Link to 1st FGC-IPM Joint Number Theory Meeting website at FGC

Link to 2nd FGC-IPM Joint Mini Workshop “Arithmetic of Local and Global Fields”

Link to 3rd FGC-IPM Joint Number Theory Workshop "Around Langlands"

Link to IPM School of Mathematics

Link to IPM

IPM Number Theory website

Link to HRI
 

Starting on 31.01.2021, Harish-Chandra Research Institute (HRI) has joined in the FGC-IPM number theory network. From now on the number theory seminars being conducted will be named FGC-HRI-IPM Number theory seminars.
FGC and IPM jointly organized a Number Theory Meeting on March 15 - 17, 2021. There is a link above to this event which formed the genesis of biweekly FGC-IPM number theory seminars.
Number Theory meetings will reconvene for Fall 2021 semester starting on September 28 2021.

2025 -2026

Spring Term

Speaker: Özge Ülkem (Academia Sinica, Taipei)

Title: Drinfeld's Elliptic Sheaves and Generalizations

Time: February 25, Wednesday, 2026 at 13:00 (Istanbul LT), 13:30 (Tehran LT), 15:30 (Allahabad LT)--NOTE THE UNUSUAL TIME!

Place: Online

Meeting ID: 997 1547 1656
Passcode: 848084

Abstract: In this talk, we will explore the area of function field arithmetic, with a focus on Drinfeld's elliptic sheaves and their generalizations, as well as analogies to the number field setting. Drinfeld modules, introduced in 1974 as analogues of elliptic curves in the function field setting, play a central role in this context. To establish a Langlands correspondence, Drinfeld studied moduli spaces of elliptic sheaves, or equivalently, shtukas. After a brief introduction to the function field framework, we will examine some well-known generalizations of elliptic sheaves, concentrating on generalized D-elliptic sheaves and presenting results on their moduli spaces. In the final part of the talk, we will explore the connections between (generalized) shtukas and (generalized) elliptic sheaves.

Speaker: Oleg German (National Research University Higher School of Economics, Moscow)

Title: On the transference principle in Diophantine approximation

Time: February 11, Wednesday, 2026 at 22:00 (Istanbul and Moscow LT), 22:30 (Tehran LT), Feb. 12 at 00:30 (Allahabad LT)--NOTE THE UNUSUAL TIME!

Place: Online

Meeting ID: 997 1547 1656
Passcode: 848084

Abstract: In 1842, Dirichlet published his famous theorem which became the foundation of Diophantine approximation. The phenomenon he found inspired Liouville to study how well algebraic numbers can be approximated by rationals, and thus, to come up with a method of constructing transcendental numbers explicitly. The development of these ideas led to the concepts of irrationality measure and transcendence measure. Thanks to Minkowski, it became clear that many problems arising in the theory of Diophantine approximation could be addressed quite effectively using the tools of geometry of numbers. In particular, the geometric approach naturally offers a wide variety of multidimensional analogues of the concept of irrationality measure — so called Diophantine exponents. In the talk, we will discuss various Diophantine exponents and the geometry that arises when studying them. We will pay special attention to the phenomenon discovered by Khintchine, which he called the transference principle.

Fall Term

Speaker: Pierre Lochak (Sorbonne Université)

Title: A historical introduction to Grothendieck-Teichmüller theory

Time: January 28, Wednesday, 2026 at 19:00 (Istanbul LT), 19:30 (Tehran LT), 21:30 (Allahabad LT)

Place: Online

Meeting ID: 997 1547 1656
Passcode: 848084

Abstract: Starting with the statement of Belyi's theorem, I will explain how Grothendieck-Teichmüller theory was born, then move to a (necessarily incomplete) exposition of its main tenets, the already existing results and the main conjectures.

