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Number Theory seminar abstracts, Term 3 2017-18

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B

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.

H

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.

L

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.

M

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.
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.