Why is it interesting to study the sign distribution of Hecke eigenvalues?

Question

Let $f$ be a primitive form of an even weight $k$ and level $N\geq 1.$ By the theory of Hecke operators, $$\lambda_f(n)=\frac{\hat{f}(n)}{n^{\frac{k-1}{2}}}$$ is a real number. Studying the distribution of signs of Hecke eigenvalues has become an active area of research in recent past. Many authors investigated the problem of sign changes of the sequence $(\lambda_f(n))$ like Landau, Murty and Kohnen....

My question is: What's the reason of studying the sign changes problem, I mean are there any algebraic or geometric purposes behind this study?

Many thanks

Answer 1

Kowalski, Lau, Soundararajan and Wu have a very nice explanation of why one might want to study the sign changes of Fourier coefficients of primitive cusp forms in the introduction of the paper On modular signs.

The idea is that the Fourier coefficients of a primitive cusp form are multiplicative and one might hope to study these coefficients in much the same way that other arithmetic functions are studied in analytic number theory. The particular question (on the analytic number theory side) that Kowalski et al are interested in is that of bounding the number of primitive real Dirichlet characters $\chi$ of modulus $q\leq D$ for which the least $n$ such that $\chi(n)=-1$ is $\gg \log D$. They explain that this problem is unlikely to have a good analogue in the modular forms setting because Hecke eigenvalues can take on many more than two values. By narrowing their attention to primitive cusp forms with real eigenvalues and instead looking at the ${\it signs}$ of the Fourier coefficients, they are able to recover their analogy. For instance, they give a bound on the least $n\geq 1$ such that $\lambda_f(n)<0$ (which is an analogue of the least quadratic non-residue problem).

Answer 2

Generally speaking sign changes of multiplicative functions are interesting because they highlight the decorelation between the additive and multiplicative structure of the integers (i.e the event $\lambda_f(n) \lambda_f(n + 1) < 0$ involves understanding the factorizations of consecutive integers and typically involves showing that the factorization of $n$ and $n + 1$ do not conspire).

As far as I know there are no geometric reasons, however it is a pretty natural question given that first people asked about tight upper bounds for $\lambda_f(n)$ (Ramanujan conjecture), then non-vanishing (Serre), then the finer variation of the size (Sato-Tate) and the signs on the primes (Sato-Tate again), so then asking about the sign on the integers is a pretty natural thing to ask if you are looking for anything like a complete picture of the behavior of $\lambda_f(n)$.