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.
Organiser:Dr Byoung Du Kim
List of Participants and Speakers
- Byoung Du Kim (Victoria University of Wellington)
- Karl Rubin (the University of California, Irvine)
- Cristian Popescu (the University of California, San Diego)
- Hershy Kisilevsky (Concordia University)
- Chang Heon Kim (Hanyang University)
- Massimo Bertolini (University of Milan)
- Daniel Delbourgo (Waikato University)
- Mihran Papikian (Pennsylvania State University)
- Daeyeol Jeon (Kongju National University)
- Keisuke Arai
- Shun Ohkubo (University of Tokyo)
- Heiko Knospe (Cologne University of Applied Sciences)
Abstracts and Titles:
Karl Rubin - Higher rank Kolyvagin systemsAbstract: 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 extensionAbstract: 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-functionsAbstract: 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 fieldsAbstract: 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 $\Gamma_0(p)$-typeAbstract: 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 cyclesAbstract: 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 groupsAbstract: 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 caseAbstract: 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 applicationsAbstract: 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.
|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|
|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|
|Fri 21 Dec|
|10:00 - 10:25||Refreshments|
|10:30 - 12.30||Workshop Discussion|