Gergely Zábrádi's homepage

Email: zger 'at' cs.elte.hu

I am an assistant professor at the Department of Algebra and Number Theory, Institute of Mathematics, Eötvös Loránd University, Budapest. My CV, publication list (including citations), Google Scholar Profile, and Mathematics Genealogy.

Bulletin board for my students (in Hungarian)

Research interests:

My field of interest lies in Algebraic Number Theory, however it is twofold.

I wrote my PhD in Cambridge under the supervision of John Coates on noncommutative Iwasawa theory for elliptic curves. The arithmetic of elliptic curves and especially the conjectures of Birch and Swinnerton-Dyer have been lying in the centre of research in Arithmetic Algebraic Geometry. One of the most powerful tools known at present attacking these conjectures is Iwasawa theory. The main idea of Iwasawa theory is to relate various arithmetic objects to complex L-functions via a so-called p-adic L-function. This arithmetic object could be the ideal class group of a number field, the Selmer group of an elliptic curve, or more generally of an abelian variety, or even of a motive. The p-adic L-function - in most cases conjecturally - interpolates special values of the twisted complex L-functions of the arithmetic object. On the other hand, it is supposed to be - by the Main Conjecture - a characteristic element of the Selmer groups.

I did a post-doc at the Westfälische Wilhelmsuniversität Münster with Peter Schneider. There I learnt a lot about representation theory of p-adic groups and the p-adic Langlands programme. The (global) Langlands programme is a huge web of conjectures that relates Galois representations of number fields to automorphic representations (which are - in a certain sense - generalizations of modular forms). The local Langlands conjectures (which are now theorems for GL_n) relate the (continuous) representation theory of the absolute Galois group (or in fact the Weil-Deligne group) of local fields (such as the field Q_p of p-adic numbers) in finite dimensional vectorspaces over C to the smooth representations of reductive algebraic groups over the local field in (infinite dimensional) vectorspaces over C. However, if we allow continuous representations in vectorspaces over other fields (such as Q_p or F_p) on both the Galois and reductive group side, we obtain much more representations. The precisely formulated conjectures, how these should correspond to each other (if at all), are still missing. However, (through the work of Fontaine, Colmez, Breuil, Paskunas, Berger, Kisin, Emerton, Schneider, Vigneras, and others) it has become increasingly clear that some kind of p-adic (and also mod p) Langlands correspondences should exist. In fact, Colmez managed to prove such a correspondence for GL₂(Q_p).

Papers and preprints:

7. From étale P₊-modules to G-equivariant sheaves on G/P (jt. with P. Schneider and M.-F. Vigneras), in preparation
6. (φ,Γ)-modules over noncommutative overconvergent and Robba rings, preliminary version of 27th March 2012, comments welcome pdf
5. Exactness of the reduction on étale modules, J. of Algebra 331 (2011), 400-415, available on the arxiv
4. Generalized Robba rings (with an Appendix by Peter Schneider), Israel J. Math. (2012), available on the arxiv
3. Pairings and functional equations over the GL₂-extension, Proc. London Math. Soc. (2010) 101 (3), 893-930, pdf
2. Characteristic elements, pairings and functional equations over the false Tate curve extension, Math. Proc. Camb. Phil. Soc. 144 (2008), 535-574, pdf.
1. On irregularities in the graph of generalized divisor functions, Acta Arith., 110 (2003), 165-171.

PhD thesis, Trinity College, University of Cambridge (2008): pdf.