Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least $2$. The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.

This is the second of a series of papers dealing with an analog in Arakelov geometry of the holomorphic Lefschetz fixed point formula. We use the main result of the first paper to prove a residue formula "à la Bott" for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable torus; recent results of Bismut- Goette on the equivariant (Ray-Singer) analytic torsion play a key role in the proof.

Let $A$ be an abelian variety defined over a number field $k$. In this short Note we give a characterization of the endomorphisms that preserve the height pairing associated to a polarization. We also give a functorial interpretation of this result.

We prove an algebraicity criterion for leaves of algebraic foliations defined over number fields. Namely, consider a number field $K$ embedded in $C$, a smooth algebraic variety $X$ over $K$, equipped with a $K-$rational point $P$, and $F$ an algebraic subbundle of the its tangent bundle $T_X$, defined over $K$. Assume moreover that the vector bundle $F$ is involutive, i.e., closed under Lie bracket. Then it defines an holomorphic foliation of the analytic manifold $X\left(C\right)$, and one may consider its leaf $F$ through $P$. We prove...

We give an Arakelov theoretic proof of the equality of conductor and discriminant.

Number fields can be viewed as analogues of curves over fields. Here we use metrized line bundles as analogues of divisors on curves. Van der Geer and Schoof gave a definition of a function $h^0$ on metrized line bundles that resembles properties of the dimension $l\left(D\right)$ of $H^0(X,ℒ\left(D\right))$, where $D$ is a divisor on a curve $X$. In particular, they get a direct analogue of the Rieman-Roch theorem. For three theorems of curves, notably Clifford’s theorem, we will propose arithmetic analogues.

Let $(𝒪,\sum ,F_{\infty })$ be an arithmetic ring of Krull dimension at most $1,S=\text{Spec}\left(𝒪\right)$ and $(𝒳\to S;{\sigma }_1,...,{\sigma }_n)$ a pointed stable curve. Write $𝒰=𝒳\setminus {\cup }_j{\sigma }_j\left(S\right)$. For every integer $k>0$, the invertible sheaf ${\omega }_{𝒳/S}^{k+1}(k{\sigma }_1+...+k{\sigma }_n)$ inherits a singular hermitian structure from the hyperbolic metric on the Riemann surface ${𝒰}_{\infty }$. In this article we define a Quillen type metric ${∥·∥}_Q$ on the determinant line ${\lambda }_{k+1}=\lambda {\omega }_{𝒳/S}^{k+1}$$(k...$

Let $(𝒪,\Sigma ,F_{\infty })$ be an arithmetic ring of Krull dimension at most 1, $𝒮=\mathrm{Spec}𝒪$ and $(\pi :𝒳\to 𝒮;{\sigma }_1,...,{\sigma }_n)$ an $n$-pointed stable curve of genus $g$. Write $𝒰=𝒳\setminus {\cup }_j{\sigma }_j\left(𝒮\right)$. The invertible sheaf ${\omega }_{𝒳/𝒮}({\sigma }_1+\cdots +{\sigma }_n)$ inherits a hermitian structure ${\parallel ·\parallel }_{\mathrm{hyp}}$ from the dual of the hyperbolic metric on the Riemann surface ${𝒰}_{\infty }$. In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of ${\omega }_{𝒳/𝒮}{({\sigma }_1+...+{\sigma }_n)}_{\mathrm{hyp}}$. The theorem is applied to modular curves $X\left(\Gamma \right)$, $\Gamma ={\Gamma }_0\left(p\right)$ or ${\Gamma }_1\left(p\right)$, $p\ge 11$ prime, with sections given by the cusps. We show $Z^{\text{'}}(Y\left(\Gamma \right),1)\sim e^a{\pi }^b{\Gamma }_2{(1/2)}^{c}L(0,{ℳ}_{\Gamma })$, with $p\equiv 11mod12$ when $\Gamma ={\Gamma }_0\left(p\right)$. Here $Z\left(Y\right(\Gamma ),s)$ is the Selberg zeta...

We prove an analog in Arakelov geometry of the Grothendieck-Riemann-Roch theorem.

Arakelov invariants of arithmetic surfaces are well known for genus 1 and 2 ([4], [2]). In this note, we study the modular height and the Arakelov self-intersection for a family of curves of genus 3 with many automorphisms:$$C_n:Y^4=X^4-(4n-2)X^2{Z}^2+Z^4.$$Arakelov calculus involves both analytic and arithmetic computations. We express the periods of the curve $C_n$ in terms of elliptic integrals. The substitutions used in these integrals provide a splitting of the jacobian of $C_n$ as a product of three elliptic curves. Using the corresponding...