On the Exceptional Zeros of Rankin–Selberg L-Functions

By Dinakar Ramakrishnan, Song Wang. Compositio Mathematica, 2001

54 citations recorded by the source corpus.

The main objects of study in this article are two classes of Rankin–Selberg L-functions, namely L(s,f×g) and L(s, sym2(g)× sym2(g)), where f,g are newforms, holomorphic or of Maass type, on the upper half plane, and sym2(g) denotes the symmetric square lift of g to GL(3). We prove that in general, i.e., when these L-functions are not divisible by L-functions of quadratic characters (such divisibility happening rarely), they do not admit any LandauSiegel zeros. Such zeros, which are real and close to s=1, are highly mysterious and are not expected to occur. There are corollaries of our result, one of them being a strong lower bound for special value at s=1, which is of interest both geometrically and analytically. One also gets this way a good bound onthe norm of sym2(g).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2283f87a0bff4ee6bf4c63fc6fa888184de1bbdb

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In Geometry and Topology

1. V. Blomer
2. Djordje Milićević
3. J. Cogdell
4. P. Michel
5. M. Young
6. Akshay Venkatesh
7. R. Munshi
8. Y. Lau

In Mathematical Physics

1. V. Blomer
2. P. Michel
3. Djordje Milićević
4. J. Cogdell
5. M. Young
6. R. Munshi
7. Emmanuel Kowalski
8. Gergely Harcos

Citations in the corpus

• On the complex moments of symmetric power $L$-functions at $s = 1$ (2004, background): “Remarkably (by comparison with the exceptional zero problem for L-functions of quadratic characters) Hypothesis LSZk(q) is known unconditionally for a few k’s: for k = 1, 2 by the works of Hoffstein/Lockhart, Goldfeld/Hoffstein/Lieman and Hoffstein/Ramakrishnan [10, 6, 11] and for k = 4 by the work ”
• The Second Moment of Twisted Modular L-Functions (2014, background): “(1) = (loglogq)O(1) and P′(1)/P(1) = O(loglogq), and that the leading coefficients L(1,sym2f1) and L(1,f1 × f2) do not vanish by the lower bounds of Hoffstein and Lockhart [HL] and Ramakrishnan and Wang [RW] (see also [Br]), so that the term M(f1,f2,q) is not far from a linear polynomial in logqor a co”