73 research outputs found
CoCaml: Functional Programming with Regular Coinductive Types
Functional languages offer a high level of abstraction, which results in programs that are elegant and easy to understand. Central to the development of functional programming are inductive and coinductive types and associated programming constructs, such as pattern-matching. Whereas inductive types have a long tradition and are well supported in most languages, coinductive types are subject of more recent research and are less mainstream.
We present CoCaml, a functional programming language extending OCaml, which allows us to define recursive functions on regular coinductive datatypes. These functions are defined like usual recursive functions, but parameterized by an equation solver. We present a full implementation of all the constructs and solvers and show how these can be used in a variety of examples, including operations on infinite lists, infinitary γ-terms, and p-adic numbers
Normal origamis of Mumford curves
An origami (also known as square-tiled surface) is a Riemann surface covering
a torus with at most one branch point. Lifting two generators of the
fundamental group of the punctured torus decomposes the surface into finitely
many unit squares. By varying the complex structure of the torus one obtains
easily accessible examples of Teichm\"uller curves in the moduli space of
Riemann surfaces. The p-adic analogues of Riemann surfaces are Mumford curves.
A p-adic origami is defined as a covering of Mumford curves with at most one
branch point, where the bottom curve has genus one. A classification of all
normal non-trivial p-adic origamis is presented and used to calculate some
invariants. These can be used to describe p-adic origamis in terms of glueing
squares.Comment: 21 pages, to appear in manuscripta mathematica (Springer
Variation of Tamagawa numbers of Jacobians of hyperelliptic curves with semistable reduction
We study how Tamagawa numbers of Jacobians of hyperelliptic curves vary as one varies the base field or the curve, in the case of semistable reduction. We find that there are strong constraints on the behaviour that appears, some of which are unexpected and specific to hyperelliptic curves. Our methods are explicit and allow one to write down formulae for Tamagawa numbers of infinite families of hyperelliptic curves, of the kind used in proofs of the parity conjecture for Jacobians of curves of small genus
An Eisenstein ideal for imaginary quadratic fields and the Bloch-Kato conjecture for Hecke characters
For certain algebraic Hecke characters chi of an imaginary quadratic field F
we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal
automorphic forms of GL_2/F. By finding congruences between Eisenstein
cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a
lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms
of the special L-value L(0,chi). We further prove that its index is bounded
from above by the order of the Selmer group of the p-adic Galois character
associated to chi^{-1}. This uses the work of R. Taylor et al. on attaching
Galois representations to cuspforms of GL_2/F. Together these results imply a
lower bound for the size of the Selmer group in terms of L(0,chi), coinciding
with the value given by the Bloch-Kato conjecture.Comment: 26 page
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