Abstract
In this note, we derive a non-cusp form of weight \(k+1/2\) (\(k\ge 2\), even) for \(\Gamma _0 (4)\) in the Kohnen plus space whose Petersson scalar product with a cuspidal Hecke eigenform f is equal to a constant times the L value \(L(f,k-1/2).\) We also prove that for such a form f and the associated form F under the \(D{\text {th}}\) Shimura–Kohnen lift the quantity \(\frac{a_f(D)L(F,2k-1)}{\pi ^{k-1}\langle f,f\rangle L(D,k)}\) is algebraic.
Dedicated to Murugesan Manickam on the occasion of his 60th birthday.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Apostol, T.M.: Introduction to Analytic Number Theory, Springer International Student Edition. Narosa Publishing House (1998)
Cohen, H.: Sums involving the values at negative integers of L-functions of quadratic characters. Math. Ann. 217(3), 271–285 (1975)
Cohen, H., Strömberg, F.: Modular Forms - A Classical Approach. Graduate Studies in Mathematics, vol. 179. American Mathematical Society, Providence (2017)
Iwaniec, H.: Topics in Classical Automorphic Forms. Graduate Studies in Mathematics, vol. 17. American Mathematical Society, Providence (1997)
Koblitz, N.I.: Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics, 2nd edn. Springer, Berlin (1993)
Kohnen, W.: Modular forms of half-integral weight on \(\Gamma _0(4)\). Math. Ann. 248, 249–266 (1980)
Kohnen, W., Zagier, D.: Values of L-series of modular forms at the center of the critical strip. Invent. Math. 64, 175–198 (1980)
Manickam, M., Ramakrishnan, B., Vasudevan, T.C.: Hecke operators on modular forms of half-integral weight. Arch. Math. 51, 343–352 (1988)
Manin, Y.: Periods of parabolic forms and p-adic Hecke series. Math. USSR Sb. 21, 371–393 (1973)
Serre, J.P., Stark, H.M.: Modular forms of weight \(1/2\). Modular Functions of One Variable VI. Lecture Notes in Mathematics, vol. 627, pp. 27–67. Springer, Berlin (1976)
Shimura, G.: The special values of the zeta functions associated with cusp forms. Commun. Pure Appl. Math. 29, 783–804 (1976)
Acknowledgements
The author thanks his supervisor M. Manickam for framing the problem and guidance. The author would like to thank IMSc, Chennai for the hospitality, and the author also thanks referee for carefully reading the article and valuable comments made which improved the style of the article. The author is supported by INSPIRE Fellowship (IF170843).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Sreejith, M.M. (2020). A Certain Kernel Function for L-Values of Half-Integral Weight Hecke Eigenforms. In: Ramakrishnan, B., Heim, B., Sahu, B. (eds) Modular Forms and Related Topics in Number Theory. ICNT 2018. Springer Proceedings in Mathematics & Statistics, vol 340. Springer, Singapore. https://doi.org/10.1007/978-981-15-8719-1_14
Download citation
DOI: https://doi.org/10.1007/978-981-15-8719-1_14
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-8718-4
Online ISBN: 978-981-15-8719-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)