scholarly journals p-adic modular forms of non-integral weight over Shimura curves

2012 ◽  
Vol 149 (1) ◽  
pp. 32-62 ◽  
Author(s):  
Riccardo Brasca
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Hecke Operators ◽  
Analytic Space ◽  
Shimura Curves ◽  
Rigid Analytic Space ◽  
Integral Weight ◽  
Set Up ◽  
Totally Real ◽  
Totally Real Fields

AbstractIn this work, we set up a theory of p-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of kth invariant differentials over the Shimura curves we are interested in, for any p-adic character. In this way, we are able to introduce the notion of overconvergent modular form of any p-adic weight. Moreover, our sheaves can be put in p-adic families over a suitable rigid analytic space, that parametrizes the weights. Finally, we define Hecke operators, including the U operator, that acts compactly on the space of overconvergent modular forms. We also construct the eigencurve.

scholarly journals Construction of siegel modular forms of degree three and commutation relations of Hecke operators

1985 ◽  
Vol 100 ◽  
pp. 83-96 ◽  
Author(s):  
Yoshio Tanigawa
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Siegel Modular Forms ◽  
Hecke Operators ◽  
Weil Representation ◽  
Commutation Relations ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Shimura Correspondence

In connection with the Shimura correspondence, Shintani [6] and Niwa [4] constructed a modular form by the integral with the theta kernel arising from the Weil representation. They treated the group Sp(1) × O(2, 1). Using the special isomorphism of O(2, 1) onto SL(2), Shintani constructed a modular form of half-integral weight from that of integral weight. We can write symbolically his case as “O(2, 1)→ Sp(1)” Then Niwa’s case is “Sp(l)→ O(2, 1)”, that is from the halfintegral to the integral. Their methods are generalized by many authors. In particular, Niwa’s are fully extended by Rallis-Schiffmann to “Sp(l)→O(p, q)”.

scholarly journals The Modularity of Special Cycles on Orthogonal Shimura Varieties over Totally Real Fields under the Beilinson–Bloch Conjecture

2020 ◽  
pp. 1-15
Author(s):  
Yota Maeda
Keyword(s):  
Modular Form ◽  
Chow Group ◽  
Siegel Modular Form ◽  
Real Field ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Generating Series ◽  
Totally Real ◽  
Totally Real Fields ◽  
Real Places

Abstract We study special cycles on a Shimura variety of orthogonal type over a totally real field of degree d associated with a quadratic form in $$n+2$$ variables whose signature is $$(n,2)$$ at e real places and $$(n+2,0)$$ at the remaining $$d-e$$ real places for $$1\leq e <d$$ . Recently, these cycles were constructed by Kudla and Rosu–Yott, and they proved that the generating series of special cycles in the cohomology group is a Hilbert-Siegel modular form of half integral weight. We prove that, assuming the Beilinson–Bloch conjecture on the injectivity of the higher Abel–Jacobi map, the generating series of special cycles of codimension er in the Chow group is a Hilbert–Siegel modular form of genus r and weight $$1+n/2$$ . Our result is a generalization of Kudla’s modularity conjecture, solved by Yuan–Zhang–Zhang unconditionally when $$e=1$$ .

ARITHMETIC TRACES OF NON-HOLOMORPHIC MODULAR INVARIANTS

2010 ◽  
Vol 06 (01) ◽  
pp. 69-87 ◽  
Author(s):  
ALISON MILLER ◽  
AARON PIXTON
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Algebraic Numbers ◽  
Heegner Points ◽  
Positive Weight ◽  
Poincare Series ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Modular Invariants

We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

scholarly journals All modular forms of weight 2 can be expressed by Eisenstein series

2020 ◽  
Vol 6 (3) ◽  
Author(s):  
Martin Raum ◽  
Jiacheng Xia
Keyword(s):  
Modular Form ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Linear Combinations ◽  
Integral Weight ◽  
Elliptic Modular Form

Abstract We show that every elliptic modular form of integral weight greater than 1 can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central $$\mathrm{L}$$ L -values present in all previous work. For weights greater than 2, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice.

scholarly journals On the divisibility of r2(n)

1977 ◽  
Vol 18 (1) ◽  
pp. 109-111 ◽  
Author(s):  
E. J. Scourfield
Keyword(s):  
Modular Form ◽  
Algebraic Number ◽  
Modular Forms ◽  
Congruence Subgroup ◽  
Algebraic Number Field ◽  
The Past ◽  
Divisibility Property ◽  
Integral Weight ◽  
Image Position ◽  
Positive Integers

During the past few years, some papers of P. Deligne and J.-P. Serre (see [2], [9], [10] and other references cited there) have included an investigation of certain properties of coefficients of modular forms, and in particular Serre [10] (see also [11]) obtained the divisibility property (1) below. Letbe a modular form of integral weight k ≧ 1 on a congruence subgroup of SL2(Z), and suppose that each cn belongs to the ring RK of integers of an algebraic number field K finite over Q. For c ∈ RK and m ≧ 1 an integer, write c ≡ 0 (mod m) if c ∈ m RK and c ≢ 0 (mod m) otherwise. Then Serre showed that there exists α > 0 such thatas x → ∞, where throughout this note N(n ≦ x: P) denotes the number of positive integers n ≦ x with the property P.

scholarly journals On Ramanujan and Dirichlet series with Euler products

1984 ◽  
Vol 25 (2) ◽  
pp. 203-206 ◽  
Author(s):  
S. Raghavan
Keyword(s):  
Modular Form ◽  
Dirichlet Series ◽  
Modular Forms ◽  
Hecke Operators ◽  
Euler Product ◽  
Arithmetical Functions ◽  
In Trans ◽  
Euler Products ◽  
Unpublished Manuscripts ◽  
Profound Insight

In his unpublished manuscripts (referred to by Birch [1] as Fragment V, pp. 247–249), Ramanujan [3] gave a whole list of assertions about various (transforms of) modular forms possessing naturally associated Euler products, in more or less the spirit of his extremely beautiful paper entitled “On certain arithmetical functions” (in Trans. Camb. Phil. Soc. 22 (1916)). It is simply amazing how Ramanujan could write down (with an ostensibly profound insight) a basis of eigenfunctions (of Hecke operators) whose associated Dirichlet series have Euler products, anticipating by two decades the famous work of Hecke and Petersson. That he had further realized, in the event of a modular form f not corresponding to an Euler product, the possibility of restoring the Euler product property to a suitable linear combination of modular forms of the same type as f, is evidently fantastic.

scholarly journals QUATERNIONIC MODULAR FORMS OF ANY WEIGHT

2014 ◽  
Vol 10 (01) ◽  
pp. 31-53 ◽  
Author(s):  
RICCARDO BRASCA
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quaternion Algebra ◽  
Hecke Operators ◽  
Line Bundles ◽  
Analytic Variety ◽  
Test Objects ◽  
Finite Slope ◽  
Geometric Definition

In this work we give a geometric definition, as sections of line bundles, of p-adic analytic families of overconvergent modular forms attached to an indefinite quaternion algebra over ℚ. As a consequence of this, we obtain the existence of an eigencurve in this context. Our theory includes the interpretation of a modular form as a rule on test objects. We introduce the Hecke operators U and T l, both in families and for a single weight. We show that the U -operator acts compactly on the space of overconvergent modular forms. We finally construct the eigencurve, a rigid analytic variety whose points correspond to systems of eigenvalues associated to overconvergent eigenforms of finite slope with respect to the U -operator.

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