# Workshop on Algebraic Number Theory - 17 Dec 2012

#### Location: School of Mathematics, Statistics and Operations Research, Victoria University of Wellington.

Even a casual observer cannot fail to be impressed by the dizzying speed and scope of the development of algebraic number theory for the past hundred years. Yet our search for mathematical truth and beauty always leaves us wanting more. In this workshop, we will discuss some of the most amazing developments in Iwasawa theory and related areas. We hope that this event will lead to deeper appreciation of many old and new results, and hopefully another breakthrough.

Prominent mathematicians who are leading their respective fields will gather in Victoria University in December, 2012 for Workshop in Number Theory. The honors and awards given to the speakers in the past include

AMS Cole Prize

Guggenheim Fellowship

Humboldt Award

The central coordinatorship of the European Marie Curie RTN Network

Membership of the Scientific Advisory Board of Max Planck Institute for Mathematics

Simion Stoilow Prize of the Romanian Academy of Science

Fellowship of Max-Planck institute for Mathematics at Bonn

and also, our speakers include editors of three mathematical journals by my quick count, including the prestigious Journal of AMS.

It is also noteworthy that Karl Rubin, Massimo Bertolini, Cristian Popescu, and Daniel Delbourgo will all attend the workshop and give presentations. They are leading experts in what is called "Iwasawa Theory", a major subfield of number theory. I am close to their field, and familiar with their recent and past works. I expect that discussions and collaborations with them which will come out of this workshop will be fruitful.

Dr Byoung Du Kim

#### List of Participants and Speakers

1. Byoung Du Kim (Victoria University of Wellington)
2. Karl Rubin (the University of California, Irvine)
3. Cristian Popescu (the University of California, San Diego)
4. Hershy Kisilevsky (Concordia University)
5. Chang Heon Kim (Hanyang University)
6. Massimo Bertolini (University of Milan)
7. Daniel Delbourgo (Waikato University)
8. Mihran Papikian (Pennsylvania State University)
9. Daeyeol Jeon (Kongju National University)
10. Keisuke Arai
11. Shun Ohkubo (University of Tokyo)
12. Heiko Knospe (Cologne University of Applied Sciences)

### Karl Rubin - Higher rank Kolyvagin systems

Abstract: Both a rank-one Euler system and a rank-one Kolyvagin system consist of families of cohomology classes with appropriate properties and interrelationships. Given such an Euler system, Kolyvagin's derivative construction produces a rank-one Kolyvagin system, and a rank-one Kolyvagin system gives a bound on the size of a Selmer group. Ideas and conjectures of Perrin-Riou show that in some situations (for example, starting with an abelian variety of dimension r, or the global units in a totally real field of degree r) an Euler system is more naturally a collection of elements in the r-th exterior powers of cohomology groups. In this situation, Barry Mazur and I define a Kolyvagin system of rank r also to be a suitable collection of elements in r-th exterior powers, and we show how a Kolyvagin system of rank r bounds the size of the corresponding Selmer group.

### Chang Heon Kim - Families of elliptic curves over cubic number fields with prescribed torsion subgroups (joint work with Daeyeol Jeon and Yoonjin Lee)

Abstract: Eight years ago Jeon, Kim and Schweizer determined those finite groups which appear infinitely often as torsion groups of elliptic curves over cubic number fields. In this talk we will construct such elliptic curves and cubic number fields for each possible torsion group.

### Hershy Kisilevsky - Mordell-Weil groups of elliptic curves under field extension

Abstract: Let E be an elliptic curve defined over the rational field \Q with L-function L(E,s). We are interested in studying E(K)as K varies over finite extensions of \Q. Analytically this questions translates (under the Birch and Swinnerton-Dyer conjecture) into when L(E,1, \chi)=0 for Artin characters \chi. For [K:\Q]=2, there is an extensive literature on this question. We present our results and conjectures when [K:\Q]>2.

### Daniel Delbourgo - Congruences between Hasse-Weil L-functions

Abstract: I shall first explain some general conjectures in non-commutative
Iwasawa theory, which predict the existence of special elements in the
K-theory of an appropriate Iwasawa algebra. I'll then focus on a
specific d+1 dimensional Lie group, and discuss how the shape of its
K_1 suggests certain p-power congruences linking together Hasse-Weil
L-values, for each fixed elliptic curve.

