| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | April 5, 2017 |
| Latest Amendment Date: | July 17, 2020 |
| Award Number: | 1701638 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
James Matthew Douglass
mdouglas@nsf.gov (703)292-2467 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | July 1, 2017 |
| End Date: | June 30, 2023 (Estimated) |
| Total Intended Award Amount: | $410,002.00 |
| Total Awarded Amount to Date: | $410,002.00 |
| Funds Obligated to Date: |
FY 2018 = $118,731.00 FY 2019 = $86,868.00 FY 2020 = $87,341.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
10889 WILSHIRE BLVD STE 700 LOS ANGELES CA US 90024-4200 (310)794-0102 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
6363 Math Science Building Los Angeles CA US 90095-1555 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | ALGEBRA,NUMBER THEORY,AND COM |
| Primary Program Source: |
01001819DB NSF RESEARCH & RELATED ACTIVIT 01001920DB NSF RESEARCH & RELATED ACTIVIT 01002021DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
This research project concerns number theory, the oldest branch of mathematics. More specifically, it focuses on the study of automorphic forms. Automorphic forms are a very special class of functions that form an important bridge connecting the discrete objects of algebraic number theory and the continuous objects of analytic number theory. Automorphic forms are used as tools to study number theoretic functions, e.g. by measuring their rates of growth, discovering formulas for them, or proving relations that they satisfy. Classically, special values of automorphic forms provided the solution to Kronecker's problem for extensions of the rational numbers that do not lie in the real line. More recently, automorphic forms played a pivotal role in the proof of Fermat's Last Theorem. This project aims to further explore the connection of automorphic forms to other mathematical structures.
The main object of the proposed research is to further understand and exploit the relationship between automorphic forms and quadratic number fields. This relationship is exceedingly rich and unites the study of diverse objects such as Heegner points, closed geodesics, the hyperbolic Laplacian, Kloosterman sums, L-functions and class fields. The theory for imaginary quadratic fields is in general better developed and simpler, especially in relation to class field theory. The investigators will concentrate mostly on the real quadratic case. In one direction, they will study some new geometric invariants associated to real quadratic fields that were introduced recently. These invariants are certain surfaces that are bounded by modular closed geodesics. The investigators will study the distribution of the areas of the surfaces, especially as this relates to ideal classes and also investigate some new geometric problems about the closed geodesics. They plan to express various invariants for real quadratic fields (such as surface integrals of modular functions) in terms of the Fourier coefficients of automorphic forms. This naturally leads to problems involving sums of Kloosterman sums and to extensions of recent work on uniform estimates for such sums.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
The grant supported research that resulted in several significant advances in knowledge in one of the oldest and most challenging areas on Mathematics, namely number theory. This knowledge was disseminated to the community in more than fourteen articles in peer reviewed journals and reported on in numerous lectures, including a recent colloquium at Princeton University.
Four main areas of Number theory were addressed: 1) the theory of modular forms and associated L-functions, 2) Diophantine approximations, 3) the analytic theory of quadratic forms and 4) arithmetic invariant theory.
Contributions to the first area were made in two joint papers with the co-PI Andersen on the asymptotics of singular moduli and in a recent paper on higher Rademacher symbols, which has stimulated quite a bit of research by others already.
The second area was contributed to in four papers with Andersen and several graduate students. Perhaps the most important gives a serious generalization of a classical theorem of Davenport and Schmidt on improvements of Dirichlet's approximation result.
A paper on the analytic theory of isotropic ternary quadratic forms gives probably the most important set of results supported by the grant. A classical result of Siegel (Siegel's main theorem on quadratic forms) is adapted to apply to integral ternary quadratic forms that represent zero (are isotropic). This case includes a fundamental and well-known result of Legendre on ternary diagonal forms. The main result supported by the grant relates the count of locally defined orbits of solutions to a count of orbits of globally defined integral solutions. The result is similar but actually simpler than Siegel's theorem. In fact, Siegel's theorem does not apply to this case. This is the first basic advance in the arithmetic theory of indefinite ternary quadratic forms in many years.
Finally, the grant supported research in the arithmetic theory of binary forms of degree higher than two, in particular binary quartic forms. Here the most important paper gives an analogue of Dirichlet's class number formula for binary quadratic forms. This is a class number formula for certain positive definite binary quartic forms given in terms of a residue of a Dirichlet series defined using the numbers of integral points on the twists of an associated elliptic curve.
Last Modified: 09/29/2023
Modified by: William D Duke
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