Abstract. It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N = 4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X1(16), whose rational points had been previously computed. We prove there are none with N = 5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Gal(Q/Q)-stable 5-cycles, and show that there exist Gal(Q/Q)-stable N-cycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N = 6. 1.

. 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...

this paper we deal only with #-curves with no complex multiplication, the complex multiplication case requiring di#erent techniques. An abelian variety of GL 2 -type is an abelian variety A defined over # such that the #-algebra of #-endomorphisms E = ## End# (A) is a number field of degree equal to the dimension of the variety; the reason for the name is that the Galois action on the #-adic Tate module of the variety gives rise to a representation of G# with values in GL 2 (E# # # ). The main source of abelian varieties of GL 2 -type is a construction by Shimura (see [13, Theorem 7.14]) of abelian varieties A f attached to newforms f for the congruence subgroups # 1 (N ). Recent interest in #-curves with no complex multiplication has been motivated by the works of Elkies [1] and Ribet [10] on the subject. In [1], Elkies shows that every isogeny class of #-curves with no complex multiplication contains a curve whose j-invariant corresponds to a rational noncusp non-CM point of the modular curve X # (N) quotient of the curve X 0 (N) by all the Atkin-Lehner involutions, for some squarefree integer N . In [10], Ribet characterizes #-curves as the elliptic curves defined over # that are quotients of some abelian variety of GL 2 - type. In the same paper, he gives evidence for the conjecture that the varieties A f constructed by Shimura exhaust (up to isogeny) all the abelian varieties of GL 2 -type; in particular, and as a consequence, one has the conjectural characterization of #-curves as those elliptic curves over # that are quotients of some J 1 (N ). The condition of being a #-curve is invariant by isogeny. It is then natural to investigate some of their arithmetic properties up to isogeny; in particular their fields of definition. In [11] Ribet attaches to a #...

Abstract. It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of abelian surfaces of GL2type over Q by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by local and global methods, including Chabauty techniques on explicit equations of Shimura curves. 1.

Abstract. It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a fixed number field. We explore this conjecture when particularized to quaternion endomorphism algebras of abelian surfaces of GL2-type over Q by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by local and global methods, including Chabauty techniques on explicit equations of Shimura curves. 1.

Let C be a #-curve with no complex multiplication. In this note we characterize the number fields K such that there is a curve C # isogenous to C having all the isogenies between its Galois conjugates defined over K, and also the curves C # isogenous to C defined over a number field K such that the abelian variety ResK/# (C # /K) obtained by restriction of scalars is a product of abelian varieties of GL2 -type. 1 Definitions, notation and basic facts We work in the category of abelian varieties up to isogeny. End k (A) will denote the #-algebra of endomorphisms defined over a field k of an abelian variety A.

Abstract. We investigate the fields of definition up to isogeny of the abelian varieties known as building blocks. These varieties are defined as the Q-varieties admitting real or quaternionic multiplications of the maximal possible degree allowed by their dimensions (cf. Pyle (2004)). The Shimura-Taniyama conjecture predicts that every such variety is isogenous to a non-CM simple factor of a modular Jacobian J1(N). The obstruction to descend the field of definition of a building block up to isogeny is given by Ribet in 1994 as an element in a Galois cohomology Galois-cohomological point of view, and obtain results and formulas for the computation of invariants related to them. When considered for the element attached to a building block, these invariants give the structure of its endomorphism algebra, and also complete information on the possible fields of definition up to isogeny of this building block. We implemented these computations in Magma for building blocks given as

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x 4 + dy 2 = z p via modular Q-curves over polyquadratic fields

Solving Fermat-type equations via modular Q-curves over polyquadratic fields