### Byoung Du Kim - Iwasawa theory for elliptic curves for non-ordinary/supersingular primes over imaginary quadratic fields

Abstract: Often in Iwasawa theory, (for a fixed prime $p$) we want to relate some kind of $p$-Selmer groups (which can be roughly considered as the set of rational points plus the Shafarevich-Tate group) on the algebraic side and some kind of $p$-adic $L$-function (which can be thought of as a $p$-adic power series that incorporates the special values of an $L$-function) on the analytic side. It is well-known that the Iwasawa theoretic properties of the conventional Selmer groups and p-adic L-functions break down when the prime $p$ is non-ordinary/supersingular. Kobayashi and Pollack proposed the plus/minus Selmer groups and plus/minus $p$-adic$L$-functions respectively as alternatives, and it is known that they work well over the cyclotomic extensions of the rational number field $\mathbb Q$. By building upon their ideas, the works of Katz and Hida, and our previous works, we construct two-variable p-adic L-functions, and $\pm/\pm$-Selmer groups over the $\mathbbZ_p^2$-extension of imaginary quadratic fields where $p$ is non-ordinary/supersingular, and splits completely over the imaginary quadratic field. We will argue that these are good objects to study by illustrating their properties. We also present a conjecture in the spirit of the main conjecture of Iwasawa theory, which hypothetically connects the algebraic and analytic properties of elliptic curves by way of relating the two objects we construct.

### Keisuke Arai - Algebraic points on Shimura curves of $\Gamma_0(p)$-type

Abstract: We classify the characters associated to algebraic points on  Shimura curves of $\Gamma_0(p)$-type, and over number fields  not only quadratic fields but also fields of higher degree)  we show that there are few points on such a Shimura curve for every sufficiently large prime number $p$.

We also obtain an effective bound of such $p$.  This is an analogue of the study of rational points or points  over quadratic fields on the modular curve $X_0(p)$ by Mazur  and Momose.

### Massimo Bertolini - p-adic Rankin L-series and algebraic cycles

Abstract: I will report on recent work, in collaboration with Darmon and Prasanna, relating values of Rankin p-adic L-series to the p-adic Abel-Jacobi image of certain algebraic cycles. Arithmetic applications will be discussed.

### Mihran Papikian - Optimal quotients of Mumford curves and component groups

Abstract: Let $X$ be a Mumford curve. We say that an elliptic curve is an optimal quotient of $X$ is there is a finite morphism $X\to E$ such that the homomorphism $\pi: Jac(X)\to E$ induced by the Albanese functoriality has connected and reduced kernel. We consider the functorially induced map $\pi_\ast: \Phi_X\to \Phi_E$ on component groups of the Neron models of $Jac(X)$ and $E$. We show that in general this map need not be surjective, which answers negatively a question of Ribet and Takahashi. Using rigid-analytic techniques, we give some conditions under which $\pi_\ast$ is surjective, and discuss arithmetic applications to modular curves. This is a joint work with Joe Rabinoff.

### Shun Ohkubo - On differential modules associated to de Rham representations in the imperfect residue field case

Abstract: Let K be a CDVF of mixed characteristic (0,p), whose residue  field admits a finite p-basis. Denote the absolute Galois group of K by  G. For a given de Rham representation V of G, we will construct a differential module N_dR(V), which is a generalization of Laurent Berger 's N_dR(V) in the perfect residue field case. We also explain some  properties of this differential module.

### Daeyeol Jeon - Families of elliptic curves over quartic number fields with prescribed torsion subgroups (joint work with Chang Heon Kim and Yoonjin Lee)

Abstract: Jeon, Kim and Schweizer determined those finite groups which apprear infinitely often as torsion groups of elliptic curves over quartic number fields. In this talk we will construct infinite families of elliptic curves over quartic number fields for each possible torsion group.

### Cristian Popescu - Explicit models for Tate sequences and applications

Abstract: I will describe applications of my joint work with Greither in equivariant Iwasawa theory to the construction of explicit models for Tate sequences and its consequences for the theory of special values of Artin L-functions.

#### Workshop Timetable:

 Mon 17th Dec 10:00 - 11:30 Registration & Light Lunch 12:00 - 13:30 Weta Workshop* 14:00 - 16:00 Te Papa* 16:00 - late Free time
* Weta Workshop is the multi Academy Award winning special effects company that helped create movies like the Lord of the Rings, Avatar, District 9, King Kong and The Hobbit. They are currently at work on movies including The Adventures of Tintin. The Weta Cave is a unique visitor experience showcasing the creativity of Weta. Opened one year ago, it has hosted over 70,000 people from all over the world, including VIPs from TV, movies, gaming and sports, such as Guillermo del Toro (Director, Hellboy), John Stevenson (Director, Kung Fu Panda), and the All Blacks (New Zealand’s national rugby team).

