Gergely Zábrádi's homepage


Email: zger 'at' cs.elte.hu



Bulletin board for my students (in Hungarian)

Research interests:

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

I completed 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 \(\mathrm{GL}_n\) ) relate the (continuous) representation theory of the absolute Galois group (or in fact the Weil-Deligne group) of local fields (such as the field \(\mathbb{Q}_p\) of \(p\)-adic numbers) in finite dimensional vectorspaces over \(\mathbb{C}\) to the smooth representations of reductive algebraic groups over the local field in (infinite dimensional) vectorspaces over \(\mathbb{C}\). However, if we allow continuous representations in vectorspaces over other fields (such as \(\overline{\mathbb{Q}_p}\) or \(\overline{\mathbb{F}_p}\) ) on both the Galois and reductive group sides, 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 \(\mathrm{GL}_2(\mathbb{Q}_p)\).

Events:

Past Events:

Slides:

Talk at ELTE on Representations of p-adic linear groups (in Hungarian).

Papers and preprints:

13. The \(p\)-adic Hodge decomposition after Beilinson (jt. with Tamás Szamuely), preliminary version, pdf

12. Multivariable (φ,Γ)-modules and products of Galois groups, preprint, also available on the arxiv

11. On twists of modules over non-commutative Iwasawa algebras (jt. with T. Ochiai and S. Jha), to appear in Algebra & Number Theory (2016), available on the arxiv

10. Multivariable (φ,Γ)-modules and smooth o-torsion representations, preprint, also available on the arxiv

9. Links between generalized Montréal functors (jt. with M. Erdélyi), preprint, also available on the arxiv

8. Algebraic functional equations and completely faithful Selmer groups (jt. with T. Backhausz), International Journal of Number Theory 11(4) (2015), 1233-1257, arxiv.

7,5. A note on central torsion Iwasawa modules , not intended for publication, incorporated into the paper "Algebraic functional equations and completely faithful Selmer groups", pdf

7. From étale P+-representations to G-equivariant sheaves on G/P (jt. with P. Schneider and M.-F. Vigneras), in: LMS Lecture Note Series 415 `Automorphic Forms and Galois Representations' (eds.: F. Diamond, P. Kassaei, M. Kim) Volume 2 (2014), 248-366, preprint version, arxiv

6. (φ,Γ)-modules over noncommutative overconvergent and Robba rings, Algebra & Number Theory 8(1) (2014), 191-242, preprint version, also available on the arxiv

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. 191(2) (2012), 817-887, 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, pdf.

PhD thesis, Trinity College, University of Cambridge (2008): Characteristic elements, pairings, and functional equations in non-commutative Iwasawa theory.

Students:

Papers under my supervision

1. Tibor Backhausz (ELTE): Ranks of GL2 Iwasawa modules of elliptic curves, (2013) arxiv, Functiones et Approximatio, Commentarii Mathematici 52(2) (2015), 283-298, 1st prize at Hungarian student research competition (OTDK)
2. Tamás Csige (ELTE): \(K_0\)-invariance of the completely faithful property of Iwasawa-modules, (2014) arxiv
3. Márton Erdélyi (CEU): On the Schneider-Vigneras functor for principal series, Journal of Number Theory 162 (2016), 68-85, arxiv
4. Tamás Csige (ELTE): The Grothendieck group of completed distribution algebras, (2016) arxiv

Graduate Level

PhD theses advised:
1. Márton Erdélyi (CEU), 2011-2015, thesis: Computations and comparison of generalized Montréal functors
2. Tamás Csige (ELTE), 2012-

Master's theses advised:
1. Siddharth Mathur (CEU): Local Class Field Theory and Lubin-Tate Extensions: An Explicit Construction of the Artin Map (2012) pdf
2. Tamás Csige (ELTE): Fields of norms (Normák testei), in Hungarian (2012) pdf
3. Péter Kutas (ELTE): Galois representations (2013) pdf
4. Dávid Szabó (ELTE): p-adic Galois representations and (φ,Γ)-modules (2015) pdf

Undergraduate Level

BSc theses advised:
1. Szabolcs Mészáros (ELTE): Localisation of rings (Gyűrűk lokalizáltja), in Hungarian (2012) pdf
2. Bertalan Bodor (ELTE): Torsion points of elliptic curves (Elliptikus görbék torziópontjai), in Hungarian (2013) pdf
3. Barna Bognár (ELTE): The Hasse-Minkowski Theorem (A Hasse-Minkowski tétel), in Hungarian (2013) pdf
4. Tibor Backhausz (ELTE): p-adic Banach space representations of p-adic groups (p-adikus csoportok p-adikus Banach-tér reprezentációi), in Hungarian (2014) pdf
5. Donát Nagy (ELTE): Semilinear maps over local fields (Szemilineáris leképezések lokális testek felett), in Hungarian (2014) pdf

Visiting students:
1. Lucia Mocz (from Harvard to ELTE): reading mod p representation theory of p-adic groups (May-August 2011)