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Number Theory seminar abstracts, Term 1 2017-18



Congruences and the local Jacquet-Langlands correspondence, by Shaun Stevens (joint work with Vincent S├ęcherre)

The local Jacquet—Langlands correspondence is a bijection between certain irreducible complex representations of a general linear group over a p-adic field and an inner form of such a group, defined by a character relation; equivalently, it is the bijection between these representations induced via the local Langlands correspondence. While the existence of the Jacquet—Langlands correspondence has been known since the 1980s -- before the local Langlands correspondence -- it is not yet known how to make it explicit in general, even though there are classifications of the irreducible representations on both sides (and more); moreover, all results so far (mostly due to Bushnell—Henniart) have concentrated on the ``cuspidal'' case, where the character relation is more amenable to computation.

As well as trying to explain what these words mean, I will report on work where we use mod-l ``congruences'' between representations (for l a prime different from p) to bear on this question, reducing most of the problem to the cuspidal case. Subsequent work of Dotto, using the same ideas, has made all but the ``unramified part'' of the correspondence explicit.


Equidistribution of rational points on the sphere, Sarnak's Conjecture, and the twisted Linnik Conjecture, by Raphael Steiner
It is a classical theorem in the theory of modular forms that the points $\boldsymbol{x}/\sqrt{N}$, where $\boldsymbol{x} \in \mathbb{Z}^n$ runs over all the solutions to $\sum_{i=1}^n x_i^2=N$, equidistribute on $S^{n-1}$ for $n \ge 4$ as $N$ (odd) tends to infinity. The rate of equidistribution poses however a more challenging problem. Due to its Diophantine nature the points inherit a repulsion property, which opposes equidistribution on small sets. Sarnak conjectures that this Diophantine repulsion is the only obstruction to the rate of equidistribution. Using the smooth delta-symbol circle method, developed by Heath-Brown, Sardari was able to show that the conjecture is true for $n\ge 5$ and recovering Sarnak's progress towards the conjecture for $n=4$. Building on Sardari's work, Browning, Kumaraswamy, and myself were able to reduce the conjecture to correlation sums of Kloosterman sums of the following type:
$$ \sum_{q \le Q} \frac{1}{q}S(m,n;q)exp(4 \pi i \alpha \sqrt{mn}/q). $$
Assuming the twisted Linnik conjecture, which states that the above sum is $O((Qmn)^{\epsilon})$ for $|\alpha|\le 2$, we are able to verify Sarnak's Conjecture. I will further discuss progress towards the twisted Linnik conjecture and the techniques involved. If the time permits I will lose some words about the differences in the automorphic and the circle method approach and how they may be combined.


Quartic orders of D4-type with monogenic cubic resolvent, by Stanley Xiao

In a seminal paper, M. Bhargava showed that quartic orders are parametrized by $\operatorname{GL}_2(\mathbb{Z}) \times \operatorname{SL}_3(\mathbb{Z})$-orbits of pairs of ternary quadratic forms. Later, M. Wood showed that $\operatorname{GL}_2(\mathbb{Z})$-orbits of integral binary quartic forms parametrize pairs $(Q,C)$ where $Q$ is a quartic order and $C$ a monogenic cubic resolvent ring of $Q$. A quartic order $Q$ is said to be $D_4$-type if the Galois closure of the field of fractions of $Q$ has Galois group isomorphic to $D_4$. In a recent paper, Altug, Shankar, Varma, and Wilson enumerated quartic orders of $D_4$-type when counting by conductor. In this talk, we give a report on recent progress made on counting maximal pairs $(Q,C)$ with $Q$ a quartic order of $D_4$-type and $C$ a monogenic cubic resolvent ring of $Q$. This is joint work with C. Tsang.


Recent Progress on the Lind-Lehmer Problem for p-groups, by Chris Pinner
In 2005 Doug Lind generalized the concept of Mahler measure to an arbitrary compact abelian group. As in Lehmer's problem for the classical Mahler measure one can ask for the minimal non-trivial measure. For a finite abelian group this corresponds to the smallest non-trivial integral group determinant. After a brief survey of existing results I will present some new congruences satisfied by the Lind Mahler measure for p-groups. These enable us to determine the minimal measure when the p-group has one particularly large component and to compute the minimal measures for many new families of small p-groups.

This is joint work with Mike Mossinghoff of Davison College.
If there is time I will also mention some 3-group results from a summer undergraduate research project with Stian Clem which may suggest what is going on in general.


Some algebras associated to genus one curves, by Tom Fisher
Haile, Han and Kuo have studied certain non-commutative algebras associated to a binary quartic or ternary cubic form. These give an explicit realisation of an isomorphism relating the Weil-Chatelet and Brauer groups of an elliptic curve. I will describe how I expect their constructions to generalise to other genus one curves.


The Kakeya conjecture and number theory, by Ben Green

The Kakeya conjecture asserts that every subset of R^n containing a unit line in every direction has dimension n. Whilst the Kakeya conjecture itself does not have any number-theoretic implications (so far as I know), many arithmetic questions reside nearby. For example, there is a purely arithmetic (and very simple-to-state) conjecture which would imply the Kakeya conjecture. In another direction, it is my belief that questions about the distribution of arithmetic functions in progressions, such as the Elliott-Halberstam conjecture, are strictly harder than the Kakeya conjecture. I will discuss these issues.