CONFERENCES AND SEMINARS ORGANIZED
Organizers: Cristian D. Popescu, Romyar Sharifi,
Organizers:
Manfred Kolster, Cristian D. Popescu, Romyar Sharifi,
Theme:
"The Arithmetic of L-functions"
Organizers: Cristian D. Popescu, Karl Rubin, Alice Silverberg.
Co-organized with the Number Theory group at UCSD, with financial
support from the UCSD and the NSA.
An informal seminar for faculty and gradute students - schedule.
Organizers:
Adebisi Agboola, UCSB; Cristian
Popescu, UCSD.
An informal seminar for graduate students and faculty.
RESEARCH
1)
My
main mathematical interests are in the fields of algebraic number
theory
and arithmetic algebraic geometry, with a focus on special values of p-adic and global
L-functions.
My most recent
results and research
projects are joint with Cornelius Greither
(Münich) and have led to
the proof an Equivariant
Main Conjecture in Iwasawa Theory
(unconditionally, in char.
p>0 and
away from 2 and under the assumption that a certain Iwasawa
mu-invariant vanishes, in char. 0.)
By using
Iwasawa co-descent, we have been able to derive the following type of
results on special
values of global
L-functions:
- A proof of the Equivariant
Tamagawa Number Conjecture (ETNC) of
Bloch-Kato-Burns-Flach for Dirichlet motives (in full generality, in
char. p>0, and on the "minus side", in char.
0);
- A proof of Rubin's integral
refinement and Gross's p-adic
refinement of the abelian Stark conjecture (in full generality, in
char. p and on the ''minus side'', in char. 0) ;
- A proof of a refinement of
the Brumer-Stark Conjecture (in full generality, in char. p and
away from 2, in char. 0);
- A proof of a refined version of the
étale cohomological
Coates-Sinnott Conjecture on special values of Dirichlet L-functions at
negative integers (in general, in char. p and away from 2, in char. 0);
- A construction of l-adic canonical
models for Tate
sequences (arising in the theory of multiplicative Galois module
structure and playing a central role in the statement of the ETNC for
Artin motives.)
In char. p, our Equivariant Main Conjecture (now, a theorem)
relates the Fitting ideal of an l-adic realization of a Picard
1-motive over a profinite group-ring to an equivariant l-adic
L-function.
In char. 0, the role of the l-adic realizations of
Picard
1-motives is played by our newly constructed class of Iwasawa modules,
which are of finite projective dimension over the appropriate
profinite group-rings and behave well under Iwasawa co-descent. The
EMC
in char. 0 (now, a theorem, if a certain mu-invariant
vanishes)
links the Fitting ideals of these new Iwasawa modules to
the appropriate equivariant l-adic L-functions.
See my recent lecture at "The First Abel
Conference honoring John Tate" (IMA, Minneapolis,
January 2011) for a
detailed account of the above
results.
Possible extensions of the above results and techniques to the study of
special values of L-functions of higher dimensional arithmetic schemes
(e.g. elliptic curves) are also under investigation.
2) In recent joint work with Grzegorz Banaszak (Poznan, Poland),
we have
began a systematic study of the Galois
module structure of Quillen's K-groups associated to
rings of algebraic integers,
with the following goals in mind:
- A proof of the K-theoretic
Coates-Sinnott conjecture on
special values of Dirichlet L-functions at negative integers;
- A systematic construction of Euler
systems (or Kolyvagin
systems à la Mazur and Rubin) in the Quillen K-groups of rings
algebraic integers;
- Using these Euler Systems to study the
structure of the
groups of divisible elements in the even K-theory of number fields.
(The structure of these groups is closely related
to various central open problems in algebraic
number theory,
such as the Kummer-Vandiver and Iwasawa conjectures on class-groups of
cyclotomic fields.)
My research is currently funded through the National Science Foundation award
DMS
- 0901447.
Recent
papers and preprints
- On a Refined Stark
Conjecture for
Function Fields ( stark.pdf ); Compositio
Mathematica,
Vol. 116 (1999), No.3 , pp. 321-367. See Featured Review .
- Gras-type Conjectures for Function
Fields ( gras.pdf ); Compositio
Mathematica,
Vol. 118 (1999), No. 3, pp. 263-290. See Featured Review .
