scholarly journals Eichler Maps and Hyperbolic Fourier Expansion

1970 ◽  
Vol 40 ◽  
pp. 173-192 ◽  
Author(s):  
Toyokazu Hiramatsu
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Fourier Expansion ◽  
Automorphic Forms ◽  
Hilbert Modular Forms ◽  
Ternary Quadratic Forms ◽  
Hilbert Modular ◽  
Lecture Notes

In his lecture notes ([1, pp. 33-35], [2, pp. 145-152]), M. Eichler reduced ‘quadratic’ Hilbert modular forms of dimension —k {k is a positive integer) to holomorphic automorphic forms of dimension — 2k for the reproduced groups of indefinite ternary quadratic forms, by means of so-called Eichler maps.

scholarly journals On the Bloch–Kato conjecture for Hilbert modular forms

2021 ◽  
Author(s):  
Matteo Tamiozzo
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Euler System ◽  
Central Point ◽  
Shimura Curves ◽  
Hilbert Modular Forms ◽  
Reciprocity Laws ◽  
Hilbert Modular ◽  
Special Points

AbstractThe aim of this paper is to prove inequalities towards instances of the Bloch–Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the L-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.

scholarly journals Mixed Hilbert modular forms and families of abelian varieties

1997 ◽  
Vol 39 (2) ◽  
pp. 131-140 ◽  
Author(s):  
Min Ho Lee
Keyword(s):  
Modular Forms ◽  
Fuchsian Group ◽  
Automorphic Forms ◽  
Elliptic Surface ◽  
Hilbert Modular Forms ◽  
Monodromy Representation ◽  
General Elliptic ◽  
Hilbert Modular ◽  
Upper Half Plane ◽  
Image Position

In [18] Shioda proved that the space of holomorphic 2-forms on a certain type of elliptic surface is canonically isomorphic to the space of modular forms of weight three for the associated Fuchsian group. Later, Hunt and Meyer [6] made an observation that the holomorphic 2-forms on a more general elliptic surface should in fact be identified with mixed automorphic forms associated to an automorphy factor of the formfor z in the Poincaré upper half plane ℋ, g = and χ(g) = , where g is an element of the fundamental group Γ⊂PSL(2, R) of the base space of the elliptic fibration, χ-Γ→SL(2, R) the monodromy representation, and w: ℋ→ℋ the lifting of the period map of the elliptic surface.

On adelic automorphic forms with respect to a quadratic extension

2000 ◽  
Vol 62 (1) ◽  
pp. 29-43 ◽  
Author(s):  
Ze-Li Dou
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Quadratic Extension ◽  
Algebraic Number Field ◽  
Hilbert Modular Forms ◽  
Arithmetic Properties ◽  
Theta Lift ◽  
Hilbert Modular ◽  
Algebraic Foundation ◽  
Totally Real

Let E/F be a totally real quadratic extension of a totally real algebraic number field. The author has in an earlier paper considered automorphic forms defined with respect to a quaternion algebra BE over E and a theta lift from such quaternionic forms to Hilbert modular forms over F. In this paper we construct adelic forms in the same setting, and derive explicit formulas concerning the action of Hecke operators. These formulas give an algebraic foundation for further investigations, in explicit form, of the arithmetic properties of the adelic forms and of the associated zeta and L-functions.

Overconvergent Families of Siegel–Hilbert Modular Forms

2015 ◽  
Vol 67 (4) ◽  
pp. 893-922 ◽  
Author(s):  
Chung Pang Mok ◽  
Fucheng Tan
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Galois Representations ◽  
Hilbert Modular Forms ◽  
Hilbert Modular

AbstractWe construct one-parameter families of overconvergent Siegel-Hilbert modular forms. This result has applications to the construction of Galois representations for automorphic forms of noncohomological weights.

CONGRUENCE PRIMES FOR SIEGEL MODULAR FORMS OF PARAMODULAR LEVEL AND APPLICATIONS TO THE BLOCH–KATO CONJECTURE

2020 ◽  
pp. 1-22
Author(s):  
JIM BROWN ◽  
HUIXI LI
Keyword(s):  
Functional Equation ◽  
Modular Forms ◽  
Class Number ◽  
Automorphic Forms ◽  
Siegel Modular Forms ◽  
Real Field ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
Free Level ◽  
Totally Real

Abstract It has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel–Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito–Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch–Kato conjecture for elliptic newforms of square-free level and odd functional equation.

scholarly journals Theta-functions and Hilbert modular forms

1978 ◽  
Vol 69 ◽  
pp. 97-106 ◽  
Author(s):  
Stephen S. Kudla
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Integral Kernel ◽  
Theta Functions ◽  
Cusp Forms ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
History Of ◽  
Indefinite Quadratic

The purpose of this note is to show how the theta-functions attached to certain indefinite quadratic forms of signature (2, 2) can be used to produce a map from certain spaces of cusp forms of Nebentype to Hilbert modular forms. The possibility of making such a construction was suggested by Niwa [4], and the techniques are the same as his and Shintani’s [6]. The construction of Hilbert modular forms from cusp forms of one variable has been discussed by many people, and I will not attempt to give a history of the subject here. However, the map produced by the theta-function is essentially the same as that of Doi and Naganuma [2], and Zagier [7]. In particular, the integral kernel Ω(τ, z1, z2) of Zagier is essentially the ‘holomorphic part’ of the theta-function.

scholarly journals Mixed Jacobi-like forms of several variables

2006 ◽  
Vol 2006 ◽  
pp. 1-14
Author(s):  
Min Ho Lee
Keyword(s):  
Half Plane ◽  
Modular Forms ◽  
Automorphic Form ◽  
Automorphic Forms ◽  
Hilbert Modular Forms ◽  
Equivariant Maps ◽  
Several Variables ◽  
Hilbert Modular ◽  
Upper Half Plane

We study mixed Jacobi-like forms of several variables associated to equivariant maps of the Poincaré upper half-plane in connection with usual Jacobi-like forms, Hilbert modular forms, and mixed automorphic forms. We also construct a lifting of a mixed automorphic form to such a mixed Jacobi-like form.

scholarly journals Ternary quadratic forms and half-integral weight modular forms

2012 ◽  
Vol 15 ◽  
pp. 418-435 ◽  
Author(s):  
Alia Hamieh
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Ternary Quadratic Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Shimura Correspondence

AbstractLet k be a positive integer such that k≡3 mod 4, and let N be a positive square-free integer. In this paper, we compute a basis for the two-dimensional subspace Sk/2(Γ0(4N),F) of half-integral weight modular forms associated, via the Shimura correspondence, to a newform F∈Sk−1(Γ0(N)), which satisfies $L(F,\frac {1}{2})\neq 0$. This is accomplished by using a result of Waldspurger, which allows one to produce a basis for the forms that correspond to a given F via local considerations, once a form in the Kohnen space has been determined.

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