NSF Org:	DMS Division Of Mathematical Sciences
Recipient:	UNIVERSITY OF ARIZONA
Initial Amendment Date:	July 18, 2000
Latest Amendment Date:	May 4, 2001
Award Number:	0070839
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, 2000
End Date:	July 31, 2003 (Estimated)
Total Intended Award Amount:	$60,000.00
Total Awarded Amount to Date:	$60,000.00
Funds Obligated to Date:	FY 2000 = $30,000.00 FY 2001 = $30,000.00
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-0100  01000102DB 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

Dr. Ulmer proposes three projects in arithmetical algebraic
geometry,
related to Galois representations, modular forms, and
elliptic curves,
both over number fields and over function fields. The first
project
proposed is to study the reduction modulo a prime p of
certain
representations of the Galois group of the p-adic numbers,
using the
recent work of Colmez and Fontaine on p-adic Hodge theory.
In the
second project, Dr. Ulmer has constructed a subgroup of the
local
points at suitable places on an elliptic curve over a
function field; this subgroup contains the global points.
He proposes to use these local
points to study the conjecture of Birch and Swinnerton-Dyer
for
elliptic curves over function fields. The third project Dr.
Ulmer
proposes is to study a new class of problems in the
cohomology of
varieties which are inspired by classical non-vanishing
results for
L-series. The new questions come by reinterpreting
vanishing results
using Grothendieck's analysis of L-functions and lead to
purely
geometric questions. In some instances these questions can
be treated
using monodromy results of Katz and Sarnak.

This proposal falls into the general area of arithmetical
algebraic
geometry - a subject that blends two of the oldest areas of
mathematics: number theory and geometry. This combination
has proved
extraordinarily fruitful, having recently solved problems
that
withstood the efforts of generations. Among its many
consequences are
new error correcting codes which are used in computer
storage devices
like compact disks and hard drives and secure information
transmission
schemes which are used for financial transactions on the
internet.

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