Speaker: Kübra Benli (Boğaziçi University)

Title: Sums of proper divisors with missing digits

Time: January 14, Wednesday, 2026 at 19:00 (Istanbul LT), 19:30 (Tehran LT), 21:30 (Allahabad LT)

Place: Online

Meeting ID: 997 1547 1656
Passcode: 848084

Abstract: Let $s(n)$ denote the sum of proper divisors of a positive integer $n$. In 1992, Erd\H{o}s, Granville, Pomerance, and Spiro conjectured that if $\mathcal{A}$ is a set of integers with asymptotic density zero, then the preimage set $s^{-1}(\mathcal{A})$ also has asymptotic density zero. In this talk, we will discuss the verification of this conjecture when $\mathcal{A}$ is the set of integers with missing digits (also known as ellipsephic integers) by giving a quantitative estimate on the size of the set $s^{-1}(\mathcal{A})$. This talk is based on joint work with Giulia Cesana, C'{e}cile Dartyge, Charlotte Dombrowsky, Paul Pollack, and Lola Thompson.

Speaker: Lejla Smajlovic (University of Sarajevo)

Title: On some nonholomorphic automorphic forms, their inner products and generating functions

Time: December 17, Wednesday, 2025 at 17:00 (Istanbul LT), 17:30 (Tehran LT), 19:30 (Allahabad LT)

Place: Online

Meeting ID: 997 1547 1656
Passcode: 848084

Abstract: In this talk we focus on the following three automorphic forms on a Fuchsian group of the first kind with at least one cusp: the Eisenstein series, the Niebur–Poincaré series associated to the cusp at infinity, and the resolvent kernel/Green's function. We discuss how these functions can be viewed as building blocks for describing log-norms of certain meromorphic functions in terms of their divisors and derive a generalization of a Rorlich–Jensen type formula, based on an evaluation of the Petersson inner product of the Niebur–Poincaré series with a suitably regularized Green's function. We then turn our attention to the generating functions of the Niebur–Poincaré series and its derivative at s=1. Both functions depend on two variables in the upper half-plane. We prove that, for any Fuchsian group of the first kind, the generating function of the Niebur–Poincaré series in each variable is a polar harmonic Maass form of a certain weight, describe its polar part, and explain how it serves as a building block for describing weight two meromorphic modular forms in terms of their divisors. Moreover, we show that the generating function of the derivative of the Niebur–Poincaré series at s=1 can be expressed—up to a certain function appearing in the Kronecker limit formula—as a derivative of an automorphic kernel associated with a new point-pair invariant expressed in terms of the Rogers dilogarithm. This talk is based on joint work with Kathrin Bringmann, James Cogdell, and Jay Jorgenson.

Speaker: James Borger (Australian National University)

Title: Scheme theory over semirings

Time: December 3, Wednesday, 2025 at 12:00 (Istanbul LT)

Place: Online

Meeting ID: 997 1547 1656
Passcode: 848084

Abstract: Usual scheme theory can be viewed as the syntactic theory of polynomial equations with coefficients in a ring, most importantly the ring of integers. But none of its most fundamental ingredients, such as faithfully flat descent, require subtraction. So we can set up a scheme theory over semirings (``rings but possibly without additive inverses’’, such as the non-negative integers or reals), thus bringing positivity in to the foundations of scheme theory. It is then reasonable to view non-negativity as integrality at the infinite place, the Boolean semiring as the residue field there, and the non-negative reals as the completion. In this talk, I'll discuss some recent developments in module theory over semirings. While the classical definitions of ``vector bundle'' are not all equivalent over semirings, the classical definitions of ``line bundle'' are all equivalent, which allows us to define Picard groups and Picard stacks. The narrow class group of a number field can be recovered as the reflexive class group of the semiring of its totally nonnegative integers, i.e. the arithmetic compactification of the spectrum of the ring of integers. This gives a scheme-theoretic definition of the narrow class group, as was done for the ordinary class group a long time ago. This is based mostly on arXiv:2405.18645, which is joint work with Jaiung Jun, and also on forthcoming paper with Johan de Jong and Ivan Zelich.

