Generalized jacobians and Pellian polynomials

Bertrand, Daniel

doi:10.5802/jtnb.909

Cited by 4 publications

(5 citation statements)

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“…Now, the group P ic o (A) is the underlying group of the dual abelian variety Â, and since a Jacobian is self-dual it is isomorphic to J in the present case. The extension coming from ρ is checked to correspond to the point ξ+ − ξ− in J (see Bertrand's paper [6], §2.1). This yields an extension G which is isogenous to a split one if and only if ξ+ − ξ− is a torsion point in J.…”

Section: 3mentioning

confidence: 99%

“…Our survey paper [34] points out with some examples certain generalizations of this to Pell equations with non-squarefree D(t), this time in terms of generalized Jacobians associated to the curve (as described e.g. in Serre's book [29]); see also [2], [5], [6], [7], [19] for further instances and links with other contexts.…”

Section: Introductionmentioning

confidence: 97%

“…As a matter of fact, from many viewpoints 'pellianity' is extremely rare for any given d > 1: for instance, it may be shown that inside the (2d − 2)-dimensional family of polynomials D(t) of degree 2d suitably normalized, the Pellian ones form a denumerable union of algebraic families of dimension ≤ d − 1. 6 See also e.g. the joint paper with D. Masser [21] for a proof that on 'most' 1-dimensional families of polynomials of degree 2d ≥ 6 there are only finitely many Pellian ones.…”

Section: Introductionmentioning

confidence: 99%

“…Already in the numerical case, Dirichet class-number formulae and other results indicate a strict connection of the topic with the suitable Picard groups 5. The case of polynomials over a finite field is, on the contrary, completely similar to the integer case 6. A formal proof of this is the object of work in progress, but some detail appears already in[34], especially §2.2.…”

mentioning

confidence: 97%

“…Let p/q be a convergent to √ D, for coprime polynomials p, q. Then, after choosing appropriately the sign related to ∞ + , we have (6) ord…”

mentioning

confidence: 99%

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Hyperelliptic continued fractions and generalized Jacobians

Zannier¹

2019

American Journal of Mathematics

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For a complex polynomial D(t) of even degree, one may define the continued fraction of D(t). This was found relevant already by Abel in 1826, and later by Chebyshev, concerning integration of (hyperelliptic) differentials; they realized that, contrary to the classical case of square roots of positive integers treated by Lagrange and Galois, we do not always have pre-periodicity of the partial quotients.In this paper we shall prove that, however, a correct analogue of Lagrange's theorem still exists in full generality: pre-periodicity of the degrees of the partial quotients always holds. Apparently, this fact was never noted before.This also yields a corresponding formula for the degrees of the convergents, for which we shall prove new bounds which are generally best possible (halving the known ones).We shall further study other aspects of the continued fraction, like the growth of the heights of partial quotients. Throughout, some striking phenomena appear, related to the geometry of (generalized) Hyperelliptic Jacobians. Another conclusion central in this paper concerns the poles of the convergents: there can be only finitely many rational ones which occur infinitely many times. (This is crucial for applications to a function field version of a question of McMullen.)Our methods rely, among other things, on linking Padé approximants and convergents with divisor relations in generalized Jacobians; this shall allow an application of a version for algebraic groups, proved in this paper, of the Skolem-Mahler-Lech theorem.1 When λ = a/b is rational the procedure eventually terminates and corresponds to the Euclidean algorithm for a, b.The fraction pn/qn determines the polynomials pn, qn only up to a factor; usually here we implicitly mean that pn, qn are calculated formally from the an in the well-known natural way.3 This notion heavily depends on the ground field, but here we tacitly stick to C. 4 Already in the numerical case, Dirichet class-number formulae and other results indicate a strict connection of the topic with the suitable Picard groups.5 The case of polynomials over a finite field is, on the contrary, completely similar to the integer case. 6 A formal proof of this is the object of work in progress, but some detail appears already in [34], especially §2.2. It shall anyway clearly appear later that pellianity is indeed uncommon. 7 For reasons of space, we omit a discussion of this here, which is somewhat laborious, depending on Jacobians of dimension ≥ 3 containing a translate of an elliptic curve inside the set of sums of two points of the curve. To give a specific example, the polynomial D(t) = t 8 − t 7 − (3/4)t 6 + (7/2)t 5 − (21/4)t 4 + (7/2)t 3 − (3/4)t 2 − t + 1 yields infinitely many partial quotients of degrees 1 and 2, with the periodic pattern of degrees . See O. Merkert's thesis [22] for more. We plan to publish a detailed presentation in the future. 8 The methods of this paper should probably prove this for arbitrary elements of C(t, D(t)), though for simplicity we work only with the special ...

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Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 97%

“…Let p/q be a convergent to √ D, for coprime polynomials p, q. Then, after choosing appropriately the sign related to ∞ + , we have (6) ord…”

mentioning

confidence: 99%

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Hyperelliptic continued fractions and generalized Jacobians

Zannier¹

2019

American Journal of Mathematics

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Unlikely intersections of curves with algebraic subgroups in semiabelian varieties

Barroero¹,

Kühne

Schmidt

2023

Sel. Math. New Ser.

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Let G be a semiabelian variety and $$C$$ C a curve in G that is not contained in a proper algebraic subgroup of G. In this situation, conjectures of Pink and Zilber imply that there are at most finitely many points contained in the so-called unlikely intersections of $$C$$ C with subgroups of codimension at least 2. In this note, we establish this assertion for general semiabelian varieties over $$\overline{\mathbb {Q}}$$ Q ¯ . This extends results of Maurin and Bombieri, Habegger, Masser, and Zannier in the toric case as well as Habegger and Pila in the abelian case.

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Relative Manin–Mumford in additive extensions

Schmidt

2018

Trans. Amer. Math. Soc.

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In recent papers Masser and Zannier have proved various results of “relative Manin–Mumford” type for various families of abelian varieties, some with field of definition restricted to the algebraic numbers. Typically these imply the finiteness of the set of torsion points on a curve in the family. After Bertrand, Masser, and Zannier discovered some surprising counterexamples for multiplicative extensions of elliptic families, the three authors together with Pillay settled completely the situation for this case over the algebraic numbers. Here we treat the last remaining case of surfaces, that of additive extensions of elliptic families, and even over the field of all complex numbers. In particular analogous counterexamples do not exist. There are finiteness consequences for Pell’s equation over polynomial rings and integration in elementary terms. Our work can be made effective (as opposed to most of that preceding), mainly because we use counting results only for analytic curves.

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Generalized jacobians and Pellian polynomials

Cited by 4 publications

References 18 publications

Hyperelliptic continued fractions and generalized Jacobians

Hyperelliptic continued fractions and generalized Jacobians

Unlikely intersections of curves with algebraic subgroups in semiabelian varieties

Relative Manin–Mumford in additive extensions

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