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.

scholarly journals Stark–Heegner points and the Shimura correspondence

2008 ◽  
Vol 144 (5) ◽  
pp. 1155-1175 ◽  
Author(s):  
Henri Darmon ◽  
Gonzalo Tornaría
Keyword(s):  
Modular Forms ◽  
Dirichlet Character ◽  
Heegner Points ◽  
Real Quadratic Fields ◽  
Half Integral Weight ◽  
Hida Family ◽  
Integral Weight ◽  
Similar Kind ◽  
Shimura Correspondence

AbstractLet $g = \sum c(D)q^D$ and $f=\sum a_n q^n$ be modular forms of half-integral weight k+1/2 and integral weight 2k respectively that are associated to each other under the Shimura–Kohnen correspondence. For suitable fundamental discriminants D, a theorem of Waldspurger relates the coefficient c(D) to the central critical value L(f,D,k) of the Hecke L-series of f twisted by the quadratic Dirichlet character of conductor D. This paper establishes a similar kind of relationship for central critical derivatives in the special case k=1, where f is of weight 2. The role of c(D) in our main theorem is played by the first derivative in the weight direction of the Dth Fourier coefficient of a p-adic family of half-integral weight modular forms. This family arises naturally, and is related under the Shimura correspondence to the Hida family interpolating f in weight 2. The proof of our main theorem rests on a variant of the Gross–Kohnen–Zagier formula for Stark–Heegner points attached to real quadratic fields, which may be of some independent interest. We also formulate a more general conjectural formula of Gross–Kohnen–Zagier type for Stark–Heegner points, and present numerical evidence for it in settings that seem inaccessible to our methods of proof based on p-adic deformations of modular forms.

scholarly journals THE SIGN OF FOURIER COEFFICIENTS OF HALF-INTEGRAL WEIGHT CUSP FORMS

2012 ◽  
Vol 08 (03) ◽  
pp. 749-762 ◽  
Author(s):  
THOMAS A. HULSE ◽  
E. MEHMET KIRAL ◽  
CHAN IEONG KUAN ◽  
LI-MEI LIM
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Square Root ◽  
Critical Strip ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Finite Set ◽  
Invent Math

From a result of Waldspurger [W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math.64 (1981) 175–198], it is known that the normalized Fourier coefficients a(m) of a half-integral weight holomorphic cusp eigenform 𝔣 are, up to a finite set of factors, one of [Formula: see text] when m is square-free and f is the integral weight cusp form related to 𝔣 by the Shimura correspondence [G. Shimura, On modular forms of half-integral weight, Ann. of Math.97 (1973) 440–481]. In this paper we address a question posed by Kohnen: which square root is a(m)? In particular, if we look at the set of a(m) with m square-free, do these Fourier coefficients change sign infinitely often? By partially analytically continuing a related Dirichlet series, we are able to show that this is so.

Fourier Coefficients of Modular Forms of Half-Integral Weight in Arithmetic Progressions

2020 ◽  
Author(s):  
Corentin Darreye
Keyword(s):  
Gaussian Distribution ◽  
Modular Forms ◽  
Prime Power ◽  
Fourier Coefficients ◽  
Arithmetic Progressions ◽  
Cusp Forms ◽  
Half Integral Weight ◽  
Mixed Distribution ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

Abstract We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry–Ganguly–Kowalski–Michel and Kowalski–Ricotta in the context of half-integral weight holomorphic cusp forms and for prime power modulus. We actually show that these sums follow in a suitable range a mixed Gaussian distribution that comes from the asymptotic mixed distribution of Salié sums.

scholarly journals Petersson products of bases of spaces of cusp forms and estimates for Fourier coefficients

2018 ◽  
Vol 14 (08) ◽  
pp. 2277-2290 ◽  
Author(s):  
Rainer Schulze-Pillot ◽  
Abdullah Yenirce
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Orthogonal Basis ◽  
Cusp Forms ◽  
Integral Weight ◽  
Space Of Cusp Forms

We prove a bound for the Fourier coefficients of a cusp form of integral weight which is not a newform by computing an explicit orthogonal basis for the space of cusp forms of given integral weight and level.

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)”.

