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A Version of the Grothendieck Conjecture for padic Local
 Fields, The International Journal of Math
, 1997
"... The purpose of this paper is to prove an absolute version of the Grothendieck Conjecture for local padic 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 Qpalgebra isomorphisms of K with K ′. Let OutFilt(ΓK, ΓK ′) d ..."
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Cited by 20 (13 self)
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The purpose of this paper is to prove an absolute version of the Grothendieck Conjecture for local padic 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 Qpalgebra isomorphisms of K with K ′. Let OutFilt(ΓK, ΓK ′) denote the set of outer
On the surjectivity of the Galois representations associated to nonCM elliptic curves
 Canadian Math. Bulletin
"... 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 kdivision points of E. From a celebrated 1972 result of Serre we know that φl is surjective for any sufficiently large pr ..."
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Cited by 15 (5 self)
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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 kdivision 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
Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
, 2001
"... 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 ..."
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Cited by 14 (3 self)
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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
Decidability of the isomorphism problem for stationary AFalgebras and the associated ordered simple dimension groups, Ergodic Theory Dynam
 Systems
"... Abstract. The notion of isomorphism of stable AFC ∗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 matri ..."
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Cited by 13 (3 self)
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Abstract. The notion of isomorphism of stable AFC ∗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.
A review of nonarchimedean elliptic functions
 in Elliptic Curves, Modular Forms and Fermat’s Last Theorem, 2nd ed., International
, 1997
"... This expository article consists of two parts. The first is an old manuscript dating from 1959, entitled “Rational points on elliptic curves over complete fields ” containing my first proof of the isomorphism k ∗ /t Z ≃ Et(k). The second part is a discussion of some further aspects of the theory. It ..."
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Cited by 12 (0 self)
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This expository article consists of two parts. The first is an old manuscript dating from 1959, entitled “Rational points on elliptic curves over complete fields ” containing my first proof of the isomorphism k ∗ /t Z ≃ Et(k). The second part is a discussion of some further aspects of the theory. It begins with a sketch of some topics I had hoped to add to the old manuscript before publishing it, namely, the description of the rational functions on Et as “rigid analytic ” meromorphic functions on k ∗ with multiplicative period t, the construction of these via thetafunctions, and the classification of isogenies between the Et’s. Then, after a discussion of some consequences of the isogency classification, there is a description of the kernel Et[m] of multiplication by m on Et as finite flat group scheme, and an indication of its relevance to the main theme of this conference. Finally, curves Et over more general base rings than local fields are discussed, in particular, the “universal curve ” Eq over Z[[q]][q −1], and the connection with moduli. First, here is the old manuscript. Rational Points on Elliptic Curves Over Complete Fields. Let k be a field complete with respect to a nontrivial real valued valuation of the type
Torsion points on modular curves
 Invent. Math
, 1999
"... Abstract. Let N ≥ 23 be a prime number. In this paper, we prove a conjecture of Coleman, Kaskel, and Ribet about the Qvalued 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 em ..."
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Cited by 8 (2 self)
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Abstract. Let N ≥ 23 be a prime number. In this paper, we prove a conjecture of Coleman, Kaskel, and Ribet about the Qvalued 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.
Iwasawa theory for elliptic curves
 Lecture Notes in Math. 1716
, 1999
"... The topics that we will discuss have their origin in Mazur’s synthesis of the theory of elliptic curves and Iwasawa’s theory of Zpextensions 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 ..."
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Cited by 4 (2 self)
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The topics that we will discuss have their origin in Mazur’s synthesis of the theory of elliptic curves and Iwasawa’s theory of Zpextensions 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
A MULTIFREY APPROACH TO SOME MULTIPARAMETER FAMILIES OF DIOPHANTINE EQUATIONS
, 2006
"... We solve several multiparameter 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 nonzero integers x, y and positive integers u, r, s and n ≥ 3. Our approach uses several Frey curves simultaneously, Galois representations an ..."
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Cited by 4 (3 self)
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We solve several multiparameter 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 nonzero integers x, y and positive integers u, r, s and n ≥ 3. Our approach uses several Frey curves simultaneously, Galois representations and levellowering, new lower bounds for linear forms in 3 logarithms due to Mignotte and a famous theorem of Bennett on binomial Thue equations.
TOPICS IN ABSOLUTE ANABELIAN GEOMETRY I: GENERALITIES
, 2008
"... � � � This paper forms the first part of a threepart 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 ..."
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Cited by 3 (1 self)
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� � � This paper forms the first part of a threepart 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, mixedcharacteristic local fields, or finite fields, we take a more detailed look at certain padic Hodgetheoretic aspects of the absolute Galois groups of mixedcharacteristic local fields. This allows us, for instance, to derive, from a certain result communicated orally to the author by A. Tamagawa, a “semiabsolute ” Homversion of the anabelian conjecture for hyperbolic curves over mixedcharacteristic local fields. Finally, we generalize to the case of varieties of arbitrary dimension over arbitrary subpadic fields certain techniques developed by the author in previous papers over mixedcharacteristic local fields for grouptheoretically constructing the étale fundamental group of one