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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...
A High Fibered Power Of A Family Of Varieties Of General Type Dominates A Variety Of General Type  with a few diagrams and one illustration
, 1997
"... INTRODUCTION In Which We Are Introduced to Our Main Theorem, and the Story Begins. We work over C . 0.1. Statement. We prove the following theorem: Theorem 0.1 (Fibered power theorem). Let X ! B be a smooth family of positive dimensional varieties of general type, with B irreducible. Then there ..."
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Cited by 10 (7 self)
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INTRODUCTION In Which We Are Introduced to Our Main Theorem, and the Story Begins. We work over C . 0.1. Statement. We prove the following theorem: Theorem 0.1 (Fibered power theorem). Let X ! B be a smooth family of positive dimensional varieties of general type, with B irreducible. Then there exists an integer n ? 0, a positive dimensional variety of general type W n , and a dominant rational map X n B 9 9 KW n . Specifically, let m n : X n B 9 9 KW n be the npointed birationalmoduli map. Then for sufficiently large n, W n is a variety of general type. The latter statement will be explained in section 3. This solves "Co
Lang's Conjectures, Fibered Powers, and Uniformity
 Journal of Math
, 1996
"... . We prove that the fibered power conjecture of Caporaso et al. (Conjecture H, [CHM], x6) together with Lang's conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHM]; a few applications on the arithmetic and geometry of curves are stated. In an ..."
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Cited by 10 (4 self)
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. We prove that the fibered power conjecture of Caporaso et al. (Conjecture H, [CHM], x6) together with Lang's conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHM]; a few applications on the arithmetic and geometry of curves are stated. In an opposite direction, we give counterexamples to some analogous results in positive characteristic. We show that curves that change genus can have arbitrarily many rational points; and that curves over Fp (t) can have arbitrarily many Frobenius orbits of nonconstant points. Contents 1. Introduction 21 1.1. A Few Conjectures of Lang 21 1.2. The Fibered Power Conjecture 22 1.3. Summary of Results on the Implication Side 22 1.4. Summary of Results: Examples in Positive Characteristic 24 1.5. Acknowledgments 25 2. Proof of Theorem 1.5 25 2.1. Preliminaries 25 2.2. Prolongable Points 25 2.3. Proof of Theorem 1.5 26 3. A Few Refinements and Applications in Arithmetic and Geometry 26 3.1. Proof of Theor...
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].
Lang's conjectures, conjecture H, and uniformity
"... The purpose of this note is to wish a happy birthday to Professor Lucia Caporaso. We prove that Conjecture H of Caporaso et al. [CHarM], §6 together with Lang’s conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHarM]; a few applications in arithmeti ..."
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Cited by 2 (2 self)
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The purpose of this note is to wish a happy birthday to Professor Lucia Caporaso. We prove that Conjecture H of Caporaso et al. [CHarM], §6 together with Lang’s conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHarM]; a few applications in arithmetic and geometry are stated. As in [ℵ], one uses the fact that rational points on Xn B are ntuples of points when trying to bound n, as well as the fact that the Noetherian induction used in [CHarM] can be wired into the definitions. Geometers with numbertheory anxiety should skip directly to the corollary on the last page.
BIRATIONAL GEOMETRY FOR NUMBER THEORISTS
, 2007
"... Lecture 0. Geometry and arithmetic of curves 2 ..."
LANG MAPS AND HARRIS’S CONJECTURE PRELIMINARY VERSION
, 2008
"... We work over fields of characteristic 0. Let X be a variety of general type defined over a number field K. A well known conjecture of S. Lang [L] states that the set of rational points X(K) is not Zariski dense in X. As noted in [ℵ], this implies that if X is a variety ..."
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We work over fields of characteristic 0. Let X be a variety of general type defined over a number field K. A well known conjecture of S. Lang [L] states that the set of rational points X(K) is not Zariski dense in X. As noted in [ℵ], this implies that if X is a variety
Lang’s Conjectures, Fibered Powers, and Uniformity
"... Abstract. We prove that the fibered power conjecture of Caporaso et al. (Conjecture H, [CHM], §6) together with Lang’s conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHM]; a few applications on the arithmetic and geometry of curves are stated. In ..."
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Abstract. We prove that the fibered power conjecture of Caporaso et al. (Conjecture H, [CHM], §6) together with Lang’s conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHM]; a few applications on the arithmetic and geometry of curves are stated. In an opposite direction, we give counterexamples to some analogous results in positive characteristic. We show that curves that change genus can have arbitrarily many rational points; and that curves over Fp(t) can have arbitrarily many Frobenius orbits of nonconstant points.