ON A CORRESPONDENCE BETWEEN JACOBI CUSP FORMS AND ELLIPTIC CUSP FORMS

2013 ◽  
Vol 09 (04) ◽  
pp. 917-937 ◽  
Author(s):  
B. RAMAKRISHNAN ◽  
KARAM DEO SHANKHADHAR
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Cusp Forms ◽  
Jacobi Forms ◽  
Binary Quadratic Forms ◽  
Congruence Subgroups ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Forms Of Higher Degree ◽  
Appl Math

In this paper, we prove a generalization of a correspondence between holomorphic Jacobi cusp forms of higher degree (matrix index) and elliptic cusp forms obtained by K. Bringmann [Lifting maps from a vector space of Jacobi cusp forms to a subspace of elliptic modular forms, Math. Z.253 (2006) 735–752], for forms of higher levels (for congruence subgroups). To achieve this, we make use of the method adopted by M. Manickam and the first author in Sec. 3 of [On Shimura, Shintani and Eichler–Zagier correspondences, Trans. Amer. Math. Soc.352 (2000) 2601–2617], who obtained similar correspondence in the degree one case. We also derive a similar correspondence in the case of skew-holomorphic Jacobi forms (matrix index and for congruence subgroups). Such results in the degree one case (for the full group) were obtained by N.-P. Skoruppa [Developments in the theory of Jacobi forms, in Automorphic Functions and Their Applications, Khabarovsk, 1988 (Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk, 1990), pp. 168–185; Binary quadratic forms and the Fourier coefficients of elliptic and Jacobi modular forms, J. Reine Angew. Math.411 (1990) 66–95] and by M. Manickam [Newforms of half-integral weight and some problems on modular forms, Ph.D. thesis, University of Madras (1989)].

scholarly journals ON SHIMURA'S DECOMPOSITION

2013 ◽  
Vol 09 (06) ◽  
pp. 1431-1445 ◽  
Author(s):  
SOMA PURKAIT
Keyword(s):  
Modular Forms ◽  
Theta Series ◽  
Dirichlet Character ◽  
Practical Applications ◽  
Quadratic Twists ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Sum Formula ◽  
Practical Algorithm ◽  
Definition Of

Let k be an odd integer ≥ 3 and N be a positive integer such that 4|N. Let χ be an even Dirichlet character modulo N. Shimura decomposes the space of half-integral weight cusp forms Sk/2(N,χ) as a direct sum [Formula: see text] where F runs through all newforms of weight k - 1, level dividing N/2 and character χ2, the space Sk/2(N,χ,F) is the subspace of forms that are "Shimura equivalent" to F, and the space S0(N,χ) is the subspace spanned by single-variable theta-series. The explicit computation of this decomposition is important for practical applications of a theorem of Waldspurger relating the critical values of L-functions of quadratic twists of newforms of even integral weight to coefficients of modular forms of half-integral weight. In this paper, we give a more precise definition of the summands Sk/2(N,χ,F) whilst proving that it is equivalent to Shimura's definition. We use our definition to give a practical algorithm for computing Shimura's decomposition, and illustrate this with some examples.

scholarly journals On a completed generating function of locally harmonic Maass forms

2014 ◽  
Vol 150 (5) ◽  
pp. 749-762 ◽  
Author(s):  
Kathrin Bringmann ◽  
Ben Kane ◽  
Sander Zwegers
Keyword(s):  
Differential Operator ◽  
Generating Function ◽  
Cusp Form ◽  
Quadratic Forms ◽  
Cusp Forms ◽  
Binary Quadratic Forms ◽  
Maass Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Harmonic Maass Forms

AbstractWhile investigating the Doi–Naganuma lift, Zagier defined integral weight cusp forms $f_D$ which are naturally defined in terms of binary quadratic forms of discriminant $D$. It was later determined by Kohnen and Zagier that the generating function for the function $f_D$ is a half-integral weight cusp form. A natural preimage of $f_D$ under a differential operator at the heart of the theory of harmonic weak Maass forms was determined by the first two authors and Kohnen. In this paper, we consider the modularity properties of the generating function of these preimages. We prove that although the generating function is not itself modular, it can be naturally completed to obtain a half-integral weight modular object.

scholarly journals Explicit application of Waldspurger’s theorem

2013 ◽  
Vol 16 ◽  
pp. 216-245
Author(s):  
Soma Purkait
Keyword(s):  
Modular Form ◽  
Cusp Form ◽  
Modular Forms ◽  
Critical Value ◽  
Automorphic Representations ◽  
Half Integral Weight ◽  
Dirichlet Characters ◽  
Integral Weight ◽  
Quadratic Twist

AbstractFor a given cusp form $\phi $ of even integral weight satisfying certain hypotheses, Waldspurger’s theorem relates the critical value of the $\mathrm{L} $-function of the $n\mathrm{th} $ quadratic twist of $\phi $ to the $n\mathrm{th} $ coefficient of a certain modular form of half-integral weight. Waldspurger’s recipes for these modular forms of half-integral weight are far from being explicit. In particular, they are expressed in the language of automorphic representations and Hecke characters. We translate these recipes into congruence conditions involving easily computable values of Dirichlet characters. We illustrate the practicality of our ‘simplified Waldspurger’ by giving several examples.

Sign in / Sign up

Export Citation Format

Share Document