Gergely Zábrádi's homepage

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 of Breuil. 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, Paškūnas, 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:

Colloquium at Heidelberg (06/06/2019) on p-adic Galois representations.
Colloquium at BSM (23/03/2017) on Galois representations and automorphic forms modulo p.
Talk at ELTE on Representations of p-adic linear groups (in Hungarian).

Documents/theses for my habilitation (partly in Hungarian)

Papers and preprints:

22. The sup-norm problem for automorphic cusp forms of $$\operatorname{PGL}(n,\mathbb{Z}[i])$$ (jt. with Péter Maga), Proc. Amer. Math. Soc. 152 (2024), 559-572, arxiv:2301.04405

21. Matrix Kloosterman sums modulo prime powers (jt. with Márton Erdélyi and Árpád Tóth), Math. Zeitschrift 306, 68 (2024) , arxiv:2209.08021

20. Finding nontrivial zeros of quadratic forms over rational function fields of characteristic 2 (jt. with Tímea Csahók, Péter Kutas, and Mickaël Montessinos), ISSAC '22: Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation, 235-244, arxiv:2203.04068

19. Explicit isomorphisms of quaternion algebras over quadratic global fields (jt. with Tímea Csahók, Péter Kutas, and Mickaël Montessinos), Proceedings of the Fifteenth Algorithmic Number Theory Symposium, ANTS-XV, Res. Number Theory 8, Article number: 77 (2022), arxiv:2007.06981

18. Multivariable $$(\varphi,\Gamma)$$-modules and Representations of Products of Galois Groups: The Case of Imperfect Residue Field (jt. with Jishnu Ray and Feng Wei), final version, Bull. Soc. Math. France 149(3) (2021), 521-546, arxiv:2005.11887

17. On Kuznetsov-Bykovskii's formula of counting prime geodesics (jt. with Giacomo Cherubini and Han Wu), Math. Zeitschrift 300 (2022), 881-928, arxiv:1901.03824

16. Drinfeld's lemma for perfectoid spaces and overconvergence of multivariate $$(\varphi,\Gamma)$$-modules (jt. with Annie Carter and Kiran S. Kedlaya), Documenta Math. 26 (2021), 1329-1393, arxiv:1808.03964

15. Estimating the greatest common divisor of the value of two polynomials (jt. with Péter E. Frenkel), International Journal of Number Theory 14(9) (2018), 2543-2554, arxiv:1712.01054

14. Cohomology and overconvergence for representations of powers of Galois groups (jt. with Aprameyo Pal), final pdf, Journal of the Institute of Mathematics of Jussieu 20(2) (2021), 361-421, 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 $$(\varphi,\Gamma)$$-modules and products of Galois groups, pdf, Math. Research Letters 25(2) (2018), 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 $$(\varphi,\Gamma)$$-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. $$(\varphi,\Gamma)$$-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 $$\operatorname{GL}_2$$-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 $$\operatorname{GL}_2$$ 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
5. Hoang Anh Tran (ELTE): Applications of resultant of two $$p$$-adic power series, (2021) arxiv
6. Kristóf Szabó (ELTE): Estimating the $$p$$-adic valuation of the resultant, (2021) arxiv

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

Current PhD students:
3. Dániel Seress (ELTE), 2021-
4. Bálint Mogyorósi (ELTE), 2023-
5. Xiangsheng Wei (ELTE), 2023-

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 $$(\varphi,\Gamma)$$-modules (2015) pdf
5. Tamás Krutki (ELTE): $$p$$-adic integration and its applications ($$p$$-adikus integrálás és alkalmazásai), in Hungarian (2020) pdf
6. Ádám Sagmeister (ELTE): Diophantine geometry - Faltings' theorem (Diofantikus geometria - Faltings tétele), in Hungarian (2020) pdf
7. Orsolya Lévai (ELTE): The $$p$$-adic Satake isomorphism (A $$p$$-adikus Satake-izomorfizmus), in Hungarian (2021) pdf
8. Dániel Seress (ELTE): Galois theory over the $$p$$-adics (Galois-elmélet a $$p$$-adikusok fölött), in Hungarian (2021) pdf
9. Mogyorósi Bálint (ELTE): The Selmer- and the Tate-Shafarevich group (A Selmer- és a Tate-Safarevich csoport), in Hungarian (2023) pdf
10. Jakovác Gergely (ELTE): p-adic Representations and the Montréal Functor (2023) pdf

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
9. Orsolya Lévai (ELTE): Nonarchimedean functional analysis (Nemarkhimédeszi funkcionálanalízis), in Hungarian (2019) pdf
10. Dániel Seress (ELTE): A more general reciprocity law (Egy álatalánosabb reciprocitási tétel), in Hungarian (2019) pdf
11. Szabolcs Andó (ELTE): The inverse Galois problem (Az inverz Galois probléma), in Hungarian (2020) pdf
12. Tímea Csahók (ELTE): Group cohomology (Csoportkohomológia), in Hungarian (2020) pdf
13. Barna Schefler (ELTE): Central simple algebras (Centrális egyszerű algebrák), in Hungarian (2020) pdf
14. Attila Gáspár (ELTE): $$p$$-adic Lie groups ($$p$$-adikus Lie-csoportok), in Hungarian (2021) pdf
15. Gergely Jakovác (ELTE): Weil's conjectures and étale cohomology (A Weil-sejtések és az étale-kohomológia), in Hungarian (2021) pdf
16. Ákos Kőrösi (ELTE): Linear algebraic groups (Lineáris algebrai csoportok), in Hungarian (2021) pdf
17. Csaba Anderlik (ELTE): Dwork's proof of the first conjecture of Weil (Az első Weil-sejtés Dwork-féle bizonyítása), in Hungarian (2022) pdf
18. Dénes Márton (ELTE): Monsky's theorem (Monsky tétel), in Hungarian (2022) pdf
19. Bálint Mogyorósi (ELTE): Quillen-Suslin Theorem (Quillen-Suslin Tétel), in Hungarian (2022) pdf
20. Vajk Szőri (ELTE): Hilbert's irreducibility theorem (Hilbert irreducibilitási tétele), in Hungarian (2022) pdf
21. Hoang Anh Tran (ELTE): The Robba ring, in English (2022) pdf
22. Ákos Borsányi (ELTE): Investigation of cyclotomic polynomials (Körosztási polinomok vizsgálata), in Hungarian (2023) 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
3. Tur Sofian (from ENS Lyon to ELTE): reading "Perfectoid spaces with a view towards multivariate $$(\varphi,\Gamma)$$-modules" (May-June 2020, remotely/online due to COVID-19)
4. Tubach Swann (from ENS Lyon to ELTE): reading "Multivariable $$(\varphi,\Gamma)$$-modules and trianguline representations" (May-June 2020, remotely/online due to COVID-19)