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:
- Recent developements in integral \(p\)-adic cohomology theories, workshop, Hausdorff Center Bonn (Germany), 29 February-3 March 2016.
- \(p\)-adic Hodge theory and Iwasawa theory, workshop, Bielefeld (Germany), 14-18 September 2015.
- Recent Developments in Algebraic and Arithmetic Geometry, Summer School 2014 of the IRTG "Moduli and Automorphic Forms" in collaboration with the Rényi Institute, 25-30 August 2014.
- Summer school on the applications of etale cohomology, Budapest, 16-20 June 2014.
- Summer School in Mathematics at ELTE intended for undergraduates, Budapest, 24^{th} June - 5^{th} July 2013, lecture notes for my course on "p-adic numbers and applications"
- Applications of Iwasawa algebras, workshop at Banff (Canada), 3^{rd}-8^{th} March 2013
Slides:
Talk at ELTE on Representations of p-adic linear groups (in Hungarian).
Documents/theses for my habilitation (partly in Hungarian)
Papers and preprints:
15. Estimating the greatest common divisor of the value of two polynomials (jt. with Péter E. Frenkel), to appear in International Journal of Number Theory, arxiv:1712.01054
14. Cohomology and overconvergence for representations of powers of Galois groups (jt. with Aprameyo Pal), revised pdf, arxiv:1705.03786
13. The p-adic Hodge decomposition according to Beilinson (jt. with Tamás Szamuely),
final pdf, in: Algebraic Geometry: Salt Lake City (2015), Part 2, Proceedings of Symposia in Pure Mathematics 97, AMS, Providence (2018), 495-572,
arxiv:1606.01921
12. Multivariable (φ,Γ)-modules and products of Galois groups, pdf, Math. Research Letters 25(2), 687-721, arxiv:1603.04231
11. On twists of modules over noncommutative Iwasawa algebras (jt. with T. Ochiai and S. Jha), Algebra & Number Theory 10(3) (2016), 685-694, arxiv:1512.07814
10. Multivariable (φ,Γ)-modules and smooth o-torsion representations, read-only pdf, Selecta Mathematica 24(2) (2018), 935-995, arxiv:1511.01037
9. Links between generalized Montréal functors (jt. with M. Erdélyi), read-only pdf, Mathematische Zeitschrift 286(3-4) (2017), 1227-1275, arxiv:1412.5778
8. Algebraic functional equations and completely faithful Selmer groups (jt. with T. Backhausz), International Journal of Number Theory 11(4) (2015), 1233-1257, arxiv:1405.6180.
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:1206.1125
6. (φ,Γ)-modules over noncommutative overconvergent and Robba
rings, Algebra & Number Theory 8(1) (2014), 191-242, preprint version, arxiv:1208.3347
5. Exactness of the reduction on étale modules, J. of Algebra 331 (2011), 400-415, available on the arxiv:1006.5808
4. Generalized Robba rings (with an Appendix by Peter
Schneider), Israel J. Math. 191(2) (2012), 817-887, available on the arxiv:1006.4690
3. Pairings and functional equations over the
GL_{2}-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.
Habilitation thesis, submitted at ELTE (2016): Functorial relations in the p-adic Langlands programme.
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, 1^{st} 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, Humboldt - co-supervised by Elmar Grosse-Klönne), 2012-2016, thesis: \(K\)-theoretic methods in the representation theory of \(p\)-adic analytic groups
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
6. Bence Forrás (ELTE): Kummer's congruences and the p-adic zeta-function (Kummer kongruenciái és a p-adikus zeta-függvény), in Hungarian (2017) pdf
7. Tamás Kátay (ELTE): Introduction to the theory of infinite field extensions (Betekintés a végtelen testbővítések elméletébe), in Hungarian (2018) pdf
8. Bence Hevesi (ELTE): The field of p-adic periods (A p-adikus periódusok teste), in Hungarian (2018) pdf
Visiting students:
1. Lucia Mocz (from Harvard to ELTE): reading mod p representation theory of p-adic groups (May-August 2011)
2. Ugur Dogan (from Humboldt to ELTE, PhD student of Elmar Grosse-Klönne) February-April 2017, within CENTRAL network