U of T number theory seminar (2026–27)

The local Langlands correspondence for a reductive group \(G\) over \(\mathbb{Q}_p\) posits a correspondence between certain smooth representations \(\pi\) of \(G(\mathbb{Q}_p)\) and certain \(G\)-valued (in an appropriate sense) representations \(\varphi\) of \(\operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\). While brought into clearer perspective by recent work, the precise nature of what is meant by \(\varphi\) corresponding to \(\pi\) has historically been somewhat vague but, there is a natural guess: the characters of \(\varphi\) and \(\pi\) should match. This does not quite make sense as traces for \(\varphi\) take as inputs Galois-group elements \(\tau\), and traces on \(\pi\) take as inputs certain functions \(f\) on \(G(\mathbb{Q}_p)\). Thus, it is natural to guess that to each \(\tau\) there is an \(f_\tau\) and one should require that \(\operatorname{tr}(\tau \mid \varphi) = \operatorname{tr}(f_\tau \mid \pi)\). While such a naive statement cannot be true, it is a conjecture of Scholze--Shin that an appropriate refinement does hold. The question then becomes: what are the functions \(f_\tau\)? In this talk I will try to roughly explain the definition of these functions and their relationship to Shimura varieties, the so-called Langlands--Kottwitz--Scholze method.

In this talk, I will discuss a dynamical analogue of a classical number-theoretic question concerning bounds on the greatest common divisors of two integer sequences. I will present some recent progress on this problem and highlight several open directions for future research. Finally, I will explain how this question relates to the Dynamical Mordell--Lang Conjecture, a central topic in algebraic and arithmetic dynamics.

The form $x^2+y^2$ covers $\tfrac{1}{2}$ of the primes, while the forms $x^2+y^2, x^2+2y^2$ cover $\tfrac{3}{4}$ of them. In this talk, we will show that the proportion of primes covered by the forms $x^2+dy^2$, with $1 \leqslant d \leqslant \Delta$, is $1 - \exp((\alpha(\Delta) + o(1)) \tfrac{\sqrt{\Delta}}{\log \Delta})$ for some $\tfrac{\pi}{2} \leqslant \alpha(\Delta) \leqslant \tfrac{\pi}{2} + \log 4$.

Moreover, inspired by a result of Kaplansky, we prove that any prime represented by two of the forms $x^2+17y^2, x^2+65y^2, x^2+1105y^2$ is actually represented by all three. We establish these results by studying how class groups of quadratic fields interact.

Classical holomorphic modular forms on \(\mathrm{SL}_2\) play a central role in modern number theory, and their analogues on many higher-rank groups have been extensively studied. However, many groups—including all but two exceptional Lie groups—do not admit holomorphic modular forms, motivating the search for a replacement. Quaternionic modular forms (QMFs) give a promising alternative. Gan–Gross–Savin studied QMFs on the split exceptional group \(G_2\), and more recently, a theory of Fourier expansions has been developed by Aaron Pollack for quaternionic exceptional groups and by Hilado–McGlade–Yan for unitary groups \(U(2,n)\). In this talk, I will present what is known and what is still unknown about QMFs, including dimension formulas and the arithmeticity (rationality/algebraicity) of cusp forms and certain Eisenstein series.

Let $E$ be a non-trivial elliptic curve over $\mathbb{Q}(t)$ of the form $E:y^2=x^3+f(t)$ for some polynomial $f(t)$. In this talk, I will first review some results on the rank and generators of the Mordell-Weil group $E(\mathbb{Q}(t))$ for certain families of polynomials $f(t)$, and then sketch the proof of my result for $f(t)=At^6+Bt^3+C$, which generalizes the result by Kloosterman (2026) and the result by Desjardins and Naskrecki (2024).