Lifting of Maass forms to O(1,8n+1) and applications to the sup-norm problem
2025-02-04 16:00 - 17:00 (Europe/Berlin) | Zoom
International Seminar on Automorphic Forms
Ameya Pitale (University of Oklahoma): Lifting of Maass forms to O(1,8n+1) and applications to the sup-norm problem
In a joint paper with Yingkun Li and Hiroaki Narita, we had constructed liftings from Maass forms with respect to SL_2(Z) to Maass forms on O(1,8n+1), which violated the Generalized Ramanujan conjecture. These were constructed via Borcherds theta lifts and we were able to give explicit formulas for their Fourier coefficients. In a recent joint work with Simon Marshall and Hiroaki Narita, we first computed the Petersson inner product of the lift using the Rallis inner product formula. This essentially involves an archimedean integral computation. These are usually very complicated and intractable, but in this case we are able to get an exact formula for the Petersson norm. Explicit formulas for the Fourier coefficients and Petersson norm are the essential ingredients of one of the approaches to obtain sup-norm bounds on these Maass forms. Investigations regarding sup-norm bounds for modular forms in the GL(2) case has been recently a very active area of research. Using the method mentioned above, as well as a pre-trace formula approach, we obtain the first sup-norm bounds results for these orthogonal groups.
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)

Pro-étale Q_p-cohomology of rigid analytic spaces
2025-02-06 14:15 - 15:15 | Mainz, Hilbertraum (05-432)
Arthur-César Le Bras (Strasbourg)
Abstract: The pro-étale Q_p-cohomology of rigid analytic spaces exhibits surprising features. I would like to explain a joint work with Johannes Anschütz and Lucas Mann which provides a conceptual explanation for these phenomena.

Regular subdivisions and bounds on initial ideals
2025-02-07 14:30 - 15:30 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
TGiF-Seminar: Tropical geometry in Frankfurt (Second meeting Winter Semester 2024/25)
George Balla (TU Berlin)
Abstract: We extend two known constructions that relate regular subdivisions to initial degenerations of projective toric varieties and Grassmannians. We associate a point configuration A with any homogeneous ideal I. We obtain upper and lower bounds on each initial ideal of I in terms of regular subdivisions of A. We also investigate when these bounds are exact, for example, the lower bound is exact for every initial ideal of the Plücker ideal I(2,n) with respect to points in the tropicalization. This talk is based on joint work with Dan Corey, Igor Makhlin, and Victoria Schleis.

Seminar on Arithmetic Geometry
2025-02-07 15:30 - 17:00 | Darmstadt, Room 401 and Zoom, Schlossgartenstraße 7, Darmstadt, 64289, Germany
Milton Lin (Johns Hopkins University): Geometric Casselman-Shalika in mixed characteristic
I will present joint work with Ashwin Iyengar and Konrad Zou, where we proved a geometric analog of the (local) Casselman-Shalika formula for split connected reductive groups over mixed characteristic local fields. This formula captures properties of Fourier coefficients of automorphic functions that are fundamental to the Langlands correspondence.
Zoom (635 7328 0984, Kenncode: kleinste sechsstellige Primzahl

Moduli spaces of twisted maps to smooth pairs
2025-02-07 16:00 - 17:00 | Frankfurt, Robert-Mayer-Str. 10, Raum 711 groß
TGiF-Seminar: Tropical geometry in Frankfurt (Second meeting Winter Semester 2024/25)
Robert Crumplin (Universität Heidelberg)

Abstract: The question of counting maps from marked curves with fixed tangency conditions to a divisor in the target has been studied extensively over the past 15 years. One way of formulating these enumerative problems is via twisted maps to a root stack. I will describe the geometry of moduli spaces of twisted maps using tropical techniques, in particular giving new understanding to universal structural results of orbifold Gromov–Witten invariants. If time permits, I will talk about upcoming work with Sam Johnston which relates these moduli spaces to their logarithmic counterparts and provides a splitting of the virtual class in terms of the aforementioned tropical data.

Linking numbers and non-holomorphic Siegel modular forms
2025-02-11 16:00 - 17:00 | Zoom
International Seminar on Automorphic Forms
Mads Christensen (UCL):Linking numbers and non-holomorphic Siegel modular forms
In an arithmetic hyperbolic 3-manifold there is an abundance of naturally defined closed geodesics. I will present a result which relates linking number invariants of these geodesics to the Fourier coefficients of certain non-holomorphic Siegel modular forms of genus 2. .
https://tu-darmstadt.zoom.us/j/68048280736
The password is the first Fourier coefficient of the modular j-function (as digits)
