Bianchi Modular Forms Research Articles

On the non-vanishing of theta lifting of Bianchi modular forms to Siegel modular forms

On the non-vanishing of theta lifting of Bianchi modular forms to Siegel modular forms

Sharp bounds for multiplicities of Bianchi modular forms

Sharp bounds for multiplicities of Bianchi modular forms

Bianchi period polynomials: Hecke action and congruences

Let Γ=PSL2(O)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma ={{\\,\\mathrm{\ extrm{PSL}}\\,}}_2(\\mathcal {O})$$\\end{document} be a Bianchi group associated to one of the five Euclidean imaginary quadratic fields. We show that the space of weight k period polynomials for Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} is “dual” to the space of weight k modular symbols for Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}, reflecting the duality between the first and second cohomology groups of the arithmetic group Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}. Using this result, we describe the action of Hecke operators on the space of period polynomials for Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} via the Heilbronn matrices. As in the classical case, spaces of Bianchi period polynomials are related to parabolic cohomology of Bianchi groups, and in turn, via the Eichler–Shimura–Harder Isomorphism, to spaces of Bianchi modular forms. In the second part of the paper, we numerically investigate congruences between level 1 Bianchi eigenforms via computer programs which implement the above mentioned Hecke action on spaces of Bianchi period polynomials. Computations with the Hecke action are used to indicate moduli of congruences between the underlying Bianchi forms; we then prove the congruences using the period polynomials. From this we find congruences between genuine Bianchi modular forms and both a base-change Bianchi form and an Eisenstein series. We believe these congruences are the first of their kind in the literature.

Flux vacua and modularity for $\mathbb{Z}_2$ symmetric Calabi-Yau manifolds

We find continuous families of supersymmetric flux vacua in IIB Calabi-Yau compactifications for multiparameter manifolds with an appropriate \mathbb{Z}_{2}ℤ2 symmetry. We argue, supported by extensive computational evidence, that the numerators of the local zeta functions of these compactification manifolds have quadratic factors. These factors are associated with weight-two modular forms, these manifolds being said to be weight-two modular. Our evidence supports the flux modularity conjecture of Kachru, Nally, and Yang. The modular forms are related to a continuous family of elliptic curves. The flux vacua can be lifted to F-theory on elliptically fibred Calabi-Yau fourfolds. If conjectural expressions for Deligne’s periods are true, then these imply that the F-theory fibre is complex-isomorphic to the modular curve. In three examples, we compute the local zeta function of the internal geometry using an extension of known methods, which we discuss here and in more detail in a companion paper. With these techniques, we are able to compare the zeta function coefficients to modular form Fourier coefficients for hundreds of manifolds in three distinct families, finding agreement in all cases. Our techniques enable us to study not only parameters valued in \mathbb{Q}ℚ but also in algebraic extensions of \mathbb{Q}ℚ, so exhibiting relations to Hilbert and Bianchi modular forms. We present in appendices the zeta function numerators of these manifolds, together with the corresponding modular forms.

Bianchi modular symbols and p-adic L-functions

In the present paper, we prove that the first homology group of Bianchi 3-fold is generated by the special Bianchi modular symbols. Also, we construct the integral valued p-adic L-function of p-ordinary Bianchi Hecke eigenforms by taking the Lefschetz-Poincaré Pairing of the cohomology class attached to Bianchi modular forms and Bianchi modular symbols on Bianchi 3-folds. By using the homology generation result, we prove that the μ-invariant of some isotopic components of the p-adic L-function of certain Bianchi Hecke eigenforms vanishes for a positive proportion of ordinary prime ideals. The main ingredient of the proof of the generation result is analyzing the fast convergent series expression of the integration of the Bianchi modular forms along the special modular symbols.

Functional equation of the p-adic L-function of Bianchi modular forms

Let K be an imaginary quadratic field with class number 1, in this paper we obtain the functional equation of the p-adic L-function of small slope p-stabilised Bianchi modular forms. Then, using p-adic families of Bianchi modular forms, we extend our result to Σ-smooth base-change Bianchi modular forms.

Big images of two-dimensional pseudorepresentations

Bellaïche has recently applied Pink-Lie theory to prove that, under mild conditions, the image of a continuous 2-dimensional pseudorepresentation $$\rho $$ of a profinite group on a local pro-p domain A contains a nontrivial congruence subgroup of $${{\,\mathrm{{SL}}\,}}_2(B)$$ for a certain subring B of A. We enlarge Bellaïche’s ring and give this new B a conceptual interpretation both in terms of conjugate self-twists of $$\rho $$ , symmetries that constrain its image, and in terms of the adjoint trace ring of $$\rho $$ , which we show is both more natural and the optimal ring for these questions in general. Finally, we use our purely algebraic result to recover and extend a variety of arithmetic big-image results for $${{\,\mathrm{GL}\,}}_2$$ Galois representations arising from elliptic, Hilbert, and Bianchi modular forms and p-adic Hida or Coleman families of elliptic and Hilbert modular forms.