- Base Change for Stark-type
Conjectures "over Z" ( bc.pdf); Journal fur
die Reine und Angew. Mathematik, Vol. 542 (2002), pp 85-111.
(Awarded
the Simion Stoilow Prize of the Romanian Academy of Science , 2004)
- Stark's Question and a strong form of Brumer's Conjecture
( sq.pdf); Compositio Mathematica ,
Vol. 140 (2004), pp. 631-646.
- The Rubin-Stark Conjecture for imaginary abelian fields
of odd prime power conductor ( r-st2.pdf); Mathematische
Annalen Vol.. 330 (2004), No. 2, pp. 215-233.
- On the Rubin-Stark Conjecture for a special class of CM
extensions of totally real number fields (r-st1.pdf); Mathematische
Zeitschrift, Vol. 247 (2004), pp. 529-547.
- N-torsion of Brauer groups as relative
Brauer groups of abelian extensions (with J. Sonn and A.
Wadsworth) (brauer2.pdf); Journal
of Number Theory,Vol. 125 (2007), No. 1,
Pages 26-38
- The
Galois module structure of l-adic realizations of Picard 1-motives and
applications (with C. Greither)
(GP1.pdf); to appear in Intl. Math. Res. Notices,
2011. 51
pages.
- The
Stickelberger splitting map and Euler Systems in the K-theory of number
fields (with
G. Banaszak) (cg10.pdf); to appear in Jour.
of Number Theory (David Hayes Memorial Issue) (2011).
29 pages.
- An
Equivariant Main Conjecture in Iwasawa Theory and Applications (with C.
Greither) (emc.pdf);
Submitted.
52 pages.
GRADUATE
STUDENTS
Byungchul Cha, PhD 2003, Johns Hopkins U.
(co-advised with V. Kolyvagin) - Assistant Professor (tenure-track),
Muhlenberg
College, PA.
Caleb Emmons, PhD 2006, UCSD - Assistant
Professor (tenure-track), Pacific University, OR.
Barry Smith, PhD 2007, UCSD - Assistant
Professor (tenure-track), Lebanon Valley College, PA.
Daniel Vallières, PhD 2011, UCSD -
Post-Doctoral Fellow, Universität der Bundeswehr, München,
Germany.
Joel Dodge, PhD (August 2011), UCSD -
Assistant Professor (post-doc), SUNY at Binghamton, NY.
Winter 2012 Linear algebra (Math 102); Topics in Analytic Number Theory - "Modularity" (Math 204B.)
Spring 2011 Abstract Algebra (Math
100C); Topics in
Algebraic Number Theory (Math 201C.)
Winter 2011 Abstract Algebra (Math 100B); Modern Algebra (Math 103B).
Spring 2010 Abstract Algebra (Math
100C); Multivariable Calculus
(Math 20C).
Winter 2010 Topics in Number Theory (Math 201); Multivariable Calculus (Math 20C).
Summer
2009, II Number Theory (Math 104A)
Winter 2009 Topics in Number
Theory (Math 205); Abstract
Algebra (Math 100B).
Fall
2008 Abstract Algebra (Math 100A).
Winter
2008 Graduate Number Theory
(Math 204B); Undergraduate
Number Theory (Math
104B).
Fall 2007
Number Theory
(Math 104A)
Winter
2007 Topics in Algebraic
Number Theory (Math 205B) - Global Class Field Theory. (Local/global diagram.)
Fall 2006 Topics in Algebraic Number
Theory (Math 205A) - Local and Global Class Field Theory;
Calculus (Math 20C).
Spring
2006 Topics in Algebraic
Number Theory (Math 205) - LOCAL CLASS FIELD THEORY
Winter
2006 Graduate Algebra (Math 200B);
Introduction to Differential
Equations (Math 20D)
Fall
2005 Graduate
Algebra (Math 200A); Mathematical
Reasoning (Math 109)
Spring
2005 Number Theory (Math 104C)
Winter 2005 Number
Theory (Math 104B); Calculus
(Math 20C)
Fall
2004 Mathematical
Reasoning (Math 109)
Spring 2004 Mathematical Reasoning (Math 109)
Winter
2004 Linear Algebra (Math 20F)
Fall 2003 Introduction to Differential
Equations (Math 20D)
This page was last modified on Aug 2, 2010.
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