Speaker: Annette Huber-Klawitter (Universität Freiburg)

Title: 1-Motives and transcendence

Time: November 5, Wednesday, 2025 at 17:00 (Istanbul LT)

Place: Online

Meeting ID: 997 1547 1656
Passcode: 848084

Abstract: 1-Periods are complex numbers defined by integrating algebraic differential forms over paths (on algebraic varieties) with algebraic end points. The set contains many interesting numbers like log(2) or π that have been studied intensely in transcendence theorem. By the linear version of the Period Conjecture (a theorem of Wüstholz and myself in this case), all relations between them are described in terms of 1-motives. In this expository talk, we will explain this result and give a couple of examples.

Speaker: Rajan Mehta (Smith College)

Title: Coherent 2D span-valued TQFTs

Time: October 28, Tuesday, 2025 at 18:00 (Istanbul LT)

Place: Online

Meeting ID: 935 6390 4955
Passcode: 699568

Abstract: It is well-known that 2D TQFTs taking values in a category correspond to commutative Frobenius objects in that category. If the target category is the category of spans, it is reasonable to ask if there is a lift to a coherent structure (i.e. a Frobenius pseudomonoid) in the bicategory of spans. Such structures can be neatly described by simplicial sets, equipped with some additional symmetric group actions, and satisfying some conditions known as the "2-Segal conditions". I'll describe this correspondence as well as a construction that produces examples from any commutative monoid. This is joint work with Sophia Marx, building on earlier work with Ivan Contreras and Walker Stern.

Speaker: Ferruh Özbudak (Sabancı University)

Title: Some results on covering radius of codes

Time: October 22, Wednesday, 2025 at 17:00 (Istanbul LT)

Place: Online

Meeting ID: 997 1547 1656
Passcode: 848084

Abstract: The covering radius is an important parameter in coding theory. In this talk, we present several results concerning the covering radius of various classes of codes, obtained using techniques involving algebraic curves over finite fields. Connections to algebra, number theory, and geometry will also be discussed.

Speaker: Kirti Joshi (Univ. of Arizona)

Title: Deformations of arithmetic of number fields and the abc-conjecture

Time: October 9, Thursday, 2025 at 19:00 (Istanbul LT); at 21:30 (Allahabad LT); at 19:30 (Tehran LT) (Note the unusual date & time)

Place: Online
Meeting ID: 997 1547 1656
Passcode: 848084I

Abstract: In this lecture I will provide an accessible overview of my recent work on deformation of arithmetic of number fields (as suggested by Shinichi Mochizuki) and its relationship to the abc-conjecture following Mochizuki's strategy for its proof.

Speaker: Shinichi Kobayashi (Kyushu University, Japonya)

Title: p-parity conjecture and its local analogue

Time: October 03, 2025 at 16:30

Place: TB130 (Boğaziçi University, South Campus)

Abstract: For a geometric global p-adic Galois representation V, the p-parity conjecture asserts the relation between the parity of the Selmer rank and the root number attached to V. In the case where V is the p-adic Tate module of an elliptic curve, this may be viewed as a mod 2 analogue of the Birch and Swinnerton-Dyer conjecture, conditional on the finiteness of the Tate–Shafarevich group. In this talk, I will survey the conjecture, as well as a local analogue for rank-two representations. This is joint work with A. Burungale, K. Nakamura, and K. Ota.

Speaker: Kazım Büyükboduk (University College-Dublin)

Time: October 03, 2025 at 15:00.

Place: TB130 (Boğaziçi University, South Campus)

Title: Rational points on elliptic curves and their p-adic analytic construction

Abstract: The negative answer to Hilbert's 10th problem tells us that determining whether or not an algebraic variety should carry any rational points is (literally!) impossibly hard. The same problem, even for curves, is difficult: For elliptic curves, this is the subject of the celebrated Birch and Swinnerton-Dyer conjecture. I will survey recent results on this problem, and explain briefly an explicit p-adic analytic construction of rational points of infinite order on elliptic curves of analytic rank one (resolving a conjecture of Perrin-Riou). These final bits of the talk will be a report on joint works with Rob Pollack & Shu Sasaki, and with Denis Benois.

