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17
The rational function analogue of a question of Schur and exceptionality of permutation representations
, 2008
"... ..."
A linear lower bound on the gonality of modular curves
 Internat. Math. Res. Notices
, 1996
"... 0.1. Statement of result. In this note we prove the following: Theorem 0.1. Let Γ ⊂ PSL2(Z) be a congruence subgroup, and XΓ the corresponding modular curve. Let DΓ = [PSL2(Z) : Γ] and let dC(XΓ) be the Cgonality of XΓ. Then 7 ..."
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0.1. Statement of result. In this note we prove the following: Theorem 0.1. Let Γ ⊂ PSL2(Z) be a congruence subgroup, and XΓ the corresponding modular curve. Let DΓ = [PSL2(Z) : Γ] and let dC(XΓ) be the Cgonality of XΓ. Then 7
Computational Aspects of Curves of Genus at Least 2
 Algorithmic number theory. 5th international symposium. ANTSII
, 1996
"... . This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have per ..."
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Cited by 14 (3 self)
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. This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list of possible future projects are given at the end. 1. Introduction An enormous number of people have performed an enormous number of computations on elliptic curves, as one can see from even a perfunctory glance at [29]. A few years ago, the same could not be said for curves of higher genus, even though the theory of such curves had been developed in detail. Now, however, polynomialtime algorithms and sometimes actual programs are available for solving a wide variety of problems associated with such curves. The genus 2 case especially is becoming accessible: in light of recent work, it seems reasonable to expect that within a few years, packages will be available for doing genus 2 computations analogous to the elliptic curve computations that are currently possible in PARI, MAGMA, SIMATH, apec...
Nonrigid constructions in Galois theory
 PAC. JOUR
, 1994
"... The context for this paper is the Inverse Galois Problem. First we give an if and only if condition that a finite group is the group of a Galois regular extension of R(X) with only real branch points. It is that the group is generated by elements of order 2 (Theorem 1.1 (a)). We use previous work o ..."
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Cited by 13 (8 self)
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The context for this paper is the Inverse Galois Problem. First we give an if and only if condition that a finite group is the group of a Galois regular extension of R(X) with only real branch points. It is that the group is generated by elements of order 2 (Theorem 1.1 (a)). We use previous work on the action of the complex conjugation on covers of P 1 [FrD]. We also use Fried and Völklein [FrV] and Pop [P] to show each finite group is the Galois group of a Galois regular extension of Q tr (X). Here Q tr is the field of all totally real algebraic numbers (Theorem 5.7). §1, §2 and §3 discuss consequences, generalizations and related questions. The second part of the paper, §4 and §5, concerns descent of fields of definition from R to Q. Use of Hurwitz families reduces the problem to finding Qrational point on a special algebraic curve. Our first application considers realizing the symmetric group Sm as the group of a Galois extension of Q(X), regular over Q, satisfying two further conditions. These are that the extension has four branch points, and it also has some totally real residue class field specializations. Such extensions exist for m = 4, 5, 6, 7, 10 (Theorem 4.11). Suppose that m is a prime larger than 7. Theorem 5.1 shows that the dihedral group
Torsion des courbes elliptiques sur les corps cubiques
 Ann. Inst. Fourier
"... On donne la liste (à un élément près) des nombres premiers qui sont l’ordre d’un point de torsion d’une courbe elliptique sur un corps de nombres de degré trois. Motsclés: courbes elliptiques, points rationnels, symboles modulaires (1991 Mathematics Subject Classification: 11G05, 14G05). Torsio ..."
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Cited by 7 (1 self)
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On donne la liste (à un élément près) des nombres premiers qui sont l’ordre d’un point de torsion d’une courbe elliptique sur un corps de nombres de degré trois. Motsclés: courbes elliptiques, points rationnels, symboles modulaires (1991 Mathematics Subject Classification: 11G05, 14G05). Torsion of elliptic curves over cubic fields We give the list (up to one element) of prime numbers wich are the order of some torsion point of an elliptic curve over a number field of degree 3. Key words: elliptic curves, rational points, modular symbols (1991 Mathematics Subject Classification: 11G05, 14G05). 1 Présentation des résultats
Uniformity of Stably Integral Points on Elliptic Curves
, 1995
"... this paper as a simple application of the methods of [CHM]. ..."
