Let $X$ be a smooth geometrically connected projective curve over the field of fractions of a discrete valuation ring $R$, and $\mathfrak {m}$ a modulus on $X$, given by a closed subscheme of $X$ which is geometrically reduced. The generalized Jacobian $J_\mathfrak {m}$ of $X$ with respect to $\mathfrak {m}$ is then an extension of the Jacobian of $X$ by a torus. We describe its Néron model, together with the character and component groups of the special fibre, in terms of a regular model of $X$ over $R$. This generalizes Raynaud's well-known description for the usual Jacobian. We also give some computations for generalized Jacobians of modular curves $X_0(N)$ with moduli supported on the cusps.

We give a new proof of Faltings's $p$-adic Eichler–Shimura decomposition of the modular curves via Bernstein–Gelfand–Gelfand (BGG) methods and the Hodge–Tate period map. The key property is the relation between the Tate module and the Faltings extension, which was used in the original proof. Then we construct overconvergent Eichler–Shimura maps for the modular curves providing ‘the second half’ of the overconvergent Eichler–Shimura map of Andreatta, Iovita and Stevens. We use higher Coleman theory on the modular curve developed by Boxer and Pilloni to show that the small-slope part of the Eichler–Shimura maps interpolates the classical $p$-adic Eichler–Shimura decompositions. Finally, we prove that overconvergent Eichler–Shimura maps are compatible with Poincaré and Serre pairings.

We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus $g>1$ whose Jacobians have Mordell–Weil rank $g$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients $X_0^+(N)$ of prime level $N$, the curve $X_{S_4}(13)$, as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve $X_{\scriptstyle \mathrm { ns}} ^+ (17)$.

Let $J$ be a Jacobian variety with toric reduction over a local field $K$. Let $J\,\to \,E$ be an optimal quotient defined over $K$, where $E$ is an elliptic curve. We give examples in which the functorially induced map ${{\Phi }_{J}}\,\to \,{{\Phi }_{E}}$ on component groups of the Néron models is not surjective. This answers a question of Ribet and Takahashi. We also give various criteria under which ${{\Phi }_{J}}\,\to \,{{\Phi }_{E}}$ is surjective and discuss when these criteria hold for the Jacobians of modular curves.

We describe a model-theoretic setting for the study of Shimura varieties, and study the interaction between model theory and arithmetic geometry in this setting. In particular, we show that the model-theoretic statement of a certain ${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}}$-sentence having a unique model of cardinality $\aleph _{1}$ is equivalent to a condition regarding certain Galois representations associated with Hodge-generic points. We then show that for modular and Shimura curves this ${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}}$-sentence has a unique model in every infinite cardinality. In the process, we prove a new characterisation of the special points on any Shimura variety.

Consider a Shimura curve $X_{0}^{D}\left( N \right)$ over the rational numbers. We determine criteria for the twist by an Atkin–Lenher involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan and Livné on ${{\mathbf{Q}}_{p}}$ points when $p|D$ and for the first time give criteria for ${{\mathbf{Q}}_{p}}$ points when $p|N$. We also give congruence conditions for roots modulo $p$ of Hilbert class polynomials.

A fundamental object in the theory of arithmetic surfaces is the Green's function associated to the canonical metric. Previous expressions for the canonical Green's function have relied on general functional analysis or, when using specific properties of the canonical metric, the classical Riemann theta function. In this article, we derive a new identity for the canonical Green's function involving the hyperbolic heat kernel. As an application of our results, we obtain bounds for the canonical Green's function through covers and for families of modular curves.

We give a criterion to check if, given a prime power pr with r > 1, the only rational points of the modular curve X0+ (pr) are trivial (i.e. cusps or points furnished by complex multiplication). We then prove that this criterion is verified if p satisfies explicit congruences. This applies in particular to the modular curves Xsplit (p), which intervene in the problem of Serre concerning uniform surjectivity of Galois representations associated to division points of elliptic curves.

In this paper, we prove a certain maximality property of the Shimura subgroup amongst the multiplicative-type subgroups of $J_0(N)$, and apply this to verify conjectures of Stevens on the existence of certain canonical parametrizations of rational elliptic curves by modular curves. We are also able to verify some of Stevens’s conjectures on the characterization of the elliptic curve in an isogeny class with minimal Faltings–Parshin height.

AMS 2000 Mathematics subject classification: Primary 11G05; 11G18

The mod $p$ representation associated to an elliptic curve is called split or non-split dihedral if its image lies in the normaliser of a split or non-split Cartan subgroup of $\GL_2(\f_p)$, respectively. Let $\xsplit$ and $\xnonsplit$ denote the modular curves which classify elliptic curves with split and non-split dihedral mod $p$ representation, respectively. We call such curves split and non-split{\it Cartan modular curves}. The curve $\xsplit$ is isomorphic to the curve $X_0^+(p^2)$. Using the Selberg trace formula for Hecke operators, we verify that the jacobian of $\xnonsplit$ is isogenous to the new part of the jacobian of $X_0^+(p^2)$.

1991 Mathematics Subject Classification: primary 11G18; secondary 11F72.