On the cubic Shimura lift for PGL3

2001 ◽  
Vol 126 (1) ◽  
pp. 289-307
Author(s):  
Daniel Bump ◽  
Solomon Friedberg ◽  
David Ginzburg
Keyword(s):  
Shimura Lift

scholarly journals Unitary cycles on Shimura curves and the Shimura lift II

2014 ◽  
Vol 150 (12) ◽  
pp. 1963-2002
Author(s):  
Siddarth Sankaran
Keyword(s):  
Modular Form ◽  
Recent Work ◽  
Chow Groups ◽  
Shimura Varieties ◽  
Shimura Curves ◽  
Shimura Lift ◽  
Generating Series

AbstractWe consider two families of arithmetic divisors defined on integral models of Shimura curves. The first was studied by Kudla, Rapoport and Yang, who proved that if one assembles these divisors in a formal generating series, one obtains the$q$-expansion of a modular form of weight 3/2. The present work concerns the Shimura lift of this modular form: we identify the Shimura lift with a generating series comprising divisors arising in the recent work of Kudla and Rapoport regarding cycles on Shimura varieties of unitary type. In the prequel to this paper, the author considered the geometry of the two families of cycles. These results are combined with the Archimedean calculations found in this work in order to establish the theorem. In particular, we obtain new examples of modular generating series whose coefficients lie in arithmetic Chow groups of Shimura varieties.

A magnetic modular form

2019 ◽  
Vol 15 (05) ◽  
pp. 907-924
Author(s):  
Yingkun Li ◽  
Michael Neururer
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Hecke Operators ◽  
Shimura Lift ◽  
Divisibility Property ◽  
Vector Valued

In this paper, we prove a conjecture of Broadhurst and Zudilin concerning a divisibility property of the Fourier coefficients of a meromorphic modular form using the generalization of the Shimura lift by Borcherds and Hecke operators on vector-valued modular forms developed by Bruinier and Stein. Furthermore, we construct a family of meromorphic modular forms with this property.

scholarly journals Newforms of half-integral weight

1996 ◽  
Vol 143 ◽  
pp. 147-169 ◽  
Author(s):  
Thomas R. Shemanske
Keyword(s):  
Shift Operator ◽  
Hecke Eigenvalues ◽  
Half Integral Weight ◽  
Shimura Lift ◽  
Multiplicity One ◽  
Integral Weight ◽  
Definition Of ◽  
And Shifts ◽  
Half Integral Weight Forms ◽  
Space Of Cusp Forms

Two very different definitions of a newform of half-integral weight are present and continued to be developed in the literature. The first definition originated with Serre and Stark for forms of weight 1/2 [5], and is analogous to the definition of newform for integral weight forms, which uses forms of lower level and shifts of such forms to characterize the notion of old-forms. The second definition originated with Kohnen for half-integral weight forms of squarefree level [1], who used forms of lower level and their image under the Um2 operator to define the notion of oldforms. The choice of the Um2 operator over the shift operator Bd seems a propitious one, since the U operator commutes with the action of the Shimura lift, while the shift operator B does not. More to the point, Kohnen was able to develop a newform theory on a distinguished subspace of the full space of cusp forms (now referred to as the Kohnen subspace), and obtained a multiplicity-one result (with respect to Hecke eigenvalues) for half-integral weight newforms in this subspace. Even nicer, the multiplicity-one result was established by showing that there is a one-to-one correspondence between newforms of level AN in the subspace and the newforms of integral weight of level N.

On the Shimura lift for Hilbert modular forms

1993 ◽  
pp. 561-569
Author(s):  
Thomas R. Shemanske ◽  
Lynne H. Walling
Keyword(s):  
Modular Forms ◽  
Hilbert Modular Forms ◽  
Shimura Lift ◽  
Hilbert Modular

scholarly journals Hecke structure of spaces of half-integral weight cusp forms

2000 ◽  
Vol 159 ◽  
pp. 53-85 ◽  
Author(s):  
Sharon M. Frechette
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Dirichlet Character ◽  
Cusp Forms ◽  
Half Integral Weight ◽  
Shimura Lift ◽  
Integral Weight ◽  
Structure Theorems ◽  
Space Of Cusp Forms ◽  
Shimura Correspondence

We investigate the connection between integral weight and half-integral weight modular forms. Building on results of Ueda [14], we obtain structure theorems for spaces of half-integral weight cusp forms Sk/2(4N,χ) where k and N are odd nonnegative integers with k ≥ 3, and χ is an even quadratic Dirichlet character modulo 4N. We give complete results in the case where N is a power of a single prime, and partial results in the more general case. Using these structure results, we give a classical reformulation of the representation-theoretic conditions given by Flicker [5] and Waldspurger [17] in results regarding the Shimura correspondence. Our version characterizes, in classical terms, the largest possible image of the Shimura lift given our restrictions on N and χ, by giving conditions under which a newform has an equivalent cusp form in Sk/2(4N, χ). We give examples (computed using tables of Cremona [4]) of newforms which have no equivalent half-integral weight cusp forms for any such N and χ. In addition, we compare our structure results to Ueda’s [14] decompositions of the Kohnen subspace, illustrating more precisely how the Kohnen subspace sits inside the full space of cusp forms.

Shimura lift of real analytic Poincaré series and Hilbert modular Eisenstein series

1998 ◽  
Vol 229 (3) ◽  
pp. 547-574 ◽  
Author(s):  
Roland Matthes
Keyword(s):  
Eisenstein Series ◽  
Poincaré Series ◽  
Poincare Series ◽  
Shimura Lift ◽  
Hilbert Modular ◽  
Real Analytic

scholarly journals On the Shimura lift, après Selberg

1989 ◽  
Vol 32 (1) ◽  
pp. 58-64 ◽  
Author(s):  
Barry A. Cipra
Keyword(s):  
Shimura Lift
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