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Deciding existence of rational points on curves: an experiment, accepted by Exp
 Math
"... The problem to decide whether a given algebraic variety defined over the rational numbers has rational points is fundamental in Arithmetic Geometry. Abstracting from concrete examples, this leads to the question whether there exists an algorithm that is able to perform this task for any given variet ..."
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Cited by 16 (8 self)
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The problem to decide whether a given algebraic variety defined over the rational numbers has rational points is fundamental in Arithmetic Geometry. Abstracting from concrete examples, this leads to the question whether there exists an algorithm that is able to perform this task for any given variety. This is probably
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...
Empirical evidence for the Birch and SwinnertonDyer conjectures for modular Jacobians of genus 2 curves
 Math. Comp
, 2001
"... Abstract. This paper provides empirical evidence for the Birch and SwinnertonDyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the Shafarevic ..."
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Cited by 14 (9 self)
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Abstract. This paper provides empirical evidence for the Birch and SwinnertonDyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the ShafarevichTate group. Unable to compute that, we computed the five other quantities and solved for the last one. In all 32 cases, the result is very close to an integer that is a power of 2. In addition, this power of 2 agrees with the size of the 2torsion of the ShafarevichTate group, which we could compute. 1.
Modular curves of genus 2
 Math. Comp
, 1999
"... Abstract. We prove that there are exactly 149 genus two curves C defined over Q such that there exists a nonconstant morphism π: X1(N) → C defined over Q and the jacobian of C is Qisogenous to the abelian variety Af attached by Shimura to a newform f ∈ S2(Γ1(N)). We determine the corresponding new ..."
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Cited by 12 (5 self)
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Abstract. We prove that there are exactly 149 genus two curves C defined over Q such that there exists a nonconstant morphism π: X1(N) → C defined over Q and the jacobian of C is Qisogenous to the abelian variety Af attached by Shimura to a newform f ∈ S2(Γ1(N)). We determine the corresponding newforms and present equations for all these curves. 1.
Bogomolov conjecture for curves of genus 2 over function fields
 J. Math. Kyoto Univ
, 1996
"... Abstract. In this note, we will show that Bogomolov conjecture holds for a nonisotrivial curve of genus 2 over a function field. 1. ..."
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Cited by 9 (3 self)
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Abstract. In this note, we will show that Bogomolov conjecture holds for a nonisotrivial curve of genus 2 over a function field. 1.
Models of Curves and Finite Covers
 COMPOSITIO MATHEMATICA
, 1999
"... Let K be a discrete valuation field with ring of integers OK.Letf: X → Y beafinite morphism of curves over K. In this article, we study some possible relationships between the models over OK of X and of Y. Three such relationships are listed below. Consider a Galois cover f: X → Y of degree prime t ..."
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Cited by 8 (1 self)
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Let K be a discrete valuation field with ring of integers OK.Letf: X → Y beafinite morphism of curves over K. In this article, we study some possible relationships between the models over OK of X and of Y. Three such relationships are listed below. Consider a Galois cover f: X → Y of degree prime to the characteristic of the residue field, with branch locus B. We show that if Y has semistable reduction over K, thenX achieves semistable reduction over some explicit tame extension of K(B). When K is strictly henselian, we determine the minimal extension L/K with the property that XL has semistable reduction. Let f: X → Y be a finite morphism, with g(Y) � 2. We show that if X has a stable model X over OK, then Y has a stable model Y over OK, and the morphism f extends to a morphism X → Y. Finally, given any finite morphism f: X → Y, is it possible to choose suitable regular models X and Y of X and Y over OK such that f extends to a finite morphism X → Y? As was shown by Abhyankar, the answer is negative in general. We present counterexamples in rather general situations, with f a cyclic cover of any order � 4. On the other hand, we prove, without any hypotheses on the residual characteristic, that this extension problem has a positive solution when f is cyclic of order 2 or 3.
Canonical integral structures on the de Rham cohomology of curves
 Ann. Inst. Fourier (Grenoble
, 2009
"... Abstract. For a smooth and proper curve XK over the fraction field K of a discrete valuation ring R, we explain (under very mild hypotheses) how to equip the de Rham cohomology H 1 dR(XK/K) with a canonical integral structure: i.e. an Rlattice which is functorial in finite (generically étale) Kmor ..."
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Cited by 2 (2 self)
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Abstract. For a smooth and proper curve XK over the fraction field K of a discrete valuation ring R, we explain (under very mild hypotheses) how to equip the de Rham cohomology H 1 dR(XK/K) with a canonical integral structure: i.e. an Rlattice which is functorial in finite (generically étale) Kmorphisms of XK and which is preserved by the cupproduct autoduality on H 1 dR(XK/K). Our construction of this lattice uses a certain class of normal proper models of XK and relative dualizing sheaves. We show that our lattice naturally contains the lattice furnished by the (truncated) de Rham complex of a regular proper Rmodel of XK and that the index for this inclusion of lattices is a numerical invariant of XK (we call it the de Rham conductor). Using work of Bloch and of LiuSaito, we prove that the de Rham conductor of XK is bounded above by the Artin conductor, and bounded below by the Efficient conductor. We then study how the position of our canonical lattice inside the de Rham cohomology of XK is affected by finite extension of scalars. 1.
Admissible constants for genus 2 curves
"... Abstract. S.W. Zhang recently introduced a new adelic invariant ϕ for curves of genus at least 2 over number fields and function fields. We calculate this invariant when the genus is equal to 2. 1. ..."
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Abstract. S.W. Zhang recently introduced a new adelic invariant ϕ for curves of genus at least 2 over number fields and function fields. We calculate this invariant when the genus is equal to 2. 1.
Iterated Endomorphisms of Abelian Algebraic Groups
"... Abstract. Given an abelian algebraic group A over a global field F, α ∈ A(F), and a prime ℓ, the set of all preimages of α under some iterate of [ℓ] has a natural tree structure. Using this data, we construct an “arboreal ” Galois representation ω whose image combines that of the usual ℓadic repres ..."
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Abstract. Given an abelian algebraic group A over a global field F, α ∈ A(F), and a prime ℓ, the set of all preimages of α under some iterate of [ℓ] has a natural tree structure. Using this data, we construct an “arboreal ” Galois representation ω whose image combines that of the usual ℓadic representation and the Galois group of a certain Kummertype extension. For several classes of A, we give a simple characterization of when ω is surjective. The image of ω also encodes information about the density of primes p in K such that the order of the reduction mod p of α is prime to ℓ. We compute this density in the general case for several A of interest. For example, if F is a number field, A/F is an elliptic curve with surjective 2adic representation and α ∈ A(F), with α ̸ ∈ 2A(F(A[4])), then the density of primes p with α mod p having odd order is 11/21. 1.
CHABAUTY’S METHOD PROVES THAT MOST ODD DEGREE HYPERELLIPTIC CURVES HAVE ONLY ONE RATIONAL POINT
"... Abstract. Consider the smooth projective models C of curves y 2 = f(x) with f(x) ∈ Z[x] monic and separable of degree 2g + 1. We prove that for g ≥ 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower bound on this fraction that tends to 1 as g → ∞. ..."
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Abstract. Consider the smooth projective models C of curves y 2 = f(x) with f(x) ∈ Z[x] monic and separable of degree 2g + 1. We prove that for g ≥ 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower bound on this fraction that tends to 1 as g → ∞. Finally, we show that C(Q) can be algorithmically computed for such a fraction of the curves, via Chabauty’s method at the prime 2. 1.