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It follows from the Grothendieck-Ogg-Shafarevich formula that the rank of an abelian variety (with trivial trace) defined over the function field of a curve is bounded by a quantity which depends on the genus of the base curve and on bad reduction data. Using a function field version of classical $\ell$-descent techniques, we derive an arithmetic refinement of this bound, extending previous work of the second and third authors from elliptic curves to abelian varieties, and improving on their result in the case of elliptic curves. When the abelian variety is the Jacobian of a hyperelliptic curve, we produce a more explicit $2$-descent map. Then we apply this machinery to studying points on the Jacobians of certain genus $2$ curves over $k(t)$, where $k$ is some perfect base field of characteristic not $2$., Comment: 19 pages. Minor changes in the exposition

We consider elliptic curves defined by an equation of the form $y^2=x^3+f(t)$, where $f\in k[t]$ has coefficients in a perfect field $k$ of characteristic not $2$ or $3$. By performing $2$ and $3$-descent, we obtain, under suitable assumptions on the factorization of $f$, bounds for the number of integral points on these curves. These bounds improve on a general result by Hindry and Silverman. When $f$ has degree at most $6$, we give exact expressions for the number of integral points of small height in terms of certain subgroups of Picard groups of the $k$-curves corresponding to the $2$ and $3$-torsion of our curve. This allows us to recover explicit results by Bremner, and gives new insight into Pillai's equation., Comment: 43 pages. Minor changes. Added a reference to Lang in the introduction. Corrected a harmless sign error in the 2 and 3-descent maps

We study the vanishing of Massey products of order at least $3$ for absolutely irreducible smooth projective curves over a perfect field with coefficients in $\mathbb{Z}/\ell$. We mainly focus on elliptic curves, for which we obtain a complete characterization of when triple Massey products do not vanish., Comment: 25 pages. In the fourth version, we removed the assumption that F is a perfect field and we added a new Theorem 1.2. Moreover, we changed Example 6.1 and added Examples 7.6 and 7.8

For each finite subgroup $G$ of $PGL_2(\mathbb{Q})$, and for each integer $n$ coprime to $6$, we construct explicitly infinitely many Galois extensions of $\mathbb{Q}$ with group $G$ and whose ideal class group has $n$-rank at least $\#G-1$. This gives new $n$-rank records for class groups of number fields., Comment: 24 pages; added Lemma 3.8. To appear in International Journal of Number Theory

We construct \'etale generalized Heisenberg group covers of hyperelliptic curves over number fields. We use these to produce infinite families of quadratic extensions of cyclotomic fields that admit everywhere unramified generalized Heisenberg Galois extensions., Comment: 11 pages; added Remark 3.4, extended Section 3.3, shortened Section 4.2 and added Section 4.3

Given a non-isotrivial elliptic curve over $\mathbb{Q}(t)$ with large Mordell-Weil rank, we explain how one can build, for suitable small primes $p$, infinitely many fields of degree $p^2-1$ whose ideal class group has a large $p$-torsion subgroup. As an example, we show the existence of infinitely many cubic fields whose ideal class group contains a subgroup isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{11}$., Comment: 10 pages, LaTeX. Minor improvements following the referee's suggestions. To appear in Int. J. Number Theory

We introduce the use of $p$-descent techniques for elliptic surfaces over a perfect field of characteristic not $2$ or $3$. Under mild hypotheses, we obtain an upper bound for the rank of a non-constant elliptic surface. When $p=2$, this bound is an arithmetic refinement of a well-known geometric bound for the rank deduced from Igusa's inequality. This answers a question raised by Ulmer. We give some applications to rank bounds for elliptic surfaces over the rational numbers., Comment: 22 pages, LaTeX. Minor improvements in the statement of Theorem 1.1. Added Theorem 1.7 and its proof. To appear in Algebra and Number Theory

Let $C$ be a hyperelliptic curve defined over $\mathbb{Q}$, whose Weierstrass points are defined over extensions of $\mathbb{Q}$ of degree at most three, and at least one of them is rational. Generalizing a result of R. Soleng (in the case of elliptic curves), we prove that any line bundle of degree $0$ on $C$ which is not torsion can be specialised into ideal classes of imaginary quadratic fields whose order can be made arbitrarily large. This gives a positive answer, for such curves, to a question by Agboola and Pappas., Comment: 28 pages, LaTeX. Minor improvements following the referee's suggestions. New proof of Lemma 3.10. To appear in Trans. Amer. Math. Soc

