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Number Theory seminar abstracts, Term 3 2017-18



Betina, Adel

On the failure of Gorensteinness at weight 1 Eisenstein points of the eigencurve, by Adel Betina


Coleman and Mazur Introduced the p-adic eigencurve, a rigid analytic space parametrizing the system of Hecke eigenvalues of p-adic families of finite slope. We know from the results of Hida and Coleman that the eigencurve is étale over the weight space at classical non critical points of cohomological weight. Moreover, Bellaiche-Dimitrov proved that the eigencurve is smooth at classical p-regular weight one forms and they gave a precise criterion for etalness over the weight space. However, the geometry of the eigencurve is still misterious at classical irregular weight one forms. I will present in this talk a joint work with Dimitrov and Pozzi in which we describe the geometry of the eigencurve at irregular weight one Eisenstein series. Such forms are not cuspidal in a classical sense, but they become cuspidal when viewed as p-adic modular forms. Thus, they give rise to points that belong to the intersection of the Eisenstein locus and the cuspidal locus of the eigencurve. We proved that the cuspidal p-adic eigencurve is etale over the weight space at any irregular classical weight 1 Eisenstein point, and that cuspidal locus meets transversely each of the two Eisenstein components of the eigencurve passing through that point and give a new proof of Greenberg–Ferrero’s theorem on the order of vanishing of the Kubota-Leopoldt p-adic L-function. Finally, we prove that the local ring of C at f is Cohen-Macaulay but not Gorenstein and compute the q-expansions of a basis of overconvergent weight 1 modular forms lying in the same generalised eigenspace as f.


Dietmann, Rainer

Lines on cubic hypersurfaces

Abstract: Wooley has shown that every rational cubic hypersurface defined by a cubic form in at least 37 variables contains a rational line. In joint work with Julia Brandes we were able to improve this in the generic case of non-singular cubic forms, reducing the required number of variables to 31, applying recent work of Browning, Dietmann and Heath-Brown on intersections of cubic and quadric hypersurfaces. Time permitting, we also briefly want to discuss the related problem of finding lines on cubic hypersurfaces over p-adic fields.


Harron, Piper

Title: Equidistribution of Shapes of Number Fields of degree 3, 4, and 5

Abstract: In her talk, Piper Harron will introduce the ideas that there are number fields, that number fields have shapes, and that these shapes are everywhere you want them to be. This result is joint work with Manjul Bhargava and uses his counting methods which currently we only have for cubic, quartic, and quintic fields. She will sketch the proof of this result and leave the rest as an exercise for the audience. (Check your work by downloading her thesis!)

Heuer, Ben

On elliptic curves over perfectoid fields

In his work on torsion in the cohomology of locally symmetric varieties, Scholze constructed a perfectoid modular curve at infinite level. In this talk, we discuss how perfectoid modular curves can be used to "tilt" elliptic curves: To an elliptic curve over a perfectoid field $K$ of characteristic 0, equipped with certain extra data, one can associate an elliptic curve over the tilt $K^{\flat}$. This construction is functorial and can be extended to an equivalence of categories between elliptic curves with extra data over perfectoid algebras of characteristics 0 and p, respectively.

We will describe how various objects attached to elliptic curves can be translated into each other via the correspondence, and discuss a few consequences for another perfectoid space attached to elliptic curves over perfectoid fields: The inverse limit of the p-multiplication tower.

Humphries, Peter

Title: The Conductor and the Newform for Representations of GL_n(R) and GL_n(C)

Abstract: There is a well-known theory of decomposing spaces of automorphic forms into subspaces spanned by newforms and oldforms, and associated to a newform is its conductor. This theory can be reinterpreted as a local statement, and generalised to GL_n, as distinguishing certain vectors in a representation of GL_n(F), where F is a nonarchimedean local field, and associating to this representation a conductor. Such a local theory was previously not well understood for archimedean fields. In this talk, I will introduce this theory in this hitherto unexplored setting.


Lee, Min

Twist-minimal trace formulas and applications

Abstract: One of the most-well known examples of L-functions of degree 2 are L-functions of modular forms. Less well known, but equally important are the L-funcitons of Maass forms. A Maass form is a function on a hyperbolic surface which is also an eigenfunction of the Laplace-Beltrami operator. Named after H. Maass, who discovered some examples in the 1940s, Maass forms remain largely mysterious.

Fortunately, there are concrete tools to study Maass forms: trace formulas, which relate the spectrum of the Laplace operator on a hyperbolic surface to its geometry. After Selberg introduced his famous trace formula in 1956, his ideas were generalised, and various trace formulas have been constructed and studied. However, there are few numerical results from trace formulas, the main obstacle being their complexity. Various types of trace formulas are investigated,
constructed and used to understand automorphic representations and their L-functions from theoretical point of view, but most of them are not explicit enough to implement in computer code.

In this talk, we present a fully explicit version of the Selberg trace formula for twist-minimal Maass forms of weight 0, and and its applications.

This is a joint work with Andrew Booker and Andreas Strömbergsson.


Maynard, James
Fractional parts of polynomials
Let $f_1$,..., $f_k$ be real polynomials with no constant term and degree at most $d$. We will talk about work in progress showing that there are integers $n$ such that the fractional part of each of the $f_i(n)$ is very small, with the quantitative bound being essentially optimal in the k-aspect. This is based on the interplay between Fourier analysis, Diophantine approximation and the geometry of numbers. In particular, the key idea is to find strong additive structure in Fourier coefficients.
Milinovich, Micah
Title: Fourier Analysis and the zeros of the Riemann zeta-function
Abstract: I will show how the classical Beurling-Selberg extremal problem in harmonic analysis arises naturally when studying the vertical distribution of the zeros of the Riemann zeta-function and other L-functions. Using this relationship, along with techniques from Fourier analysis and reproducing kernel Hilbert spaces, we can prove the sharpest known bounds for the number of zeros in an interval on the critical line and we can also study the pair correlation of zeros. Our results on pair correlation extend earlier work of P. X. Gallagher and give some evidence for the well-known conjecture of H. L. Montgomery. This talk is based on a series of papers that are joint with E. Carneiro, V. Chandee, and F. Littmann.
Mocanu, Andreea

On Eisenstein series for Jacobi forms of lattice index, by Andreea Mocanu

Jacobi forms arise naturally in number theory in several ways: theta functions arise as functions of lattices and Siegel modular forms give rise to Jacobi forms through their Fourier-Jacobi expansion, for example. Jacobi forms of lattice index appear in the theory of reflective modular forms and that of vertex operator algebras. In this talk, we introduce Eisenstein series for Jacobi forms of lattice index and we discuss some of their properties, such as their orthogonality to cusp forms and their Fourier expansion. In particular, we give an explicit formula for the Fourier coefficients of the trivial Eisenstein series. We also discuss the relation between Eisenstein series and so-called level raising operators in the context of newforms.


Walker, Aled
Additive energy and the metric poissonian property
Abstract: Let A be a set of natural numbers. The metric poissonian property, which concerns the distribution of dilates of A modulo 1, was first introduced to pure mathematics by Rudnick and Sarnak in 1998, motivated by applications to quantum mechanics. It has recently received renewed attention, owing to the discovery of a strong link between the additive energy of A (a crude measure of the additive structure of A) and whether or not A enjoys the metric poissonian property. In this talk we will discuss our recent work on the quantitative dependence between these two notions, and discuss to what extent a Khintchine-type threshold may be expected to hold: that is, to what extent the metric poissonian property is determined by whether or not a certain sum of additive energies is convergent or divergent.