Seminars Sorted by Series

In joint work with Jacob Tsimerman we study when the primitive of a given algebraic function can be constructed using primitives from some given finite set of algebraic functions, their inverses, algebraic functions, and composition. When the given...

Motivated by work of Cantat-Dujardin, we study orbits by translations in K3 surfaces with two elliptic fibrations. We prove in particular that all orbits are infinite away from a proper Zariski-closed subset.

Among the tools, beyond the Pila-Wilkie...

We will describe emerging understanding of the structures related to the arithmetic of Zeta and Multizeta values for function fields through various results and conjectures.

In a recent joint work with S. Iyengar, C. Khare and J. Manning, we use their notion of congruence modules in higher codimension to give a new construction of the bottom class of the rank d=[F:\Q] Euler system attached to nearly ordinary Hilbert...

Spherical varieties play an important role in the study of periods of automorphic forms. But very closely related varieties can lead to very distinct arithmetic problems. Motivated by applications to relative trace formulas, we discuss the natural...

It is known that a p-adic family of modular forms does not necessarily specialize into a classical modular form at weight one, unlike the modular forms of weight 2 or higher. We will explain how this obstruction to classicality leads to a "derived"...

In this talk, I will report a recent joint work with Shouwu Zhang about a generic positivity of the Beilinson-Bloch height for the Gross-Schoen and Ceresa cycles of curves of genus at least 3. We also construct a Zariski open dense subset U of the...

The second and fourth moments of the Riemann zeta function have been known for about a century, but the sixth moment remains elusive.  

The sixth moment of zeta can be thought of as the second moment of a GL_3 Eisenstein series, and it is natural to...

In the early 2000's, Bertolini and Darmon introduced a new technique to bound Selmer groups of elliptic curves via level raising congruences. This was the first example of what is now termed a "bipartite Euler system", and over the last decade we...

I will first discuss the notion of automorphic representations on a unitary group that are $P$-ordinary (at $p$), where $P$ is some parabolic subgroup. In the “ordinary” setting (i.e. when $P$ is minimal), such a representation $\pi$ has a...

In the 1940s Mahler initiated the program of determining the bass note spectrum $$\mathrm{Spec}(P):=\left\{\inf_{\underline{x} \in\Lambda \setminus \underline{0}}\left\vert P(\underline{x})\right\vert, \Lambda \subset \mathbb{R}^k \text{ a...

The Cohen-Lenstra heuristics are influential conjectures in arithmetic statistics from 1984 which predict the average number of p-torsion elements in class groups of quadratic fields, for p an odd prime. So far, this average number has only been...

We prove an effective equidistribution theorem for semisimple
closed orbits on compact adelic quotients. The obtained error depends
polynomially on the minimal complexity of intermediate orbits and the
complexity of the ambient space. As an application...

Are there infintely many quadratic characters (for instance, the Legendre symbol mod p) for which the partial sums are always non-negative? Although only 0% of characters can have this property, numerical work (most recently by Kalmynin) suggests...

In the 1960s, Kirillov’s orbit method provided a striking correspondence between irreducible representations of a Lie group $G$ and certain geometric objects called coadjoint orbits. In 2021, Nelson and Venkatesh profitably adapted this method to...

We will discuss an inductive approach to determining the asymptotic number of G-extensions of a number field with bounded discriminant, and outline the proof of Malle's conjecture in numerous new cases. This talk will include discussions of several...

We resolve Manin's conjecture for all Châtelet surfaces over $Q$ (surfaces given by equations of the form $x^2 + ay^2 = f(z)$) -- we establish asymptotics for the number of rational points of increasing height. The key analytic ingredient is...

Quaternionic modular forms (QMFs) are a type of non-holomorphic automorphic function that exist on certain forms of the exceptional groups, and on orthogonal groups SO(4,n) with n at least 3.  They have a robust notion of Fourier coefficients...

We show that for every quadratic extension of number fields K/F, there exists an abelian variety A/F of positive rank whose rank does not grow upon base change to K. By work of Shlapentokh, this implies that Hilbert's tenth problem over the ring of...

I will discuss recent results for algebraicity of critical values of Spin L-functions for GSp_6. I will also discuss ongoing work toward the construction of p-adic L-functions interpolating these values. I will explain how this work fits into the...

I will discuss joint work in progress with Will Sawin on function field multiple Dirichlet series constructed based on a set of axioms from algebraic geometry. These series have applications to moments of Dirichlet L-functions for characters of...

A construction of Thurston assigns a hyperbolic 3-manifold to any polyhedron; a natural question is: which such are arithmetic? We report on ongoing work aiming to answer this question. 

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The main source for the works of Archimedes is a medieval manuscript, written originally around in 975 CE and then erased and covered with a prayerbook in the year 1229. Briefly studied in 1906, the full contents of the manuscripts were finally...