 Author

C.P. Debry
 Title
 Towards a class number formula for Drinfeld modules
 Supervisors
 Award date
 15 November 2016
 Number of pages
 115
 Document type
 PhD thesis
 Faculty
 Faculty of Science (FNWI)
 Institute
 Kortewegde Vries Institute for Mathematics (KdVI)
 Abstract

Let F be the function field of an irreducible, smooth, projective curve over a finite field. Let A be the ring of functions on the curve which are regular away from a fixed closed point ∞. Let F_{∞} be the completion of F at ∞.
Consider an integral model φ of a Drinfeld Amodule over a finite extension of F. We associate to such a model an element (the Lvalue of φ of F_{∞}, mimicking the residue at 1 of the Dedekind zeta function of a number field, or the top coefficient at 1 of the Lfunction of an elliptic curve over Q. The class module of φ is an Amodule of finite cardinality, which serves as an analogue of the class group of a number field, or the TateShafarevich group of an elliptic curve. The regulator of φ is an invertible subAmodule of F_{∞} and looks like the regulators of number fields and elliptic curves. The aim of this thesis is to present a conjectural formula which relates the Lvalue, the class module and the regulator of φ. It could be regarded as a function field analogue of the class number formula for number fields or the BSD conjecture for elliptic curves; hence the name “a class number formula for Drinfeld modules.”
Our conjecture is a direct generalisation of a theorem by Taelman which states that the formula holds if A=F_{q} [t]. The current manuscript generalises his work, by introducing the analogues of his Lvalues, class modules and regulators for general A, by stating a conjectural formula, and by proving it for all A which are principal ideal domains. We also show how our techniques can be used to prove the class number formula in specific cases, for example, when the curve has genus zero.  Note
 This thesis was prepared within the partnership between the University of Amsterdam and KU Leuven with the purpose of obtaining
a joint doctorate degree.
Research conducted at: Universiteit van Amsterdam  Permalink
 http://hdl.handle.net/11245/1.545161
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