Time | Speaker | Title | Resources | |
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09:00 to 10:00 | Haruzo Hida - I (UCLA, USA) |
Non-vanishing Modulo p of Values of a Modular Form at CM Points Abstract: LINK |
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16:00 to 17:00 | Samit Dasgupta - I (Duke Univ, USA) |
On the Brumer-Stark Conjecture (Lecture 1) The Brumer-Stark conjecture is a generalisation of the classical theorem of Stickelberger on annihilation of class groups of abelian CM fields. We will first see a formulation of the Brumer-Stark conjecture. A consequence of the Brumer-Stark conjecture is existence of special elements called the Brumer-Stark units. These units generate abelian CM extensions of totally real number fields. A conjecture of Gross known as "the tower of fields conjecture" takes the first step towards giving an explicit description of the Brumer-Stark units. A precise conjectural p-adic analytic formula for the Brumer-Stark units was given by Dasgupta. We will review statement of the tower of fields conjecture as well as the explicit p-adic analytic formula and describe a precise relationship between the two. Furthermore, we will give an application towards explicit class field theory. |
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17:30 to 18:30 | Arul Shankar - I (Univ of Toronto, Canada) |
Average rank of elliptic curves: overview of the results In this talk, I will describe the Goldfeld and Katz--Sarnak conjectures on the average rank of all elliptic curves, and the Poonen--Rains conjectures on the distribution of Selmer groups of elliptic curves. I will provide some intuitive reasons on why these conjectures are true, and how the two conjectures are related. Finally, I will detail recent progress towards these conjectures. This is joint work with Manjul Bhargava. |
Time | Speaker | Title | Resources | |
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09:00 to 10:00 | Matthias Flach - I (Caltech, USA) |
Special values of Motivic L-functions I We discuss some history of special value conjectures with Dedekind Zeta functions as the main example. We give some algebraic background on determinant functors and introduce the concept of the fundamental line due to Fontaine and Perrin-Riou. |
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10:30 to 11:30 | Yifeng Liu - I (Zhejiang University, China) |
Recent advances on Beilinson-Bloch-Kato conjecture In this mini series of lectures, we will introduce the recent advances on the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives coming from automorphic representations of unitary groups, obtained by Liu-Tian-Xiao-Zhang-Zhu. We will tour the main path of the proof, using U(4) \times U(5) as a guiding example. In particular, we will explain how different ingredients like geometry of integral modules of Shimura varieties and period integrals of automorphic forms are used in the proof. |
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14:30 to 15:30 | Haruzo Hida - II (UCLA, USA) |
Non-vanishing Modulo p of Values of a Modular Form at CM Points Abstract: LINK |
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16:00 to 17:00 | Samit Dasgupta - II (Duke Univ, USA) |
On the Brumer-Stark Conjecture (Lecture 2) The Brumer-Stark conjecture is a generalisation of the classical theorem of Stickelberger on annihilation of class groups of abelian CM fields. We will first see a formulation of the Brumer-Stark conjecture. A consequence of the Brumer-Stark conjecture is existence of special elements called the Brumer-Stark units. These units generate abelian CM extensions of totally real number fields. A conjecture of Gross known as "the tower of fields conjecture" takes the first step towards giving an explicit description of the Brumer-Stark units. A precise conjectural p-adic analytic formula for the Brumer-Stark units was given by Dasgupta. We will review statement of the tower of fields conjecture as well as the explicit p-adic analytic formula and describe a precise relationship between the two. Furthermore, we will give an application towards explicit class field theory. |
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17:30 to 18:30 | Antonio Lei - I (Univ of Ottawa/Laval Univ, Canada) |
Euler systems and Beilinson--Flach elements The first goal of this series of talks is to give a general introduction to the theory of Euler systems and provide certain motivations of the subject. In particular, we will review the definition and basic properties of Euler systems. We will study cyclotomic units as a first example of Euler systems. The second goal is to describe the construction of Beilinson--Flach elements and explain how they are used to define Euler systems for Rankin--Selberg convolutions of two modular forms. The third goal is to discuss arithmetic applications derived from Beilinson--Flach Euler systems, including Iwasawa main conjectures in both ordinary and non-ordinary settings. If time permits, we will also discuss some recent developments in the subject. |
Time | Speaker | Title | Resources | |
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09:00 to 10:00 | Matthias Flach - III (Caltech, USA) |
Special values of Zeta functions I This and the next talk cover joint work with B. Morin. We introduce a special value conjecture for the Zeta function of a proper, regular arithmetic scheme which has three equivalent formulations: In the style of the analytic class number formula, via an integral fundamental line and via Weil-Arakelov cohomology complexes. |
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10:30 to 11:30 | Francesc Castella - I (UC Santa Barbara, USA) |
On the Iwasawa theory of elliptic curves at Eisenstein primes The goal of this mini-course is to explain some recent advances on the Iwasawa theory of rational elliptic curves at Eisenstein primes p, and their applications. Topics include the extension of the method of Greenberg-Vatsal to the anticyclotomic setting (joint with Grossi, Lee, and Skinner), and the related proof of Mazur's (cyclotomic) main conjecture for Eisenstein primes p (joint with Grossi and Skinner). |
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11:45 to 12:45 | Haruzo Hida - III (UCLA, USA) |
Non-vanishing Modulo p of Values of a Modular Form at CM Points Abstract: LINK |
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14:30 to 15:30 | Mahesh Kakde - II (IISc, India) |
On the Brumer-Stark Conjecture (Lecture 4) The main aim of these lectures will be to give a proof of the Brumer-Stark conjecture. We will start with a number of refinements of the Brumer-Stark conjecture. To this end we will see the definition of Ritter-Weiss module. We then formulate refinements of Brumer-Stark conjecture in terms of these Ritter-Weiss modules. We then describe our refinements of Ribet’s method to prove strengthening of the Brumer-Stark conjecture. |
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17:30 to 18:30 | Arul Shankar - III (Univ of Toronto, Canada) |
Geometry-of-numbers techniques in arithmetic statistics Let V be a coregular representation of a reductive group G. We will impose a natural height function on V(R) and see how analytic techniques can be used to count G(Z)-orbits on V(Z) having bounded height. We will apply this method to the representations discussed in the previous lecture, and obtain an explicit bound on the average rank of elliptic curves. This is joint work with Manjul Bhargava. |
Time | Speaker | Title | Resources | |
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09:00 to 10:00 | Haruzo Hida (UCLA, USA) |
Non-vanishing Modulo p of Values of a Modular Form at CM Points (Lecture 4) Abstract: LINK |
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10:30 to 11:30 | Francesc Castella (UC Santa Barbara, USA) |
On the Iwasawa theory of elliptic curves at Eisenstein primes (Lecture 3) The goal of this mini-course is to explain some recent advances on the Iwasawa theory of rational elliptic curves at Eisenstein primes p, and their applications. Topics include the extension of the method of Greenberg-Vatsal to the anticyclotomic setting (joint with Grossi, Lee, and Skinner), and the related proof of Mazur's (cyclotomic) main conjecture for Eisenstein primes p (joint with Grossi and Skinner). |
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11:45 to 12:15 | Pratiksha Shingavekar (IIT Madras, India) |
3 Selmer group, ideal class group and cube sum problem Let E be an elliptic curve over \mathbb{Q} with a rational 3-isogeny \phi. We discuss the bounds on the \phi-Selmer group of E in terms of ideal class groups of certain number fields. We also give some applications of these bounds towards Sylvester's conjecture on the cube sum problem. |