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.

In previous articles, we showed that for number fields in a certain large class, there are at most elliptic points on a Shimura curve of Γ₀(p)-type for every sufficiently large prime number p. In this article, we obtain an effective bound for such p.

We prove some new effective results of André-Oort type. In particular, we state certain uniform improvements of the main result in [L. Kühne, Ann. of Math. 176 (2012), 651-671]. We also show that the equation X + Y = 1 has no solution in singular moduli. As a by-product, we indicate a simple trick rendering André's proof of the André-Oort conjecture effective. A significantly new aspect is the usage of both the Siegel-Tatuzawa theorem and the weak effective lower bound on the class number of an...