Algebraic number theory concerns numbers that are solutions to polynomial equations. The number systems in which these solutions live are called algebraic number fields. One of the central motivating questions in number theory for over a century has been whether one can construct certain special algebraic number fields using analytic techniques. Hilbert stated this problem as the 12th in his famous list of 23 problems at the 1900 International Congress of Mathematicians. The problem has been solved in only the simplest cases. These solutions use special values of modular forms, which are certain analytic functions that have a rich supply of symmetries. This project outlines a program to give a solution to Hilbert's 12th problem in an infinite family of new situations. The key ideas are to use a modern form of analysis, called p-adic analysis, in conjunction with other advanced techniques in number theory including modular forms, Galois representations, and Iwasawa theory. Solving Hilbert's 12th problems in these new cases will be a major advance in our understanding of algebraic number systems.

The PI has stated a conjecture with Spiess for an exact p-adic analytic formula for Gross-Stark units. These units, along with other easily written elements, generate the maximal abelian extension of totally real fields. Therefore, solving this problem can be viewed as providing a solution to Hilbert's 12th problem for totally real fields. Such a solution to Hilbert's 12th problem is not provided by the usual framework of conjectures for L-functions, such as Stark's conjectures. The PI will continue his work with Kakde on attacking his conjecture. Two new ideas relative to previous work on this topic are the use of the Taylor-Wiles "horizontal Iwasawa theory" method, as well as the introduction of group-ring families of modular forms. Next, in joint work with Spiess, the PI has stated a conjecture for the principal minors and characteristic polynomial of Gross' regulator. The PI plans to work together with Spiess and Kakde to generalize the techniques described above to higher rank (in particular the application of the Taylor-Wiles method) and thereby prove his conjecture on principal minors, which again goes beyond the usual framework of p-adic L-functions. Finally, the PI will work with Guido Kings to apply the Eisenstein cocycle to the study of abelian L-functions of general number fields. Two important test cases to consider are ground fields that are almost totally real and ground fields that are abelian extensions of CM fields.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Application #
1901939
Program Officer
Michelle Manes
Project Start
Project End
Budget Start
2019-08-01
Budget End
2022-07-31
Support Year
Fiscal Year
2019
Total Cost
$124,027
Indirect Cost
Name
Duke University
Department
Type
DUNS #
City
Durham
State
NC
Country
United States
Zip Code
27705