* Te Papa is New Zealand’s national museum, renowned for being bicultural, scholarly, innovative, and fun. The collections span five areas: Art, History, Pacific, Māori, and Natural Environment. It provides exhibition highlights for first-time visitors, covering the story of Māori in New Zealand and the interactive exhibition OurSpace. Te Papa also offers a daily 60-minute tour – Introducing Te Papa for individuals and small groups. You can simply buy your ticket ($14) on the day at the Information Desk, Level 1. * Transport will be provided from outside the cotton building near “Wishbone” to take you to Weta Workshop and Te Papa. You will need to make your own arrangements back to the hotel.  Tue 18th Dec Wed 19th Dec Thur 20th Dec 10:00 - 10:25 Morning Tea 10:30 - 11:30 Karl Rubin Cristian Popescu Massimo Bertolini 11:45 - 12:45 Hershy Kisilevsky Mihran Papikian Keisuke Arai 13:00 - 14:25 Lunch 14:30 - 15:30 Chang Heon Kim Daeyeol Jeon Shun Ohkubo 15:35 - 15:55 Coffee Break 16:00 - 17:00 Daniel Delboourgo Byoung Du Kim --- 19:00 - late --- Dinner at Osteria Del Toro for all participants Dinner at Dockside for plenaries only All guests will be required to pay NZ$39 for their own dinner and drinks on Wed 19 Dec except the plenary speakers.

Note: during the workshop and dinner catering for vegetarians will be provided. If you have a dietary request please email Ping (Kelly) Shen.

 Fri 21 Dec 10:00 - 10:25 Refreshments 10:30 - 12.30 Workshop Discussion 12:30 Workshop Ends

#### Contact:

Byoung.Du.Kim@msor.vuw.ac.nz - Chair

Ginny Whatarau, Workshop Organising Committee, ginnyn@msor.vuw.ac.nz

Ping (Kelly) Shen, Workshop Organising Administrator, kelly.shen@msor.vuw.ac.nz

#### Accommodation

Mercure Wellington - 345 The Terrace - Te Aro, Wellington 6011, NZ - (04) 385 9829 - Email H1991-RE01@accor.com

There is a special rate of NZ$135 per night including breakfast. Note: There is a credit card surcharge paid in addition. #### Location & Workshop Venue The school of Mathematics, Statistics and Operations Research is located in the Cotton building, Kelburn campus of Victoria University. Please refer to the map of the Kelburn campus. The workshop will be held in the Cotton Building, Level 3, Room 350. #### Facts about New Zealand Time Zones New Zealand is one of the first places in the world to see the new day. In Summer, New Zealand uses 'Daylight Saving', with clocks put forward one hour to GMT +13 New Zealand Currency New Zealand's unit of currency is the dollar (NZ$). There is no restriction on the amount of foreign currency that can be brought in or taken out of New Zealand. However, every person who carries more than NZ\$10,000 in cash in or out of New Zealand is required to complete a Border Cash Report.

Foreign currency can easily be exchanged at banks, some hotels and Bureau de Change kiosks, which are found at international airports and most city centres.

All major credit cards can be used in New Zealand, with Visa and MasterCard accepted most widely, followed by American Express and Diners Club.

Driving in New Zealand

If you're used to driving in the city, you should take care when driving on New Zealand's open country roads. We have a good motorway system but weather extremes, the terrain and narrow secondary roads and bridges require drivers to be very vigilant.

New Zealand Climate and Weather

New Zealand has mild temperatures, moderately high rainfall, and many hours of sunshine.

New Zealand has a largely temperate climate. The average New Zealand temperature decreases as you travel south. In summer (December – February), the average maximum temperature ranges between 20-30ºC (70-90°F). In most places you can wear shorts and a t-shirt or singlet during the day, adding a light jumper at night.

New Zealand weather can change unexpectedly. Be prepared for sudden changes in weather and temperature if you're going hiking or doing other outdoor activities.

Wellington

Situated at the southern end of the North Island, nestled between a sparkling harbour and rolling green hills, Wellington is New Zealand’s capital city. Lonely Planet named Wellington ‘the coolest little capital in the world’ (2011), and the city is renowned for its arts, culture and native beauty.

#### Acknowledgment:

This workshop is supported by the Royal Society of New Zealand and the School of Mathematics, Statistics and Operations Research - Victoria University, NZ

### Events.EventForm edit

 Name Workshop on Algebraic Number Theory WebPage StartDate 17 Dec 2012 EndDate 21 Dec 2012 Time Type Workshop
This topic: Events > Main > EventList > WorkshoponAlgebraicNumberTheory
Topic revision: 12 Dec 2012, kelly