2024 -2025

Fall Term

Speaker: Ali Partofard (IPM)
Title: Prismatic higher displays
Time: 09 October 2024 at 17:00
Abstract: In this talk, we define the notion of (G,μ)-prismatic displays over quasi-syntomic rings. In the case of GL_n, prismatic displays are the same as admissible Dieudonné modules defined by Anschutz and Lebras and therefore classify p-divisible groups. We discuss the deformation theory for prismatic displays and their relation with the points of an integral Shimura variety over a quasi-syntomic ring.

Speaker: Ahmad El-Guindy (Cairo Univ.)
Title: Some ℓ-adic properties of modular forms with quadratic nebentypus and ℓ-regular partition congruences
Time: 24 October 2024 at 17:00 (NOTE THE UNUSUAL DATE! It will be on Thursday!)
Abstract: In this talk, we discuss a framework for studying ℓ-regular partitions by defining a sequence of modular forms of level ℓ and quadratic character which encode the ℓ-adic behavior of the so-called ℓ-regular partitions. We show that this sequence is congruent modulo increasing powers of ℓ to level 1 modular forms of increasing weights. We then prove that certain modules generated by our sequence are isomorphic to certain subspaces of level 1 cusp forms of weight independent of the power of ℓ, leading to a uniform bound on the ranks of those modules and consequently to ℓ-adic relations between ℓ-regular partition values. This generalizes earlier work of Folsom, Kent and Ono on the partition function, where the relevant forms had no nebentypus, and is joint work with Mostafa Ghazy.

Speaker: Erman Işık (Univ. Ottowa)
Title: The growth of Tate-Shafarevich groups of p-supersingular elliptic curves over anticyclotomic ℤ_p- extensions at inert primes
Time: 06 November 2024
Abstract: In this talk, we will discuss the asymptotic growth of both the Mordell-Weil ranks and the Tate– Shafarevich groups for an elliptic curve E defined over the rational numbers, focusing on its behaviour along the anticyclotomic ℤ_p-extension of an imaginary quadratic K. Here, p is a prime at which E has good supersingular reduction and is inert in K. We will review the definitions and properties of the plus and minus Selmer groups from Iwasawa theory and discuss how these groups can be used to derive arithmetic information about the elliptic curve.

Speaker:  Amina Abdurrahman (IHES)
Title: A formula for symplectic L-functions and Reidemeister torsion
Abstract: We give a global cohomological formula for the central value of the L-function of a symplectic representation on a curve up to squares. The proof relies crucially on a similar formula for the Reidemeister torsion of 3-manifolds together with a symplectic local system. We sketch both analogous arithmetic and topological pictures. This is based on joint work with A. Venkatesh.

Speaker:  Andrea Ferragutti (Univ. di Torino)
Title: Frobenius and settled elements in iterated Galois extensions
Time: 04 December 2024
Abstract: Understanding Frobenius elements in iterated Galois extensions is a major goal in arithmetic dynamics. In 2012 Boston and Jones conjectured that any quadratic polynomial f over a finite field that is different from x^2 is settled, namely the weighted proportion of f-stable factors in the factorization of the n-th iterate of f tends to 1 as n tends to infinity. This can be rephrased in terms of Frobenius elements: given a quadratic polynomial f over a number field K, an element \alpha in K and the extension K_\infty generated by all the f^n-preimages of \alpha, the Frobenius elements of unramified primes in K_\infty are settled. In this talk, we will explain how to construct uncountably many non-conjugate settled elements that cannot be the Frobenius of any ramified or unramified prime, for any quadratic polynomial. The key result is a description of the critical orbit modulo squares for quadratic polynomials over local fields.

This is joint work with Carlo Pagano.

