Grothendieck Conjecture for local p-adic fields (given as Theorem 4.2 in the text): Theorem: Let K and K ′ be finite extensions of Qp. LetIsomQp (K, K ′ ) denote the set of Qp-algebra isomorphisms of K with K ′. Let OutFilt(ΓK, ΓK ′) denote the set of outer

1 Let E be an elliptic curve defined over Q, of conductor N and without complex multiplication. For any positive integer k, let φk be the Galois representation associated to the k-division points of E. From a celebrated 1972 result of Serre we know that φl is surjective for any sufficiently large prime l. In this paper we find conditional and unconditional upper bounds in terms of N for the primes l for which φl is not surjective. 1

1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.-P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1

Abstract not found

Abstract. Let N ≥ 23 be a prime number. In this paper, we prove a conjecture of Coleman, Kaskel, and Ribet about the Q-valued points of the modular curve X0(N) which map to torsion points on J0(N) via the cuspidal embedding. We give some generalizations to other modular curves, and to noncuspidal embeddings of X0(N) into J0(N). 1.

The topics that we will discuss have their origin in Mazur’s synthesis of the theory of elliptic curves and Iwasawa’s theory of Zp-extensions in the early 1970s. We first recall some results from Iwasawa’s theory. Suppose that F is a finite extension of Q and that F ∞ is a Galois extension of F such that

Abstract. We solve several multi-parameter families of binomial Thue equations of arbitrary degree; for example, we solve the equation 5 u x n − 2 r 3 s y n = ±1, in non-zero integers x, y and positive integers u, r, s and n ≥ 3. Our approach uses several Frey curves simultaneously, Galois representations and level-lowering, new lower bounds for linear forms in 3 logarithms due to Mignotte and a famous theorem of Bennett on binomial Thue equations. 1.

� � � This paper forms the first part of a three-part series in which we treat various topics in absolute anabelian geometry from the point of view of developing abstract algorithms, or“software”, that may be applied to abstract profinite groups that “just happen ” to arise as [quotients of] étale fundamental groups from algebraic geometry. In the present paper, after studying various abstract combinatorial properties of profinite groups that typically arise as absolute Galois groups or geometric fundamental groups in anabelian geometry over number fields, mixed-characteristic local fields, or finite fields, we take a more detailed look at certain p-adic Hodgetheoretic aspects of the absolute Galois groups of mixed-characteristic local fields. This allows us, for instance, to derive, from a certain result communicated orally to the author by A. Tamagawa, a “semi-absolute ” Hom-version of the anabelian conjecture for hyperbolic curves over mixed-characteristic local fields. Finally, we generalize to the case of varieties of arbitrary dimension over arbitrary sub-p-adic fields certain techniques developed by the author in previous papers over mixed-characteristic local fields for group-theoretically constructing the étale fundamental group of one

A modular quintic Calabi-Yau threefold of level 55