# Bounds on shifted convolution sums for Hecke eigenforms

@inproceedings{Nordentoft2021BoundsOS, title={Bounds on shifted convolution sums for Hecke eigenforms}, author={Asbj{\o}rn Christian Nordentoft and Yiannis N. Petridis and Morten Skarsholm Risager}, year={2021} }

play an important rôle in analytic number theory, especially when λ is of arithmetic significance, see e.g. [16, 37, 6, 12, 18, 4, 2, 14] and the references therein. The case where λ(n) is the nth Hecke eigenvalue of an automorphic object is maybe the most interesting, and in this case the above sum is called a shifted convolution sum or sometimes a generalized additive divisor sum. Here are some examples:

## References

SHOWING 1-10 OF 42 REFERENCES

Refined estimates towards the Ramanujan and Selberg conjectures, appendix to H

- J. Amer. Math. Soc
- 2003

Some applications of large sieve in Riemann surfaces

- Mathematics
- 1996

1. Introduction. In [Ch] we gave some large sieve type inequalities involving elements of harmonic analysis in Riemann surfaces and compact Riemannian manifolds. In this paper we present some of…

Risager, Small scale equidistribution of Hecke eigenforms at infinity, arXiv:2011.05810 [math] (2020)

- 2020

Central values of additive twists of modular $L$-functions

- Mathematics
- 2018

Additive twists of a modular $L$-function are important invariants associated to a cusp form, since the additive twists encode the Eichler-Shimura isomorphism. In this paper we prove that additive…

Nonvanishing of Rankin–Selberg L-functions for Hilbert modular forms

- Mathematics
- 2014

Let F be a totally real number field of degree n over $\mathbb{Q}$ with ring of integers $\mathcal{O}$ and narrow class number one. Let S2k(Γ) be the vector space of cuspidal Hilbert modular forms of…

A new bound K2/3+ɛ for Rankin-Selberg ℒ-functions for Hecke congruence subgroups

- Mathematics
- 2010

Let f be a holomorphic Hecke eigenform for Γ0(N ) of weight k, or a Maass eigenform for Γ0(N ) with Laplace eigenvalue 1/4 + k. Let g be a fixed holomorphic or Maass cusp form for Γ0(N ). A…

A Sieve Method for Shifted Convolution Sums

- Mathematics
- 2008

We study the average size of shifted convolution summation terms related to the problem of Quantum Unique Ergodicity on ${\rm SL}_2 (\mathbbm{Z})\backslash \mathbbm{H}$. Establishing an upper-bound…

The spectral decomposition of shifted convolution sums

- Mathematics
- 2007

Let pi(1), pi(2)) be cuspidal automorphic representations of PGL(2)(R) Qf conductor 1 and Hecke eigenvalues lambda(pi 1,2) (n) and let h > 0 be an integer. For any smooth compactly supported weight…