Theta lifts of Bianchi modular forms and applications to paramodularity

Tobias Berger, Lassina Dembélé, Ariel Pacetti, Mehmet Haluk Şengün

Type: Article

Publication Date: 2015-07-07

Citations: 13

DOI: https://doi.org/10.1112/jlms/jdv023

Abstract

We explain how the work of Johnson-Leung and Roberts on lifting Hilbert modular forms for real quadratic fields to Siegel modular forms can be adapted to imaginary quadratic fields. For this, we use archimedean results from Harris, Soudry and Taylor and replace the global arguments of Roberts by the non-vanishing result of Takeda. As an application of our lifting result, we exhibit an abelian surface B defined over Q, which is not a restriction of scalars of an elliptic curve and satisfies the paramodularity Conjecture of Brumer and Kramer.

Locations

Journal of the London Mathematical Society - View - PDF
arXiv (Cornell University) - View - PDF
Americanae (AECID Library) - View - PDF
White Rose Research Online (University of Leeds) - View - PDF
DataCite API - View

Similar Works

Action	Title	Year	Authors
+	On the non-vanishing of theta liftings of Bianchi modular forms to Siegel modular forms	2020	Di Zhang
+	On the non-vanishing of theta lifting of Bianchi modular forms to Siegel modular forms	2024	Di Zhang
+	Liftings and Borcherds Products	2017	Eric Hofmann
+	TWISTED BORCHERDS PRODUCTS ON HILBERT MODULAR SURFACES AND THE REGULARIZED THETA LIFT	2010	Stephan Ehlen
+ PDF Chat	On the paramodularity of typical abelian surfaces (and reduction of G-covariant bilinear forms)	2018	Armand Brumer Ariel Pacetti Cris Poor Gonzalo Tornaría John Voight David S. Yuen
+ PDF Chat	Examples of abelian surfaces with everywhere good reduction	2015	Lassina Dembélé Abhinav Kumar
+ PDF Chat	On the Divisibility Properties of the Fourier Coefficients of Meromorphic Hilbert Modular Forms	2024	Baptiste Depouilly
+	Local torsion on abelian surfaces with real multiplication by ${\bf Q}(\sqrt{5})$	2014	Adam Gamzon
+	Modularity lifting results in parallel weight one and applications to the Artin conjecture: the tamely ramified case	2012	Payman L Kassaei Shu Sasaki Yichao Tian
+	Cohomological vanishing on Siegel modular varieties and applications to lifting Siegel modular forms	2014	Alexandru Ghitza Scott Mullane
+	On the paramodularity of typical abelian surfaces	2018	Armand Brumer Ariel Pacetti Cris Poor Gonzalo Tornaría John Voight David S. Yuen
+ PDF Chat	Reciprocity laws and Galois representations: recent breakthroughs	2015	Jared Weinstein
+ PDF Chat	On the paramodularity of typical abelian surfaces	2019	Armand Brumer Ariel Pacetti Cris Poor Gonzalo Tornaría John Voight David S. Yuen
+ PDF Chat	Constructing Arithmetic Siegel Modular Forms: Theta Lifting and Explicit Methods for Real Multiplication Abelian Surfaces	2025	Robert L. Jackson
+ PDF Chat	Twisted Borcherds products on Hilbert modular surfaces and their CM values	2007	Jan Hendrik Bruinier Tonghai Yang
+	Examples of abelian surfaces failing the local-global principle for isogenies	2020	Barinder S. Banwait
+	Examples of abelian surfaces failing the local-global principle for isogenies	2020	Barinder S. Banwait
+ PDF Chat	Paramodular groups and theta series	2023	Siegfried Böcherer Rainer Schulze‐Pillot
+ PDF Chat	Borcherds lifts of harmonic Maass forms and modular integrals	2019	Markus Schwagenscheidt
+	On the Bloch-Kato conjecture for the Asai L-function	2015	Tobias Berger

Works That Cite This (13)

Action	Title	Year	Authors
+ PDF Chat	Certain abelian varieties bad at only one prime	2018	Armand Brumer Kenneth Kramer
+ PDF Chat	Genuine Bianchi modular forms of higher level at varying weight and discriminant	2019	Alexander Rahm Panagiotis Tsaknias
+ PDF Chat	Non-vanishing of fundamental Fourier coefficients of paramodular forms	2017	Jolanta Marzec
+	Abelian Surfaces over totally real fields are Potentially Modular	2018	George Boxer Frank Calegari Toby Gee Vincent Pilloni
+	Genus two curves with everywhere good reduction over quadratic fields	2021	Andrzej Dąbrowski Mohammad Sadek
+ PDF Chat	On the paramodularity of typical abelian surfaces	2019	Armand Brumer Ariel Pacetti Cris Poor Gonzalo Tornaría John Voight David S. Yuen
+ PDF Chat	Abelian surfaces over totally real fields are potentially modular	2021	George Boxer Frank Calegari Toby Gee Vincent Pilloni
+ PDF Chat	ON FUNDAMENTAL FOURIER COEFFICIENTS OF SIEGEL CUSP FORMS OF DEGREE 2	2021	Jesse Jääsaari Stephen Lester Abhishek Saha
+ PDF Chat	Examples of genuine QM abelian surfaces which are modular	2019	Ciaran Schembri
+	Corrigendum to “Paramodular abelian varieties of odd conductor"	2019	Armand Brumer Kenneth Kramer

Works Cited by This (43)

Action	Title	Year	Authors
+	Invariant representations of GSp(2) under tensor product with a quadratic character	2010	Ping-Shun Chan
+ PDF Chat	Hyperbolic Tessellations Associated to Bianchi Groups	2010	Dan Yasaki
+	Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields	1984	J. E. Cremona
+	On the critical values of L-functions of GL(2) and GL(2)×GL(2)	1994	Haruzo Hida
+ PDF Chat	The Cohomology of Lattices in SL(2, ℂ)	2010	Tobias Finis Fritz Grunewald Paulo Tirao
+ PDF Chat	Proving modularity for a given elliptic curve over an imaginary quadratic field	2009	Luis Dieulefait Lucio Guerberoff Ariel Pacetti
+	Critical values of the twisted tensor L-function in the imaginary quadratic case	1999	Eknath Ghate
+	The Magma Algebra System I: The User Language	1997	Wieb Bosma John Cannon Catherine Playoust
+	Module de congruences pour GL(2) d’un corps imaginaire quadratique et théorie d’Iwasawa d’un corps CM biquadratique	1998	Eric Urban
+	Eisenstein cohomology of arithmetic groups. The case GL2	1987	G. Harder