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202
On the modularity of elliptic curves over Q
 J. of the AMS
"... In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2). Theorem A. If E /Q is an elliptic curve, then E is modular. Theorem B. If ρ: Gal(Q/Q) → GL2(F5) is an irreducible continuous representation with cyclotomic ..."
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Cited by 28 (4 self)
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In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2). Theorem A. If E /Q is an elliptic curve, then E is modular. Theorem B. If ρ: Gal(Q/Q) → GL2(F5) is an irreducible continuous representation with cyclotomic
On the equations z m = F (x, y) and Ax p + By q = Cz r
 Bull. London Math. Soc
, 1995
"... Abstract: We investigate integer solutions of the superelliptic equation (1) zm = F (x, y), where F is a homogenous polynomial with integer coefficients, and ..."
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Cited by 24 (3 self)
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Abstract: We investigate integer solutions of the superelliptic equation (1) zm = F (x, y), where F is a homogenous polynomial with integer coefficients, and
Homomorphisms of Abelian varieties
 J. REINE ANGEW. MATH
, 1998
"... It is wellknown that an abelian variety is (absolutely) simple or is isogenous to a selfproduct of an (absolutely) simple abelian variety if and only if the center of its endomorphism algebra is a field. In this paper we prove that the center is a field if the field of definition of points of prim ..."
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Cited by 19 (4 self)
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It is wellknown that an abelian variety is (absolutely) simple or is isogenous to a selfproduct of an (absolutely) simple abelian variety if and only if the center of its endomorphism algebra is a field. In this paper we prove that the center is a field if the field of definition of points of prime order ℓ is “big enough”. The paper is organized as follows. In §1 we discuss Galois properties of points of order ℓ on an abelian variety X that imply that its endomorphism algebra End 0 (X) is a central simple algebra over the field of rational numbers. In §2 we prove that similar Galois properties for two abelian varieties X and Y combined with the linear disjointness of the corresponding fields of definitions of points of order ℓ imply that X and Y are nonisogenous (and even Hom(X, Y) = 0). In §3 we give applications to endomorphism algebras of hyperelliptic jacobians. In §4 we prove that if X admits multiplications by a number field E and the dimension of the centralizer of E in End 0 (X) is “as large as possible ” then X is an abelian variety of CMtype isogenous to a selfproduct of an absolutely simple abelian variety. Throughout the paper we will freely use the following observation [21, p. 174]: if an abelian variety X is isogenous to a selfproduct Z d of an abelian variety Z then a choice of an isogeny between X and Z d defines an isomorphism between End 0 (X) and the algebra Md(End 0 (Z)) of d × d matrices over End 0 (Z). Since the center of End 0 (Z) coincides with the center of Md(End 0 (Z)), we get an isomorphism
A Lower Bound for the Canonical Height on Elliptic Curves over Abelian Extensions
 Duke Math. J
, 2003
"... Let E=K be an elliptic curve de ned over a number eld, let ^ h be the canonical height on E, and let K =K be the maximal abelian extension of K. Extending work of Baker [4], we prove that there is a constant C(E=K) > 0 so that every nontorsion ^ h(P ) > C(E=K). ..."
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Cited by 17 (2 self)
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Let E=K be an elliptic curve de ned over a number eld, let ^ h be the canonical height on E, and let K =K be the maximal abelian extension of K. Extending work of Baker [4], we prove that there is a constant C(E=K) > 0 so that every nontorsion ^ h(P ) > C(E=K).
Pseudo Algebraically Closed Fields Over Rings
 ISRAEL JOURNAL OF MATHEMATICS
, 1994
"... We prove that for almost all oeoe oe 2 G(Q) the field oe) has the following property: For each absolutely irreducible affine variety V of dimension r and each dominating . We then say that oe) is PAC over Z. ..."
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Cited by 16 (8 self)
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We prove that for almost all oeoe oe 2 G(Q) the field oe) has the following property: For each absolutely irreducible affine variety V of dimension r and each dominating . We then say that oe) is PAC over Z.
Abelian Varieties over Q and modular forms
 Progress in Math. 224, Birkhäusser
, 2004
"... conjecture asserts that there is a nonconstant map of algebraic curves Xo(N) → C which is defined over Q. Here, Xo(N) is the standard modular curve associated with the problem of classifying elliptic curves E together with cyclic subgroups of E ..."
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Cited by 16 (0 self)
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conjecture asserts that there is a nonconstant map of algebraic curves Xo(N) → C which is defined over Q. Here, Xo(N) is the standard modular curve associated with the problem of classifying elliptic curves E together with cyclic subgroups of E
Topology of Diophantine sets: remarks on Mazur’s conjectures. In Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent
 of Contemp. Math
, 1999
"... Abstract. We show that Mazur’s conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers Z in the rational numbers Q, i.e., there is no diophantine set D in some cartesian power Q i such that there exist two binary relations S ..."
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Cited by 15 (1 self)
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Abstract. We show that Mazur’s conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers Z in the rational numbers Q, i.e., there is no diophantine set D in some cartesian power Q i such that there exist two binary relations S, P on D whose graphs are diophantine in Q 3i (via the inclusion D 3 ⊂ Q 3i), and such that for two specific elements d0, d1 ∈ D the structure (D, S, P, d0, d1) is a model for integer arithmetic (Z,+, ·,0, 1). Using a construction of Pheidas, we give a counterexample to the analogue of Mazur’s conjecture over a global function field, and prove that there is a diophantine model of the polynomial ring over a finite field in the ring of rational functions over a finite field. 1.