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Cited by 6 (3 self)
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this paper as a simple application of the methods of [CHM].
Torsion in rank 1 Drinfeld modules and the uniform boundedness conjecture, Mathematische Annalen 308
, 1997
"... Abstract. It is conjectured that for fixed A, r ≥ 1, and d ≥ 1, there is a uniform bound on the size of the torsion submodule of a Drinfeld Amodule of rank r over a degree d extension L of the fraction field K of A. We verify the conjecture for r = 1, and more generally for Drinfeld modules having ..."
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Abstract. It is conjectured that for fixed A, r ≥ 1, and d ≥ 1, there is a uniform bound on the size of the torsion submodule of a Drinfeld Amodule of rank r over a degree d extension L of the fraction field K of A. We verify the conjecture for r = 1, and more generally for Drinfeld modules having potential good reduction at some prime above a specified prime of K. Moreover, we show that within an Lisomorphism class, there are only finitely many Drinfeld modules up to isomorphism over L which have nonzero torsion. For the case A = Fq[T], r = 1, and L = Fq(T), we give an explicit description of the possible torsion submodules. We present three methods for proving these cases of the conjecture, and explain why they fail to prove the conjecture in general. Finally, an application of the Mordell conjecture for characteristic p function fields proves the uniform boundedness for the pprimary part of the torsion for rank 2 Drinfeld Fq[T]modules over a fixed function field. 1. Conjectures and Theorems In a 1977 paper, Mazur [13] proved that if E is an elliptic curve over Q, its torsion subgroup is one of the following fifteen groups:
Formal Finiteness And The Torsion Conjecture On Elliptic Curves, a footnote to a paper of Kamienny and Mazur
 Columbia University Number Theory Seminar (New York
, 1995
"... 23> S(d) is finite for d 12 (theorem 3). A simple computer program should verify the conjecture for the next few degrees (I have checked this for degrees 13 and 14 using Mathematica). Acknowledgements. It is a pleasure to thank Sheldon Kamienny and Barry Mazur, whose work inspired this note and wh ..."
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23> S(d) is finite for d 12 (theorem 3). A simple computer program should verify the conjecture for the next few degrees (I have checked this for degrees 13 and 14 using Mathematica). Acknowledgements. It is a pleasure to thank Sheldon Kamienny and Barry Mazur, whose work inspired this note and who had the patience to hear my ideas at their infancy. Thanks to S. David, A. Silverberg and J. F. Voloch, and to the participants of Coleman's seminar at Berkeley, who had commented on the paper. Thanks to Arthur Ogus for pointing to a mistake in an earlier version. I am also thankful to Noam Elkies, who initiated me into the mysteries of modular curves. This work was partially supported by NSF grant DMS 92 07285. 0.1. Notation. We follow [Maz77] for the basics. Let N be a prime and let X 1 (N) be the modular curve parametrizing pairs (E; P
FAMILIES OF ELLIPTIC CURVES OVER CUBIC NUMBER FIELDS WITH PRESCRIBED TORSION SUBGROUPS
"... Abstract. In this paper we construct infinite families of elliptic curves with given torsion group structures over cubic number fields. This result provides explicit examples of the theoretical result recently developed by the first two authors and A. Schweizer; they determined all the group structu ..."
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Abstract. In this paper we construct infinite families of elliptic curves with given torsion group structures over cubic number fields. This result provides explicit examples of the theoretical result recently developed by the first two authors and A. Schweizer; they determined all the group structures which occur infinitely often as the torsion of elliptic curves over cubic number fields. In fact, this paper presents an efficient way of constructing such families of elliptic curves with prescribed torsion group structures over cubic number fields. 1.