General Information

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Project Description

Triple product L-functions are of great interest in number theory and quantum physics. This project will study these functions over finite fields in order to gain insight on the behavior of these functions more generally.

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Weekly Log

Week 1 (5/29 - 5/31)

This week was shorter and consisted of mostly orientation and getting to know my mentor, the other people in the program, and the campus. I met with my mentor a couple of times and got an overview of the project as well as the first steps I need to accomplish. These include looking through previous work on the topic and some textbooks for background reading, of which I have briefly started doing this week. I also created this website and took care of some other minor administrative tasks.

Week 2 (6/3 - 6/7)

This week I did a lot of further background reading required for the project. I studied induced representations, group algebras, p-adic numbers, and Gelfand pairs. Earlier in the week we had the introductory presentations on our projects as well.

Week 3 (6/10 - 6/14)

This week had the "Current Trends in Mathematics" lectures, which took up a majority of my time this week. I found some of these talks to be very interesting and it was nice to be exposed to many different types of mathematics. I also continued to do the background reading, mostly focusing on the Piatetski-Shapiro and the Serre texts.

Week 4 (6/17 - 6/20)

I made significant progress on the readings, specifically on representations of invertible 2x2 matrices over finite fields. I now understand how to get the irreducible representations derived from characters on the Borel subgroup, and am starting to learn about the cuspidal irreducibles. I am also starting the research component of my project now that I have sufficient background, and am working on coding some calculations on these irreducible representations. In particular, I am working on producing the actual vector space and map from the group to invertible maps on this vector space. This will require choosing a basis, and some choice might work better for the calculations, which I have to figure out.

Week 5 (6/24 - 6/28)

I did a little bit of reading this week, but I mostly focused on the coding. I figured out how to accurately describe the vector space and representationa action on that vector space. I then defined the G-invariant inner product based off of the naive dot product, and made the code producing the matrix of coefficients based off of this inner product. At first, my code had quite a few bugs that gave me wrong calculations, and it was extremely inefficient. However, eventually I fixed these bugs and made the code run much faster with some help from my mentors. However, at this point we realized that with the natural basis we chose, the naive dot product was already G-invariant and had the properties we desired. So a lot of the coding effort eventually turned out to be fruitless, and this direction of research turned out to not be as interesting as we had hoped. However there is still some directions with the other irreducible representations of GL(2, K) that we would like to see coded up, and this will provide a lot of the framework. I also need to return to some of the serious background reading, especially on cuspidal representations and Wittaker models.

Week 6 (7/1 - 7/5)

Earlier in the week, I managed to automatically find the one-dimensional invariant subspace in certain induced characters of the Borel subgroup. Next task is to find a way to represent the cuspidal representations, which will be much trickier. The way to approach this is to use Whittaker models. I first tried a naive way of playing around with combinations of eigenvectors to find the (q-1)-dimensional subspaces, but this didn't work and I realized why it couldn't work. Now I am just trying to figure out a clever way to do this decomposition, or at least find a single vector in the cuspidal representations, which would also allow me to generate the subspace.

Week 7 (7/8 - 7/12)

We had a field trip to Nokia Bell Labs on Monday, which was extremely cool. On the research side, after trying unsuccessfully to generate the cuspidal representations with the Whittalker models, I instead tried to represent them directly. This actually turned out to be much easier than expected, so I now have all the representations I need to continue on to the next part of the project. This will consist of doing sample calculations of trilinear forms using the irreducible representations that I have coded up so far.

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Presentations

Introductory Presentation

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References and Reading

Representation Theory: A First Course by William Fulton and Joe Harris
Complex Representations of GL(2, K) for Finite Fields K by Ilya Piatetski-Shapiro
A Course in Arithmetic by Jea-Pierre Serre

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Acknowledgements

This work was carried out while Dmitriy Shvydkoy was a participant in the 2024 DIMACS REU program at Rutgers University, CNS-2150186, supported by the Rutgers Department of Mathematics and Kristen Hendricks, NSF CAREER DMS-2019396.

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