Abstract
We prove a subconvexity bound in the conductor aspect for the L-function \(L(s,f,\chi )\) where f is a half integer weight modular form. This L-function has analytic continuation and functional equation, but no Euler product. Due to the lack of an Euler product, one does not expect a Riemann hypothesis for half integer weight modular forms. Nevertheless one may speculate a Lindelöf-type hypothesis, and this current subconvexity result is an indication towards its truth.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Applied mathematics series. Dover Publications (1964). http://books.google.com/books?id=MtU8uP7XMvoC
Bateman, H., Erdélyi, A.: Higher transcendental functions. In: Bateman, H., Erdélyi, A. (eds.) California Institute of Technology Bateman Manuscript Project. McGraw-Hill, New York (1953)
Blomer, V.: Shifted convolution sums and subconvexity bounds for automorphic \(L\)-functions. Int. Math. Res. Not. 2004(73), 3905–3926 (2004)
Blomer, V.: Subconvexity for a double dirichlet series. Compos. Math. 147(2), 355–374 (2011)
Blomer, V.: Period integrals and rankin-selberg \(L\)-functions on \(\operatorname{GL}(n)\). Geom. Funct. Anal. 22(3), 608–620 (2012)
Burgess, D.: On character sums and \(L\)-series. II. Proc. Lond. Math. Soc. 3(1), 524–536 (1963)
Conrey, J.B., Ghosh, A.: Remarks on the generalized Lindelöf hypothesis. Funct. Approx. Comment. Math. 36(1), 71–78 (2006)
Duke, W., Friedlander, J.B., Iwaniec, H.: The subconvexity problem for Artin \(L\)-functions. Invent. Math. 149(3), 489–577 (2002)
Gradshteyn, I.S., Jeffrey, A., Ryzhik, I.M.: Table of Integrals, Series, and Products 4th Corrected and Revised ed. Academic Press, New York (1980). With corrigenda : loose-leaf ed. [in closed Ref.] : Trans. from the 4th Russian ed., Moscow, 1963
Helgason, S.: Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions. Pure and Applied Mathematics. Academic Press, Orlando (1984)
Hoffstein, J.: Multiple Dirichlet Series and Shifted Convolutions. arXiv preprint arXiv:1110.4868 (2011)
Hulse, T.: Triple Shifted Sums of Automorphic \(L\)-Functions. In: Ph.D. Thesis, Brown University (2013)
Huxley, M.N.: Exponential sums and the Riemann zeta function V. Proc. Lond. Math. Soc. 90(1), 1-41 (2005). doi:10.1112/S0024611504014959. http://plms.oxfordjournals.org/content/90/1/1.abstract
Iwaniec, H.: Spectral methods of automorphic forms. In: Iwaniec, H. (ed.) Graduate Studies in Mathematics, vol. 53, 2nd edn. American Mathematical Society, Providence (2002)
Iwaniec, H., Kowalski, E.: Analytic Number Theory, vol. 53. Cambridge University Press, Cambridge (2004)
Sarnak, P.: Integrals of products of eigenfunctions. Int. Math. Res. Not. 1994(6), 251-260 (1994). doi:10.1155/S1073792894000280. URL: http://imrn.oxfordjournals.org/content/1994/6/251.short
Shimura, G.: On modular forms of half integral weight. Ann. Math. 97(3), 440–481 (1973)
Acknowledgments
I would like to thank my Ph.D. adviser at Brown University Jeffrey Hoffstein, for suggesting me this problem and making his manuscript available. I also would like to thank Thomas Hulse and Min Lee for fruitful discussions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kıral, E.M. Subconvexity for half integral weight L-functions. Math. Z. 281, 689–722 (2015). https://doi.org/10.1007/s00209-015-1504-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-015-1504-x