Algebra Seminar

calendars.uark.edu

Speaker: Jaclyn Lang (Temple University) Title: Quantifying level-raising congruences between modular forms via modular representation theory Abstract: Level-raising congruences between modular forms have played an important role in many modern advances in number theory, perhaps most famously in Fermat’s Last theorem. Starting from a modular form 𝑓 of level 𝑁, one asks whether, for a prime 𝑝 not dividing 𝑁, there is another (new)form 𝑔 of level 𝑁𝑝 such that 𝑓 and 𝑔 are congruent modulo another prime ℓ. Work on this problem began with Ribet and was answered decisively by Diamond and Taylor. Roughly speaking, such a 𝑔 exists whenever there is no local obstruction to its existence. These results only apply when 𝑓 is “non-Eisenstein modulo ℓ”; the Eisenstein case is more subtle and is still not entirely understood. In this talk we explain how the mod-ℓ representation theory of PGL₂(𝔽ₚ) can be used to give information about how many such forms 𝑔 exist. Our methods are agnostic about whether 𝑓 is Eisenstein modulo ℓ and gives new information in both the Eisenstein and non-Eisenstein settings. This is joint work with Robert Pollack and Preston Wake. In the evening after the talk, seminar participants are welcome to come to dinner with the speaker. Please email me (bergdall@uark.edu) if you'd like to come, or else just talk with me at the seminar. Other details about the algebra seminar can be found at https://math-colloquium.uark.edu/algebra-research-seminar/.