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April 1, 11:00AM |
Peter Koymans (MPIM) (paper)
In 2002 Malle conjectured an asymptotic formula for the number of $G$-extensions of a number field $K$ with discriminant bounded by $X$. In this talk I will discuss recent joint work with Etienne Fouvry on this conjecture. Our main result proves Malle's conjecture in the special case of nonic Heisenberg extensions. |
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April 8, 10:00AM |
Mahesh Kakde (IISc, Bangalore)
I will report on my joint work with Samit Dasgupta on the Brumer-Stark conjecture proving existence of the Brumer-Stark units and on a conjecture of Dasgupta giving a p-adic analytic formula for these units. I will present a sketch of our proof of the Brumer-Stark conjecture and also mention applications to Hilbert's 12th problem, or explicit class field theory. |
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April 15 |
Lance Miller (University of Arkansas)
Fix a prime integer $p$. Set $R$ the completed valuation ring of the maximal unramified extension of $\mathbb{Q}_p$. For $X := X_1(N)$ the modular curve with $N$ at least 4 and coprime to $p$, Buium-Poonen in 2009 showed that the locus of canonical lifts enjoys finite intersection with preimages of finite rank subgroups of $E(R)$ when $E$ is an elliptic curve with a surjection from $X$. This is done using Buium's theory of arithmetic ODEs, in particular interesting homomorphisms $E(R) \to R$ which are arithmetic analogues of Manin maps. In this talk, I will review the general idea behind this result and other applications of arithmetic jet spaces to Diophantine geometry and discuss a recent analogous result for quasi-canonical lifts, i.e., those curves with Serre-Tate parameter a root of unity. Here the ODE Manin maps are insufficient, so we introduce a PDE version of Buium's theory to provide the needed maps. All of this is joint work with A. Buium. |
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April 22 |
Owen Barrett (University of Chicago)
Nori proved in 2002 that given a complex algebraic variety $X$, the bounded derived category of the abelian category of constructible sheaves on $X$ is equivalent to the usual triangulated category $D(X)$ of bounded constructible complexes on $X$. He moreover showed that given any constructible sheaf $\mathcal F$ on $\A^n$, there is an injection $\mathcal F\hookrightarrow\mathcal G$ with $\mathcal G$ constructible and $H^i(\A^n,\mathcal G)=0$ for $i>0$. In this talk, I'll discuss how to extend Nori's theorem to the case of a variety over an algebraically closed field of positive characteristic, with Betti constructible sheaves replaced by $\ell$-adic sheaves. This is the case $p=0$ of the general problem which asks whether the bounded derived category of $p$-perverse sheaves is equivalent to $D(X)$, resolved affirmatively for the middle perversity by Beilinson. |
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April 29 |
Christian Klevdal (University of Utah)
Simpson conjectured that for a reductive group $G$, rigid $G$-local systems on a smooth projective complex variety are integral. I will discuss a proof of integrality for cohomologically rigid $G$-local systems. This generalizes and is inspired by work of Esnault and Groechenig for $GL_n$. Surprisingly, the main tools used in the proof (for general $G$ and $GL_n$) are the work of L. Lafforgue on the Langlands program for curves over function fields, and work of Drinfeld on companions of $\ell$-adic sheaves. The major differences between general $G$ and $GL_n$ are first to make sense of companions for $G$-local systems, and second to show that the monodromy group of a rigid G-local system is semisimple. All work is joint with Stefan Patrikis. |
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May 6 |
Maria Fox (University of Oregon)
Unitary Shimura varieties are moduli spaces of abelian varieties with an action of a quadratic imaginary field, and extra structure. In this talk, we'll discuss specific examples of unitary Shimura varieties whose supersingular loci can be concretely described in terms of Deligne-Lusztig varieties. By Rapoport-Zink uniformization, much of the structure of these supersingular loci can be understood by studying an associated moduli space of p-divisible groups (a Rapoport-Zink space). We'll discuss the geometric structure of these associated Rapoport-Zink spaces as well as some techniques for studying them. |
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May 13 |
Sean Howe (University of Utah)
A classical transcendence result of Schneider on the modular $j$-invariant states that, for $\tau \in \mathbb{H}$, both $\tau$ and $j(\tau)$ are in $\overline{\mathbb{Q}}$ if and only if $\tau$ is contained in an imaginary quadratic extension of $\mathbb{Q}$. The space $\mathbb{H}$ has a natural interpretation as a parameter space for $\mathbb{Q}$-Hodge structures (or, in this case, elliptic curves), and from this perspective the imaginary quadratic points are distinguished as corresponding to objects with maximal symmetry. This result has been generalized by Cohen and Shiga-Wolfart to more general moduli of Hodge structures (corresponding to abelian-type Shimura varieties), and by Ullmo-Yafaev to higher dimensional loci of extra symmetry (special subvarieties), where bialgebraicity is intimately connected with the Pila-Zannier approach to the Andre-Oort conjecture. In this talk, I will discuss work in progress with Christian Klevdal on an analogous bialgebraicity characterization of special subvarieties in Scholze's local Shimura varieties and more general diamond moduli of $p$-adic Hodge structures. There will be a pretalk! |
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May 20 |
Nahid Walji (University of British Columbia)
Let $\Pi$ be a cuspidal automorphic representation for GL(3) over a number field. We fix an integer $k \geq 2$ and we assume that the symmetric $m$th power lifts of $\Pi$ are automorphic for $m \leq k$, cuspidal for $m < k$, and that certain associated Rankin–Selberg products are automorphic. In this setting, we bound the number of cuspidal isobaric summands in the $k$th symmetric power lift. In particular, we show it is bounded above by 3 for $k \geq 7$, and bounded above by 2 when $k \geq 19$ with $k$ congruent to 1 mod 3. We will also discuss the analogous problem for GL(4). |
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May 27 |
Evan O'Dorney (Princeton University) (paper)
Coclasses in a Galois cohomology group $H^1(K, T)$ parametrize extensions of a number field with certain Galois group. It is natural to want to count these coclasses with general local conditions and with respect to a discriminant-like invariant. In joint work with Brandon Alberts, I present a novel tool for studying this: harmonic analysis on adelic cohomology, modeled after the celebrated use of harmonic analysis on the adeles in Tate's thesis. This leads to a more illuminating explanation of a fact previously noticed by Alberts, namely that the Dirichlet series counting such coclasses is a finite sum of Euler products; and we nail down the asymptotic count of coclasses in satisfying generality. |
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June 3 |
Kelly Isham (University of California Irvine)
Let $K$ be a number field of degree $n$ and $\mathcal{O}_K$ be its ring of integers. An order in $\mathcal{O}_K$ is a finite index subring that contains the identity. A major open question in arithmetic statistics asks for the asymptotic growth of orders in $K$. In this talk, we will give the best known lower bound for this asymptotic growth. The main strategy is to relate orders in $\mathcal{O}_K$ to subrings in $\mathbb{Z}^n$ via zeta functions. Along the way, we will give lower bounds for the asymptotic growth of subrings in $\mathbb{Z}^n$ and for the number of index $p^e$ subrings in $\mathbb{Z}^n$. We will also discuss analytic properties of these zeta functions. |