The purpose of this note is twofold. First, we survey results on the construction of large class groups of number fields by specialization of finite covers of curves. Then we give examples of applications of these techniques., Comment: 14 pages, LaTeX. Minor revisions. arXiv admin note: text overlap with arXiv:0805.1361

We give a positive answer to a Conjecture by Manjul Bhargava, Daniel M. Kane, Hendrik W. Lenstra Jr., Bjorn Poonen and Eric Rains, concerning the cohomology of torsion subgroups of elliptic curves over global fields. This implies that, given a global field $k$ and an integer $n$, for $100\%$ of elliptic curves $E$ defined over $k$, the $n$-th Selmer group of $E$ is the intersection of two direct summands of the adelic cohomology group $H^1(\mathbf{A},E[n])$. We also give examples of elliptic curves for which the conclusion of this conjecture does not hold., Comment: 11 pages, LaTeX. A new statement, Theorem 1.4, has been added, together with a new section containing its proof. This shows that the conclusion of Theorem 1.1 is false if its hypotheses are dropped

We prove the splitting of the Kummer exact sequence and related exact sequences in arithmetic geometry., Comment: 8 pages, LaTeX. Minor changes in the exposition. New version of \S{}1.4 (in the previous version, the counterexample was wrong)

We prove, under some mild hypothesis, that an \'etale cover of curves defined over a number field has infinitely many specializations into an everywhere unramified extension of number fields. This constitutes an "absolute" version of the Chevalley-Weil theorem. Using this result, we are able to generalize the techniques of Mestre, Levin and the second author for constructing and counting number fields with large class group., Comment: Minor improvements following the referee's suggestions. To appear in Israel J. of Math

Let X be a projective curve over Q and t a non-constant Q-rational function on X of degree n>1. For every integer a pick a points P(a) on X such that t(P(a))=a. Dvornicich and Zannier (1994) proved that for large N the field Q(P(1), ..., P(N)) is of degree at least exp(cN/log N) over Q, where c>0 depends only on X and t. In this note we extend this result, replacing Q by an arbitrary number field., Comment: Minor inaccuracies corrected following the referee's suggestions. To appear in Math. Z

The class-invariant homomorphism allows one to measure the Galois module structure of extensions obtained by dividing points on abelian varieties. In this paper, we consider the case when the abelian variety is the Jacobian of a Fermat curve. We give examples of torsion points whose associated Galois structure is trivial, as well as points of infinite order whose associated Galois structure is non-trivial., Comment: 13 pages, LaTeX

We seize the opportunity of the publication of selected papers from the \emph{Logic, categories, semantics} workshop in the \emph{Journal of Applied Logic} to survey some current trends in logic, namely intuitionistic and linear type theories, that interweave categorical, geometrical and computational considerations. We thereafter present how these rich logical frameworks can model the way language conveys meaning., Comment: Survey on the occasion of a special issue of the journal of applied logic

The bending behaviour of woven perform was investigated in order to better understand the formation of some defects during sheet forming such as wrinkling and tow buckling. The fabric composition considering hybrid and pure flax fabrics as well as some test conditions like relative humidity were examined. On the one hand, the results show a drop of the bending stiffness with flax/PLA commingled fabric. On the other hand, the study points out that moisture enhances the bending rigidity especially in the case of pure flax fabric. However, an excess of humidity, for instance 100% relative humidity, leads to an opposite effect., Comment: TexComp-11, Leuven : Belgium (2013)

We compare tame actions in the category of schemes with torsors in the category of log schemes endowed with the log flat topology. We prove that actions underlying log flat torsors are tame. Conversely, starting from a tame cover of a regular scheme that is an fppf torsor on the complement of a divisor with normal crossings, it is possible to build a unique log flat torsor that dominates this cover. In brief, the theory of log flat torsors gives a canonical approach to the problem of extending torsors into tame covers., Comment: 20 pages, minor changes: rewording of Theorem 2.8; new Definition 3.1; added missing assumption in the statement of Proposition 3.2. To appear in Algebraic Geometry

