Equidistribution of Hecke points and $p = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$

Question

I have seen two versions of a result called "Hecke Equidistribution" and I wanted to know if they were the same or different.

---

#1 Let $p = 4k+1 = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$. Then $\theta$ equidistributed as $p \to \infty$.

#2 Let $\displaystyle T_n.z = \sum_{ad=n, 0 \leq b < d} \frac{az+b}{d}$ and extend to functions $\displaystyle [T_n ]f(y) = \sum_{y \in T_n.z} f(y)$. Then we have equidistribution of Hecke points over the modular surface: $$ \lim_{n \to \infty} [T_n f](z) = \int_{X(1)} f(z) \, dz$$

---

Both of these results have Hecke's name attached. The first deals with Hecke characters and L-functions the second deals with Hecke operators and modular forms.

Is result #1 considered an instance #2 of the equidistribution of Hecke points?

Possibly I consider the theta function: $ \displaystyle \theta (z) = \sum_{(m, n) \in \mathbb{Z}^2} e^{2\pi i (m^2 + n^2) z} $

However I don't see how to act the Hecke operator in such a way as to prove the result.

Answer 1

These are two very different results. Briefly, the first result depends on the nonvanishing of (certain) Hecke $L$-functions at the edge of the critical strip, while the second result depends on a nontrivial bound for Hecke eigenvalues of Maass forms on the modular surface. As you can see, both are related to objects named after Hecke, but the objects themselves are quite different.

P.S. I see now that you mentioned Hecke characters and Hecke operators in your post. At any rate, I don't know of a way of deducing one result from the other (apart from proving both results in their own way).

Answer 2

One common feature of these two equidistribution examples is that they have proofs (perhaps among others) thinking in terms of Weyl's equidistribution criterion: show that the corresponding measures (sums of point-mass measures) go to the constant function $1$ as distributions. On a nice (e.g., homogeneous) space, the question of this limit can be recast in terms of the appropriate harmonic analysis, which in interesting cases turns out to ask for estimates on $L$-functions or automorphic forms, as @GHfromMO noted.

In the first example, the relevant automorphic forms are harmonic binary theta series with a fixed binary form and varying grossencharacters. As @GH noted, the relevant estimate boils down to non-vanishing on the edge of the critical strip. In the second example, the relevant automorphic forms are Hecke-eigenfunction waveforms (of fixed level), along with corresponding Eisenstein series (for the continuous spectrum) and the relevant estimate reduces to a non-trivial estimate on the Fourier coefficients, since these mirror Hecke operators and eigenvalues.

(In my modular forms course 2013-14, I did some easier equidistribution examples: http://www.math.umn.edu/~garrett/m/mfms/, the least cliched of which might be "section 10" about equidistribution problems on spheres, treating higher-dimension, even-dimension cases already too easy to be of interest to Kloosterman and his contemporaries, but perhaps educational.)