Big principal series, -adic families and -invariants

In earlier work, the first named author generalized the construction of Darmon-style$\mathcal {L}$-invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial coefficient system and Steinberg at a fixed prime. In this paper, assuming that the Archimedean component of the group has discrete series we show that these automorphic$\mathcal {L}$-invariants can be computed in terms of derivatives of Hecke eigenvalues in$p$-adic families. Our proof is novel even in the case of modular forms, which was established by Bertolini, Darmon and Iovita. The main new technical ingredient is the Koszul resolution of locally analytic principal series representations by Kohlhaase and Schraen. As an application of our results we settle a conjecture of Spieß: we show that automorphic$\mathcal {L}$-invariants of Hilbert modular forms of parallel weight$2$are independent of the sign character used to define them. Moreover, we show that they are invariant under Jacquet–Langlands transfer and, in fact, equal to the Fontaine–Mazur$\mathcal {L}$-invariant of the associated Galois representation. Under mild assumptions, we also prove the equality of automorphic and Fontaine–Mazur$\mathcal {L}$-invariants for representations of definite unitary groups of arbitrary rank. Finally, we study the case of Bianchi modular forms to show how our methods, given precise results on eigenvarieties, can also work in the absence of discrete series representations.

A finiteness result for p-adic families of Bianchi modular forms

We study p-adic families of cohomological automorphic forms for GL(2) over imaginary quadratic fields and prove that families interpolating a Zariski-dense set of classical cuspidal automorphic forms only occur under very restrictive conditions. We show how to computationally determine when this is not the case and establish concrete examples of families interpolating only finitely many Bianchi modular forms.

Stark–Heegner cycles attached to Bianchi modular forms

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F, and let P be a prime of F at which f is new. Let K be a quadratic extension of F, and L(f/K,s) the L-function of the base-change of f to K. Under certain hypotheses on f and K, the functional equation of L(f/K,s) ensures that it vanishes at the central point. The Bloch--Kato conjecture predicts that this should force the existence of non-trivial classes in an appropriate global Selmer group attached to f and K. In this paper, we use the theory of double integrals developed by Barrera Salazar and the second author to construct certain P-adic Abel--Jacobi maps, which we use to propose a construction of such classes via "Stark--Heegner cycles". This builds on ideas of Darmon and in particular generalises an approach of Rotger and Seveso in the setting of classical modular forms.

Universal Fourier expansions of Bianchi modular forms

We generalize Merel’s work on universal Fourier expansions to Bianchi modular forms over Euclidean imaginary quadratic fields, under the assumption of the nondegeneracy of a pairing between Bianchi modular forms and Bianchi modular symbols. Among the key inputs is a computation of the action of Hecke operators on Manin symbols, building upon the Heilbronn–Merel matrices constructed by Mohamed.

Examples of genuine QM abelian surfaces which are modular

Let K be an imaginary quadratic field. Modular forms for GL(2) over K are known as Bianchi modular forms. Standard modularity conjectures assert that every weight 2 rational Bianchi newform has either an associated elliptic curve over K or an associated abelian surface with quaternionic multiplication over K. We give explicit evidence in the way of examples to support this conjecture in the latter case. Furthermore, the quaternionic surfaces given correspond to genuine Bianchi newforms, which answers a question posed by Cremona as to whether this phenomenon can happen.

Genuine Bianchi modular forms of higher level at varying weight and discriminant

Bianchi modular forms are automorphic forms over an imaginary quadratic field, associated to a Bianchi group. Even though modern studies of Bianchi modular forms go back to the mid 1960's, most of the fundamental problems surrounding their theory are still wide open. Only for certain types of Bianchi modular forms, which we will call non-genuine, it is possible at present to develop dimension formulas: They are (twists of) those forms which arise from elliptic cuspidal modular forms via the Langlands Base-Change procedure, or arise from a quadratic extension of the imaginary quadratic field via automorphic induction (so-called CM-forms). The remaining Bianchi modular forms are what we call genuine, and they are of interest for an extension of the modularity theorem (formerly the Taniyama-Shimura conjecture, crucial in the proof of Femat's Last Theorem) to imaginary quadratic fields. In a preceding paper by Rahm and Sengun, an extreme paucity of genuine cuspidal Bianchi modular forms has been reported, but those and other computations were restricted to level One. In this paper, we are extending the formulas for the non-genuine Bianchi modular forms to deeper levels, and we are able to spot the first, rare instances of genuine forms at deeper level and heavier weight.

P-adic L-functions of Bianchi modular forms

The theory of overconvergent modular symbols, developed by Rob Pollack and Glenn Stevens, gives a beautiful and effective construction of the $p$-adic $L$-function of a modular form. In this paper, we give an analogue of their results for Bianchi modular forms, that is, modular forms over imaginary quadratic fields. In particular, we prove control theorems that say that the canonical specialisation map from overconvergent to classical Bianchi modular symbols is an isomorphism on small slope eigenspaces of suitable Hecke operators. We also give an explicit link between the classical modular symbol attached to a Bianchi modular form and critical values of its $L$-function, which then allows us to construct $p$-adic $L$-functions of Bianchi modular forms.

Theta lifts of Bianchi modular forms and applications to paramodularity

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.

On level one cuspidal Bianchi modular forms

AbstractIn this paper, we present the outcome of vast computer calculations, locating several of the very rare instances of level one cuspidal Bianchi modular forms that are not lifts of elliptic modular forms.

Weight reduction for mod ℓ Bianchi modular forms

Let K be an imaginary quadratic field with class number one and ℓ be a rational prime that splits in K. We prove that mod ℓ, a system of Hecke eigenvalues occurring in the first cohomology group of some congruence subgroup Γ of SL 2 ( O K ) can be realized, up to twist, in the first cohomology with trivial coefficients after increasing the level of Γ by ( ℓ).