We give explicit formulae for the logarithmic class group pairing on an elliptic curve defined over a number field. Then we relate it to the descent relative to a suitable cyclic isogeny. This allows us to connect the resulting Selmer group with the logarithmic class group of the base. These constructions are explicit and suitable for computer experimentation. From a conceptual point of view, the questions that arise here are analogues of "visibility" questions in the sense of Cremona and Mazur., Comment: 28 pages, LaTeX

We prove results concerning the specialisation of torsion line bundles on a variety $V$ defined over $\mathbb{Q}$ to ideal classes of number fields. This gives a new general technique for constructing and counting number fields with large class group., Comment: 16 pages, LaTeX. Minor modifications. Accepted for publication in Math. Research Letters

Let $X$ be a fine and saturated log scheme, and let $G$ be a commutative finite flat group scheme over the underlying scheme of $X$. If $G$-torsors for the fppf topology can be thought of as being unramified objects by nature, then $G$-torsors for the log flat topology allow us to consider tame ramification. Using the results of Kato, we define a concept of Galois structure for these torsors, then we generalize the author's previous constructions (class-invariant homomorphism for semi-stable abelian varieties) in this new setting, thus dropping some restrictions., Comment: 35 pages, LaTeX. Proof of Lemme 3.3 corrected. Minor modifications. Accepted for publication in Crelle's Journal

We study, using the language of log schemes, the problem of extending biextensions of smooth commutative group schemes by the multiplicative group. This was first considered by Grothendieck in SGA 7. We show that this problem admits a solution in the category of sheaves for Kato's log flat topology, in contradistinction to what can be observed using the fppf topology, for which monodromic obstructions were defined by Grothendieck. In particular, in the case of an abelian variety and its dual, it is possible to extend the Weil biextension to the whole N\'eron model. This allows us to define a pairing on the points which combines the class group pairing defined by Mazur and Tate and Grothendieck's monodromy pairing., Comment: 24 pages, LaTeX. Minor changes. Numbering of items changed

The so-called class-invariant homomorphism $\psi$ measures the Galois module structure of torsors--under a finite flat group scheme $G$--which lie in the image of a coboundary map associated to an isogeny between (N\'eron models of) abelian varieties with kernel $G$. When the varieties are elliptic curves with semi-stable reduction and the order of $G$ is coprime to 6, is is known that the homomorphism $\psi$ vanishes on torsion points. In this paper, using Weil restrictions of elliptic curves, we give the construction, for any prime number $p>2$, of an abelian variety $A$ of dimension $p$ endowed with an isogeny (with kernel $\mu_p$) whose coboundary map is surjective. In the case when $A$ has rank zero and the $p$-part of the Picard group of the base is non-trivial, we obtain examples where $\psi$ does not vanishes on torsion points., Comment: 20 pages, LaTeX. Small changes

The class-invariant homomorphism allows one to measure the Galois module structure of torsors--under a finite flat group scheme--which lie in the image of a coboundary map associated to an exact sequence. It has been introduced first by Martin Taylor (the exact sequence being given by an isogeny between abelian schemes). We begin by giving general properties of this homomorphism, then we pursue its study in the case when the exact sequence is given by the multiplication by $n$ on an extension of an abelian scheme by a torus., Comment: 19 pages, LaTeX. Minor changes

We define here an analogue, for the N\'eron model of a semi-stable abelian variety defined over a number field, of M. J. Taylor's class-invariant homomorphism (defined for abelian schemes). Then we extend an annulation result (in the case of an elliptic curve), and an injectivity result regarding an arakelovian version of this homomorphism. This is the sequel to the paper "Invariants de classes : le cas semi-stable"., Comment: 16 pages, LaTeX. Remark 1.5 added. Accepted for publication in the Annales de l'Institut Fourier

We define here an analogue, for a semi-stable group scheme whose generic fiber is an abelian variety, of M. J. Taylor's class-invariant homomorphism (defined for abelian schemes), and we give a geometric description of it. Then we extend a result of Taylor, Srivastav, Agboola and Pappas concerning the kernel of this homomorphism in the case of a semi-stable elliptic curve., Comment: 16 pages, LaTeX. Minor changes. Accepted for publication in Compositio Mathematica