Speaker:  Tomos Parry (Bilkent University)
Title: Primes in arithmetic progressions on average
Time: 18 December 2024
Abstract

01 January 2025: No seminar talk (Happy New Year!!!)

Speaker: Ahmet Güloğlu (Bilkent University)
Title: Non-vanishing of L-functions at the central point
Time: 29 January 2025 at 17:00
Abstract: I will talk about two methods used to derive non-vanishing results for a family of L-functions; the one-level density and the moments of L-functions. I will mention what these methods are and how they are used to get non-vanishing.

Speaker: Keerthi Madapusi (Boston College)
Title: A new approach to p-Hecke correspondences and Rapoport-Zink spaces
Time: 26 February 2025 at 17:00 
Abstract: We will present a new notion of isogeny between ‘p-divisible groups with additional structure’ that employs the cohomological stacks of Drinfeld and Bhatt-Lurie—-in particular the theory of apertures developed in prior work with Gardner—-and combines it with some invariant theoretic tools familiar to the geometric Langlands and representation theory community, namely the Vinberg monoid and the wonderful compactification. This gives a uniform construction of p-Hecke correspondeces and Rapoport-Zink spaces associated with unramified local Shimura data. In particular, we give the first general construction of RZ spaces associated with exceptional groups. This work is joint with Si Ying Lee.

Speaker: Sebastian Bartling (Universität Duisburg–Essen)
Title: Rapoport-Zink spaces and close p-adic fields
Time: 07 May 2025 at 17:00 Istanbul local time/17:30 Tehran local time/19:30 Allahabad local time
Abstract: Rapoport-Zink spaces are moduli spaces of p-divisible groups (with extra structure). These are p-adic analogues of integral models of Shimura varieties. Their function field versions were introduced by Hartl-Viehmann. I want to explain a construction approximating Hartl-Viehmann spaces via Rapoport-Zink spaces using the philosophy of close p-adic fields. If time permits I want to sketch how one may use this construction to deduce the Arithmetic Fundamental Lemma in the function field case. This is joint work, partly in progress, with Andreas Mihatsch.
 

FGC-IPM Joint Number Theory Webinars for 2023 

Speaker: Yen-Tsung Chen
Title: On the Partial Derivatives of Drinfeld Modular Forms of arbitrary rank
Time: 16 January

Speaker: Ilker Inam
Title: Fast Computation of Half Integral Weight Modular Forms
Time: 15 March

Speaker: Alia Hamieh
Title: Moments of L-functions and Mean Values of Long Drichlet Polynomials
Time: 5 April

Speaker: Asgar Jamneshan
Title: On Inverse Theorems and Conjectures in Ergodic Theory
Time: 19 April

Speaker: Rahul Gapta
Title: Tame Class Field Theory
Time: 3 May

Speaker: Cristiana Bertolin
Title: Periods of 1-motives and their polynomial relations
Time: 17 May

Speaker: Carlo Pagano
Title: Abelian Arboreal Representations
Time: 31 May

Speaker: Olga Lukina
Title: Weyl Groups in Cantor Dynamics
Time: 14 June

Speaker: Turku Ozlum Celik
Title: Algebraic Curves From Polygons
Time: 11 October

Speaker: Berkay Kebeci
Title: Mixed Tate Motives and Aomoto Polylogarithms
Time: 25 October

Speaker: Farzad Aryan
Title: Cancellations in Character Sums and the Vinogradov Conjecture
Time: 8 November

Speaker: Soumya Sankar
Title: Counting Points on Stacks and Elliptic Curves with a rational N-isogeny
Time: 22 November
Video link

Speaker: Emre Sertoz
Title: Computing Linear Relations between Univariate integrals
Time: 6 December

FGC-IPM Joint Number Theory Webinars for Winter-Spring 2022 Semester

Speaker: Somnath Jha (IIT Kanpur, India)
Title: Fine Selmer group of elliptic curves over global fields
Time:Tuesday May 31, 2022, 15:00 Istanbul | 16:30 Tehran | 17:30 Allahabad.
Zoom link
Passcode: 362880


Speaker: Hamza Yesilyurt (Bilkent University, Ankara, Turkey)
Title: A Modular Equation of Degree 61
Time: Tuesday May17, 2022, 15:00 Istanbul | 16:30 Tehran | 17:30 Allahabad.
Video link


