Central Value Of Automorphic L-Functions

By E. M. Baruch, Zhengyu Mao. 2003

94 citations recorded by the source corpus.

Abstract.We prove a generalization to the totally real field case of the Waldspurger’s formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger’s formula as a combination of two ingredients – an equality between global distributions, and a dichotomy result for theta correspondence. As applications we generalize the Kohnen–Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindelöf hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/491ded59948455d4112b71adbb03ea854f670629

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. M. Young
2. V. Blomer
3. P. Michel
4. Akshay Venkatesh
5. Ian Petrow
6. Gergely Harcos
7. R. Munshi
8. D. Whitehouse

In Mathematical Physics

1. M. Young
2. V. Blomer
3. P. Michel
4. Ian Petrow
5. Akshay Venkatesh
6. Gergely Harcos
7. R. Munshi
8. D. Whitehouse

Citations in the corpus

• Twisted L-Functions Over Number Fields and Hilbert’s Eleventh Problem (2009, methodology): “Perhaps the most appealing application of Theorem 1 is to combine it with the formula of Waldspurger [Wa2] and its extensions by Shimura [Sh2], Khuri-Makdisi [KM], Kojima [Koj], Baruch{ Mao [ BM ] and others in order to bound the Fourier coecients of half-integral weight Hilbert modular forms.”
• Equidistribution, L-functions and ergodic theory: on some problems of Yu. Linnik (2006, background): “The following theorem of Cogdell/PiatetskiShapiro/Sarnak combines the formula of [1] with (3.”
• A generalized cubic moment and the Petersson formula for newforms (2016, background): “ssumption χD(p) = ηp(f) for all p| r(here ηp(f) is the eigenvalue of the AtkinLehner operator). If χD(p) = −ηp(f) for some p| rthen c(|D|) = 0 while the right hand side may not vanish. Baruch and Mao [BM] have generalized the Kohnen-Zagier/Waldspurger formula by removing the conditions on D, relatin”
• Central L-Values and Toric Periods for GL(2) (2009, background): “For a formula for L(1/2,π ) in terms of Fourier coefficients when ωπ = 1 and F is totally real, see [24], [1].”