# Reaching Hecke eigenvalues from a trace formula

I am interested in studying equidistribution of Hecke eigenvalues and proving statistical properties of arithmetical objects. On the road, I face the following problem: how to express sums of the form

$$\sum_{c(\pi)<X} a_n(\pi)$$

where $a_n(\pi)$ is the $n$-th coefficient of the Dirichlet series defining $L(s, \pi)$, and $c(\pi)<X$ is some truncating condition. The sum is over irreducible representations of a totally definite quaternion algebra $Z\backslash B^\times$. We know that those coefficients can be read as Hecke eigenvalues $\lambda_n(\pi)$, so I am thinking in applying trace formulas to Hecke operators in order to reach such a sum, as in Serre's article on equidistribution.

In simple cases, such as compact groups, I can use the Selberg's trace formula, and it remains to find a suitable test-function in order to reach those coefficients/eigenvalues. Is there a standard approach, and if so are there easy ways to make those eigenvalues appear in the spectral side of the trace formula ?

More generally, what are the common relations between trace formulas and Hecke operators (I suppose it is a fertile ground, for both Hecke eigenvalues and trace formulas are powerful tools, and both are "spectral" objects) ?

Any idea or reference would be of great help !

• Please be more specific. What kind of $\pi$'s are you looking at? There are many (reductive algebraic) groups, and many kinds of automorphic representations. Also, clarify what exactly you mean by $c(\pi)$. – GH from MO Dec 9 '16 at 11:12
• @GHfromMO I just edited my post: the $\pi$'s are representations of the group of units of a totally definite quaternion algebra (so compact modulo the center), and the $c(\pi)$ is the usual notion of conductor of Iwaniec-Sarnak, lifted from $GL_2$ to $B^\times$. – Desiderius Severus Dec 9 '16 at 11:15
• The "conductor of Iwaniec-Sarnak" is not called the conductor, but the analytic conductor. At any rate, thanks for clarifying the question. – GH from MO Dec 9 '16 at 11:32

Yes, this is a standard thing to do. If you want to look at traces of Hecke operators on a definite quaternion algebra, this is the same as what are known as "traces of Brandt matrices." These have been studied with trace formulas classically by Eichler, Shimizu, etc. E.g., see the notes

I think the cleanest way to do it is adelically, though this may be personal bias. I can't think of a reference off the top of my head that puts everything together precisely, but I can give you a couple of references that deal with the necessary components. An adelic approach to the trace formula for GL(2) and the precise connection with Hecke operators is described in detail in the book

Basically, say you want the trace for $T_n$. You choose a test function $f = \prod f_v$ such that $f_p$ gives you the local Hecke operator for $p \mid N$ and $f_\infty$ gives you the desired weight of the representation.

To modify this for a definite quaternion algebra $B$, you just need to modify the test functions $f_v$ at the places where $B$ ramifies. E.g., if you want weight 2 and squarefree level, just take $f_\infty=1$ and $f_p$ to be the characteristic function of the unit group of a local maximal order at a finite place $p$ where $B$ ramifies (so $p | N$).

Similar calculations have been done several places, e.g. in the context of the relative trace formula in the paper

which computes sums of $L$-values weighted by Hecke eigenvalues.

• Thank you very much for your enlightening answer, I was indeed consulting the book of Knightly and Li, and am very grateful for the reference of Eichler. – Desiderius Severus Dec 9 '16 at 19:29