Speaker: Chirantan Chowdhury (University of Duisburg-Essen)
Title: Motivic Homotopy Theory of Algebraic Stacks
Time: Tuesday April 19, 2022, 15:00 Istanbul; 16:30 Tehran; 17:30 Allahabad.
Video link


Speaker: Farzad Aryan (Göttingen University)
Title: On the Riemann Zeta Function
Time: Tuesday April 5, 2022, 15:00 İstanbul time 16:30 Tehran time 17:30 Allahabad time.
Video link


Speaker: Semih Özlem (Yeditepe University)
Title: On the motivic Galois group of a number field
Time: Tuesday March 15, 2022, 16:00 İstanbul time 16:30 Tehran time 18:30 Allahabad time.
Video link
Seminar Notes

Speaker: Gonzalez-Aviles (Universidad de La Serena, Chile)
Title: Totally singular algebraic groups
Time: Tuesday February 1, 2022, 16:30 Tehran time; 16:00 Istanbul time.
Slides
Video link
 

FGC-IPM Joint Number Theory Webinars for Fall 2021 Semester

First Meeting of Fall 2021 semester
Speaker: Professor Ramin Takloo-Bighash (University of Illinois at Chicago)
Title: TBA
Time: Sep 28, 2021 07:30 PM Tehran
Video link

Second Meeting of Fall 2021 semester
Speaker: Professor Mark Kisin (Harvard University)
Title: Essential dimension via prismatic cohomology
Time: Tuesday October 12, 2021, 17:30 Tehran time (17:00 Istanbul-10:00 AM Boston)
Video link

Third Meeting of Fall 2021 semester
Speaker: Reza Taleb (Shahid Beheshti University)
Title: The Coates-Sinnott Conjecture
Time: Tuesday, October 26, 2021 17:30 Tehran time (17:00 Istanbul)
Slides
Video link

Speaker: Shabnam Akhatri (University of Oregon)
Title: TBA
Time: Tuesday, November 9, 2021 17:30 Tehran time (17:00 Istanbul)
Slides
Video link

Speaker: Ali Mohammadi (IPM)
Title: Bounds on point-conic incidences over finite fields and applications
Time: Tuesday, November 23, 2021 15:00 Tehran time (14:30 Istanbul)
Slides
Video link

Speaker: Andrzej Dabrowski (University of Szczecin)
Title: On a class of generalized Fermat equations of signature (2,2n,3)
Time: Tuesday, December 7, 2021 15:00 Tehran time (14:30 Istanbul)
Video link
Seminar Notes 1
Seminar Notes 2

Speaker: Amir Ghadermarzi (University of Tehran)
Title: TBA
Time: Tuesday, December 21, 2021 17:30 Tehran time (17:00 Istanbul)
https://us06web.zoom.us/j/9086116889?pwd=WGRFOGZWZ1FOMXJrcWpJMWFqUFIvQT09
Meeting ID: 908 611 6889
Passcode: 362880

Speaker: Fatma Cicek (IIT Gandhinagar)
Title: Selberg’s Central Limit Theorem
Time: Tuesday, January 4, 2022 17:30 Tehran time (17:00 Istanbul)
Slides

