The distribution of the eigenvalues of the Hecke operators \(T_p\) for primes \(p\) on a fixed space of modular forms is a difficult problem. The author treats a similar, but more accessible problem: Let a prime \(p\) be fixed, and consider a family of spaces \(S(N,k)\) of cusp forms of even weight \(k\) and level \(N\) where \(k+N\) tends to infinity. The eigenvalues of \(T_p'=p^{-(k-1)/2}T_p\) are points in the interval \(\Omega=[-2,2]\). Is there a measure \(\mu\) on \(\Omega\) such that these points are equally distributed with respect to \(\mu\)? In the case of varying \(p\) one would expect the Sato-Tate measure \(\mu_\infty=\frac{1}{2\pi}\sqrt{4-x^2}dx\).
The main result of the paper says that there is indeed such a measure which, however, depends on \(p\) and is given by \[ \mu_p=f_p(x)\mu_\infty,\quad f_p(x)=\frac{p+1}{(\sqrt{p}+1/\sqrt{p})^2-x^2}. \] The proof is an application of the Eichler-Selberg trace formula. It is shown that the “interesting terms” in this formula are negligible compared to the “evident term”. This gives the estimate \[ |\text{Tr }T_n(N,k,\chi) - \frac{k-1}{12}\chi(\sqrt{n})n^{(1/2)k-1}\psi(N)|\ll n^{(1/2)k}\sqrt{N}d(N) \] for any positive integer \(n\), with a constant in \(\ll\) which depends only on \(n\). Here, the Hecke operator \(T_n\) acts on the cusp forms of weight \(k\), character \(\chi\) and level \(N\) relatively prime to \(n\), \(\psi(N)\) denotes the index of \(\Gamma_0(N)\) in \(\text{SL}_2(\mathbb{Z})\), \(d(N)\) is the number of divisors of \(N\), and \(\chi(\sqrt{n})=0\) if \(n\) is not a square. A corollary of the main result says that the eigenvalues of \(T_p'\) are dense in \(\Omega\). The theorem remains true if cusp forms are replaced by newforms.
An application concerns the fields of rationality of the eigenvalues of \(T_n\) on \(S(N,k)\) which are totally real algebraic integers. Let \(f_1,\dots,f_s\), with \(s=s(N,k)\), be a basis of Hecke eigenforms for \(S(N,k)\). For fixed \(i\) and variable \(n\), let \(K_i\) be the field which is generated by the eigenvalues of \(T_n\) on \(f_i\), and let \(s(N,k)_r\) denote the number of \(i\) for which \(K_i\) has degree \(r\) over \(\mathbb{Q}\). Then for any fixed \(r\) it is shown that \[ \lim_{k+N\to\infty} s(N,k)_r/s(N,k)= 0. \] Stated loosely: the majority of the fields \(K_i\) have a large degree. This has consequences for the Jacobian variety \(J_0(N)\) of the modular curve \(X_0(N)\): The dimension of the greatest simple factor of \(J_0(N)\) tends to infinity with \(N\). In particular, there are only finitely many \(N\) such that \(J_0(N)\) is isogenous to a product of elliptic curves.
In two final chapters the author shows that equidistribution with respect to \(\mu_p\) occurs also with some other objects. The first one concerns the “Frobenius angles” of families of algebraic curves over a finite field \(\mathbb{F}_q\) and the numbers of their points over extensions of \(F_q\). The second one concerns eigenvalues of incidence matrices of regular graphs.