Number Theory

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $C_l$ be the family of prime cyclic extensions of degree $l$ over $\mathbb{Q}$. Under GRH for elliptic L-functions, we give a lower bound for the probability for $K \in C_l$ such that the difference $r_K(E) − r_\mathbb{Q}(E)$ between analytic rank is less than a for $a \asymp l$. This result gives conjectural evidence that the Diophantine Stability problem suggested by Mazur and Rubin holds for most of $K \in C_l$.

Let $\pi(x; q, a)$ be the number of primes $p\leq x$ such that $p \equiv a (\mod q)$. The classical Shanks–Rényi prime number race problem asks, given positive integers $q \geq 3$ and $2 \leq r \leq \phi(q)$ and distinct reduced residue classes $a_1, a_2, . . . , a_r$ modulo $q$, whether there are infinitely many integers $n$ such that $\pi (n; q, a1) > \pi(n; q, a2) > \cdots > \pi(n; q, ar)$. In this talk, I will describe what is known on this problem when the number of competitors $r \geq 3$, and how this compares to the Chebyshev’s bias case which corresponds to $r = 2$.

After a brief introduction on the theory of p-adic groups complex representations, I will explain why tempered and generic Langlands parameters are open. I will further derive a number of consequences, in particular for the enhanced genericity conjecture of Shahidi and its analogue in terms of ABV packets. This is a joint work with Clifton Cunningham, Andrew Fiori, and Qing Zhang.

To each square-free monic polynomial $D$ in a fixed polynomial ring $\mathbb{F}_q[t]$, we can associate a real quadratic character $\chi_D$, and then a Dirichlet $L$-function $L(s,\chi_D)$. We compute the limiting distribution of the family of values $L'(1,\chi_D)/L(1,\chi_D)$ as $D$ runs through the square-free monic polynomials of $\mathbb{F}_q[t]$ and establish that this distribution has a smooth density function. Time permitting, we discuss connections of this result with Euler-Kronecker constants and ideal class groups of quadratic extensions. This is joint work with Amir Akbary.

Let $F$ be the field of $p$-adic numbers (or, more generally, a non-
archimedean local field) and let $G$ be $\mathrm{GL}_n(F)$ (or, more generally,
the group of $F$-points of a split connected reductive group). In the
framework of the local Langlands program, one is interested in studying
certain classes of representations of $G$ (and hopefully in trying to match
them with certain classes of representations of local Galois groups).

In this talk, we are going to focus on the category of smooth representations
of $G$ over a field $k$. An important tool to investigate this category is
given by the functor that, to each smooth representation $V$, attaches its
subspace of invariant vectors $V^I$ with respect to a fixed compact open
subgroup $I$ of $G$. The output of this functor is actually not just a $k$-
vector space, but a module over a certain Hecke algebra. The question we are
going to attempt to answer is: how much information does this functor preserve
or, in other words, how far is it from being an equivalence of categories? We
are going to focus, in particular, on the case that the characteristic of $k$
is equal to the residue characteristic of $F$ and $I$ is a specific subgroup
called "pro-$p$ Iwahori subgroup".