Information about previous meetings, Spring 2021

These seminars will take place on every other Tuesday starting on Tuesday April 27, 2021 at 16:00 Istanbul local time and 17:30 Tehran local time over zoom, allowing for exceptions in scheduling. See the table below for details.
Below is the needed information to participate in these seminars:
Zoom Meeting Link:
https://zoom.us/j/9299700405?pwd=QXZJYTVkeHpJWDE4SGVTbkVzZmJxQT09
Meeting ID: 929 970 0405
Passcode: 210609
Next meeting: June 29, 2021 16:00 Istanbul local time, 17:30 Tehran local time
Speaker: Emre Alkan, Koç Üniversitesi
Title: A history of Asymptotic Formulas in Prime Number Theory
The seminar lasts about an hour, (maximum two hours).
The client for zoom meetings can be downloaded at https://zoom.us/download
If you have any questions, suggestions, feedback regarding the seminars please email the organizers at
fgc.ipm.math at gmail.com
Titles and abstracts of the first seven talks are at the bottom of the page.
Talk 1 | Talk 2 | Talk 3 | Talk 4 | Talk 5 | Talk 6 | Talk 7
Slides and videos of the seminars are below:
Talk 1: Polya and Pre-Polya Groups in Dihedral Number Fields, Abbas Maarefparvar.
Slides | Video
Talk 2: Shimura varieties modulo p with many compact factors, Oliver Bueltel.
Slides | Video
Talk 3: SOLVING FERMAT TYPE EQUATIONS VIA MODULAR APPROACH, Yasemin Kara.
Slides
Talk 4: Étale construction of critical p-adic L-functions and heights on thick Selmer groups, Kazım Büyükboduk.
Slides | Video
Talk 5: Effective height bounds for odd-degree totally real points on some curves, Levent Alpöge, Columbia University.
Slides | Video | Video link at IPM
Talk 6: A history of Asymptotic Formulas in Prime Number Theory, Emre Alkan, Koç Üniversitesi.
Slides | Video
Talk 7: On Drinfeld modular forms of higher rank and quasi-periodic functions, Oğuz Gezmiş, National Tsing Hua University.
Slides | Video

List of Seminars

FGC-IPM Number Theory Seminars 2021

Titles and Abstracts of the first seven talks

Talk 1

Speaker: Abbas Maarefparvar, School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran.
Email address: a.marefparvar@ipm.ir
Seminar Date: April 27, 2021
Title: PÓLYA AND PRE-PÓLYA GROUPS IN DIHEDRAL NUMBER FIELDS
Abstract: For a number field K with the ideal class group Cl(K), Pólya group of K is the subgroup Po(K) of Cl(K) generated by the classes of Ostrowski ideals Π _q(K) , where q ≥ 1 is a prime power integer and Π _q(K) denotes the product of all maximal ideals of K with norm q. K is called a Pólya field, Whenever Po(K) is trivial. Pólya fields are a generalization of PID (class number one) number fields, and classically they are defined in terms of regular bases for rings of integer valued polynomials due to George Pólya. For Galois number fields K, investigating on Pólya-ness can be expressable in terms of the action of the Galois group on the ideal class group: Po(K) and the subgroup of Cl(K) generated by the strongly ambiguous ideal classes coincide. In particular, Zantema (whose paper is a great contribution in this subject) showed that in the Galois case, Pólya groups are controllable part of ideal class groups throughout Galois cohomology and ramification. Beside, investigating on Pólya groups in the non-Galois number fields (the more difficult situation), Chabert introduced the notion of pre-Pólya group Po(-) _nr , which is a generalization of the pre-Pólya condition, duo to Zantema. The first part of my talk would be about some results of a joint work with Ali Rajaei, where using Zantema’s result and the arithmetic in ramification theory, we found some results on Pólya groups of dihedral extensions of ℚ of order 2l, for l an odd prime. In the second part, I’ll talk about my recently results on the pre-Pólya group of a D_n-field K, for n ≥ 4 an even integer, where D_n denotes the dihedral group of order 2n.

Talk 2

Speaker: Oliver Bueltel, Universität Duisburg-Essen
Seminar Date: May 11, 2021
Title: Shimura varieties modulo p with many compact factors
Abstract: We give several new moduli interpretations of the special fibres of several Shimura varieties over certain prime numbers. As a corollary we obtain, that for every prescribed odd characteristic p every bounded symmetric domain possesses quotients by arithmetic subgroups, whose models have good reduction at a prime divisor of p.

