NSF Org:	DMS Division Of Mathematical Sciences
Recipient:	UNIVERSITY OF ARIZONA
Initial Amendment Date:	June 27, 2007
Latest Amendment Date:	April 6, 2009
Award Number:	0701053
Award Instrument:	Continuing Grant
Program Manager:	Andrew Pollington adpollin@nsf.gov  (703)292-4878 DMS  Division Of Mathematical Sciences MPS  Directorate for Mathematical and Physical Sciences
Start Date:	July 1, 2007
End Date:	February 28, 2010 (Estimated)
Total Intended Award Amount:	$119,998.00
Total Awarded Amount to Date:	$119,998.00
Funds Obligated to Date:	FY 2007 = $39,999.00 FY 2008 = $40,000.00 FY 2009 = $27,280.00
History of Investigator:	Douglas Ulmer (Principal Investigator) ulmer@math.arizona.edu
Recipient Sponsored Research Office:	University of Arizona 845 N PARK AVE RM 538 TUCSON AZ  US  85721 (520)626-6000
Sponsor Congressional District:	07
Primary Place of Performance:	University of Arizona 845 N PARK AVE RM 538 TUCSON AZ  US  85721
Primary Place of Performance Congressional District:	07
Unique Entity Identifier (UEI):	ED44Y3W6P7B9
Parent UEI:
NSF Program(s):	ALGEBRA,NUMBER THEORY,AND COM
Primary Program Source:	app-0107  01000809DB NSF RESEARCH & RELATED ACTIVIT 01000910DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s):	0000, OTHR
Program Element Code(s):	126400
Award Agency Code:	4900
Fund Agency Code:	4900
Assistance Listing Number(s):	47.049

ABSTRACT

Abstract for award DMS-0701053 of Ulmer

Dr. Ulmer proposes to work on three projects related to ranks of abelian varieties and the Birch and Swinnerton-Dyer conjecture over towers of function fields. In the first project he plans to exhibit non-abelian towers of function fields over which certain L-functions have zeroes of arbitrarily large order at the critical point; this builds on his recent work demonstrating the analogous result in abelian towers. In the second project, he plans to investigate general criteria which guarantee that Jacobians of curves satisfy the conjecture of Birch and Swinnerton-Dyer in every layer of a tower of function fields; again this extends his recent work. In the third project, Dr. Ulmer will try to extend recent results which allow one to show that certain abelian varieties have bounded ranks in the layers of a tower of function fields. All three of these projects currently involve students or post-docs and there is ample scope for their continued contribution.

Dr. Ulmer works in Arithmetical Algebraic Geometry, an area of fundamental mathematics whose motivating questions are about solving systems of polynomial equations with integers or rational numbers.
The field is curiosity-driven and was once thought to be without application. However, it is now known to be crucial to many modern technologies which affect our everyday lives, such as coding theory and cryptography. CD and DVD players, mobile telephones, and secure internet communication all rely on mathematics originally created in the pursuit of questions in arithmetical algebraic geometry. The field has deep connections to other areas of mathematics such as algebra, geometry, analysis, and topology, as well as mysterious links to other areas of science such as quantum field theory. Dr. Ulmer hopes to shed light on the connections between numbers, shapes, and calculus through his research on elliptic curves and L-functions. His work in this area also provides the basis for many education, outreach, and training activities in which he is engaged.

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