An analytic function and iterated integrals
Let be a modular elliptic curve, and let be an imaginary quadratic field. We show that the -Selmer group of over certain finite anticyclotomic extensions of , modulo the universal norms, is annihilated by the «characteristic ideal» of the universal norms modulo the Heegner points. We also extend this result to the anticyclotomic -extension of . This refines in the current contest a result of [1].
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 on metrized line bundles that resembles properties of the dimension of , where is a divisor on a curve . 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 be an arithmetic ring of Krull dimension at most and a pointed stable curve. Write . For every integer , the invertible sheaf inherits a singular hermitian structure from the hyperbolic metric on the Riemann surface . In this article we define a Quillen type metric on the determinant line
Let be an arithmetic ring of Krull dimension at most 1, and an -pointed stable curve of genus . Write . The invertible sheaf inherits a hermitian structure from the dual of the hyperbolic metric on the Riemann surface . In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of . The theorem is applied to modular curves , or , prime, with sections given by the cusps. We show , with when . Here is the Selberg zeta...
We prove an analog in Arakelov geometry of the Grothendieck-Riemann-Roch theorem.