scholarly journals On the Special Values of Certain L-Series Related to Half-Integral Weight Modular Forms

2014 ◽  
pp. 119-136 ◽  
Author(s):  
Hidenori Katsurada
Keyword(s):  
Modular Forms ◽  
Special Values ◽  
Half Integral Weight ◽  
Integral Weight

scholarly journals Quadratic congruences for Cohen–Eisenstein series

1999 ◽  
Vol 41 (1) ◽  
pp. 141-144
Author(s):  
P. GUERZHOY
Keyword(s):  
Number Theory ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Analytic Number Theory ◽  
Cusp Forms ◽  
Special Values ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Published Paper

The notion of quadratic congruences was introduced in the recently published paper [A. Balog, H. Darmon and K. Ono, Congruences for Fourier coefficients of half-integral weight modular forms and special values of L-functions, in Analytic Number Theory, Vol. 1, Progr. Math.138 (Birkhäuser, Boston, 1996), 105–128.]. In this note we present different, somewhat more conceptual proofs of several results from that paper. Our method allows us to refine the notion and to generalize the results quoted. Here we deal only with the quadratic congruences for Cohen–Eisenstein series. Similar phenomena exist for cusp forms of half-integral weight as well; however, as one would expect, in the case of Eisenstein series the argument is much simpler. In particular, we do not make use of techniques other than p-adic Mazur measure, whereas the consideration of cusp forms of half-integral weight involves a much more sophisticated construction. Moreover, in the case of Cohen–Eisenstein series we are able to obtain a full and exhaustive result. For these reasons we present the argument here.

scholarly journals A NOTE ON SPECIAL VALUES OF CERTAIN DIRICHLET L-FUNCTIONS

2010 ◽  
Vol 83 (3) ◽  
pp. 435-438
Author(s):  
B. RAMAKRISHNAN
Keyword(s):  
Modular Forms ◽  
Direct Consequence ◽  
Kronecker Symbol ◽  
Special Values ◽  
Half Integral Weight ◽  
Divisor Functions ◽  
Integral Weight ◽  
Finite Sums

AbstractIn Gun and Ramakrishnan [‘On special values of certain Dirichlet L-functions’, Ramanujan J.15 (2008), 275–280], we gave expressions for the special values of certain Dirichlet L-function in terms of finite sums involving Jacobi symbols. In this note we extend our earlier results by giving similar expressions for two more special values of Dirichlet L-functions, namely L(−1,χm) and L(−2,χ−m′), where m,m′ are square-free integers with m≡1 mod 8 and m′≡3 mod 8 and χD is the Kronecker symbol $(\frac {D}{\cdot })$. As a consequence, using the identities of Cohen [‘Sums involving the values at negative integers of L-functions of quadratic characters’, Math. Ann.217 (1975), 271–285], we also express the finite sums with Jacobi symbols in terms of sums involving divisor functions. Finally, we observe that the proof of Theorem 1.2 in Gun and Ramakrishnan (as above) is a direct consequence of Equation (24) in Gun, Manickam and Ramakrishnan [‘A canonical subspace of modular forms of half-integral weight’, Math. Ann.347 (2010), 899–916].

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].

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