Talk 3

Speaker: Yasemin Kara, Boğaziçi Üniversitesi (Bosphorus University)
Seminar Date: May 25, 2021
Title: SOLVING FERMAT TYPE EQUATIONS VIA MODULAR APPROACH
Abstract: Recent work of Freitas and Siksek showed that an asymptotic version of Fermat’s Last Theorem (FLT) holds for many totally real fields. This result was extended by Deconinck to the generalized Fermat equation of the form Ax^p + By^p + Cz^p = 0, where A, B, C are odd integers belonging to a totally real field. Later Sengun and Siksek showed that the asymptotic FLT holds over number fields assuming standard modularity conjectures. Combining their techniques we* show that the generalized Fermat’s Last Theorem (GFLT) holds over number fields asymptotically assuming the standard conjectures. We also give three results which show the existence of families of number fields on which asymptotic versions of FLT or GFLT hold. In particular, we prove that the asymptotic GFLT holds for a set of imaginary quadratic number fields of density 5/6. In a different work, we** show that for a totally real number field K with narrow class number one, the Fermat type equation x^p + y^p = z² does not have certain type of solutions in the ring of integers of K for any prime exponent p > B_K where B_K is a constant depending only on K. *joint work with E. Özman ** joint work with E.Isik, E. Özman

Talk 4

Speaker: Kazım Büyükboduk, University College Dublin
Seminar Date: June 8, 2021
Title: Étale construction of critical p-adic L-functions and heights on thick Selmer groups
Abstract: In joint work with Denis Benois, we give an étale construction of Bellaïche's p-adic L-functions about θ-critical points on the Coleman–Mazur eigencurve. I will discuss applications of this construction towards p-adic Gross–Zagier formulae in terms of p-adic heights on what we call the thick Selmer groups, which come attached to the infinitesimal deformation at the said θ-critical point along the eigencurve. Besides our interpolation of the Beilinson–Kato elements about this point (which rests upon the overconvergent étale cohomology of Andreatta–Iovita–Stevens), the key input to prove the interpolative properties of our p-adic L-function is a new p-adic Hodge-theoretic "eigenspace-transition via differentiation" principle. Other notes: For the interested: parts of this work have been announced as arXiv:2008.12536.

Talk 5

Speaker: Levent Alpöge, Columbia University
Seminar Date: June 22, 2021
Title: Effective height bounds for odd-degree totally real points on some curves.
Abstract: We use potential modularity theorems to prove, for a class of smooth projective hyperbolic curves, effective height bounds for all rational points on such curves which are defined over an odd-degree totally real field. (Over such fields and for such curves this amounts to an unconditional effectivization of Faltings’ theorem.) We do this by proving effective height bounds for S-integral K-points on Hilbert modular varieties when K is totally real of odd degree (a familiar hypothesis from the theory of Hilbert modular forms), and then deducing a height bound for rational points on complete curves inside such varieties. The curves C _t : x⁶ + 4y³ = t² (with t a nonzero totally real algebraic number of odd degree, e.g. t = 1) are examples of curves to which this method applies.

Talk 6

Speaker: Emre Alkan, Koç Üniversitesi
Seminar Date: June 29, 2021
Title: A history of Asymptotic Formulas in Prime Number Theory
Abstract: We will present a survey of asymptotic formulas for prime counting functions. Time permitting, we discuss some recent results on this topic from a function theoretic point of view together with historical remarks.

Talk 7

Speaker: Oğuz Gezmiş, National Tsing Hua University
Seminar Date:July 6, 2021
Title: On Drinfeld modular forms of higher rank and quasi-periodic functions
Abstract: In 1980s, David Goss introduced Drinfeld modular forms in the rank two case and revealed several similarities between them and the classical modular forms as well as giving a recipe for how to construct them for the higher rank setting. Later on their higher rank generalization was studied extensively by Basson, Breuer, Gekeler and Pink. In this talk, we introduce a special function on the Drinfeld period domain in the higher rank setting which also generalizes the false Eisenstein series of Gekeler. We also introduce its functional equation, its relation with quasi-periodic functions of a Drinfeld module and the transcendence of its values at CM points. This is a joint work with Yen-Tsung Chen.