Counting solutions to a_p = λ

We know that the eigenvalue of \(T_2\) on \(\Delta\) is \(24.\) Are there any other level one cusp forms with the same Hecke eigenvalue? Maeda’s conjecture in its strongest form certainly implies that there does not. But what can one prove along these lines? Conjecturally, one would certainly predict the following:

Conjecture: Fix a tame level \(N\) prime to \(p.\) If \(\lambda \ne 0,\) there are finitely many eigenforms of level \(N\) an arbitrary weight such that \(a_p = \lambda.\) If \(\lambda = 0,\) there are finitely many eigenforms with the additional condition that they do not have CM by a quadratic field in which \(p\) is inert.

I have no idea how to prove this conjecture. If one counts the number of such forms of weight \(\le X,\) then the trivial bound for eigenforms with \(a_p = \lambda\) is \(O(X^2).\) When I visited Princeton a few weeks ago, Naser Sardari, a student of Sarnak, showed me a short preprint he is writing which improves this bound by a power saving (additionally, it gives a power saving for each individual weight as well). The most interesting case of this result is when \(\lambda = 0,\) but today I want to talk about the much easier case when \(\lambda \ne 0,\) where, via some \(p\)-adic tricks, one can obtain a substantial improvement on the trivial bound. Let’s start from the following:

Proposition I: Let \(S_{\lambda}(X)\) denote the number of cuspforms of level \(N\) and weights \(\le X\) such that \(a_p = \lambda\). Assume that \(\lambda \ne 0.\) Then

\(S_{\lambda}(X) = O(X).\)

Proof: Since \(\lambda \ne 0,\) the \(p\)-adic valuation of \(\lambda\) is finite. However, all forms with bounded slope belong to one of finitely many Coleman families, so the number of such forms in any weight is bounded. Using Wan’s explicit results, one can even give an explicit bound here that depends only on \(N,\) \(p,\) and the valuation of \(\lambda.\)

The point of this post, however, is to give an improvement on this bound.

Proposition II: Let \(S_{\lambda}(X)\) denote the number of cuspforms of level \(N\) and weight \(\le X\) such that \(a_p = \lambda\). Assume that \(\lambda \ne 0.\) Then, as \(X \rightarrow \infty\),

\(S_{\lambda}(X) \ll_{\lambda} \log \log \log \log \log \log \log X.\)

The argument will (obviously) allow for an arbitrary number of logs. But then the statement would become more cumbersome.

Proof: As in the proof of the previous result, we may reduce to the case where we are considering a single Coleman family \(\mathcal{F}.\) Over this family, the function \(U_p\) is continuous, and hence so is \(U_p(U_p – \lambda).\) More importantly, over a small enough disc, it is an Iwasawa function. Let \(\Sigma\) denote an infinite set of integral weight such that, for the relevant points of \(\mathcal{F},\) we have \(T_p = \lambda,\) or

\(U_p(U_p – \lambda) = – p^{k-1}.\)

If \(s\) is a limit point of \(\Sigma,\) then certainly \(U_p(U_p – \lambda)\) will vanish at \(s.\) Since this function is a non-zero bounded function on a disc, it has only finitely many zeros, and so the set of weights \(\Sigma\) will have only finitely many limit points. Thus, we may reduce to the case where the set of weights has a single limit point. In particular, if \(S_{\lambda}(X)\) is not bounded, we may imagine that the set \(\Sigma\) consists of a sequence of integers (which we may assume to be increasing in the Archimedean norm): \(k_0, k_1, k_2, \ldots \) which converge \(p\)-adically to \(s,\) and, at the relevant point of \(\mathcal{F},\) correspond to an eigenform which satisfies the equation

\(U_p(U_p – \lambda)(k_i) = – p^{k_i – 1}.\)

Around a zero \(s,\) any Iwasawa function has an asymptotic expansion of the form

\(F(s + \epsilon) \simeq A \cdot \epsilon^m + \ldots \)

where the LHS has the same valuation as the leading term of the RHS for sufficiently small \(\epsilon.\) If \(F = U_p(U_p – \lambda),\) we deduce that, for sufficiently large integers \(k_i,\)

\(v(s – k_n) = r k_n + c\)

for some \(r = 1/m > 0,\) which implies that \(v(k_{n+1} – k_n) = v(s – k_n),\) and hence also that

\(k_{n+1} – k_n > C p^{r k_n}\)

for some \(r > 0\). This iterated exponential growth proves the result. QED.

The argument also shows that if the set \(\Sigma\) is infinite, the limit roots of \(U_p – \lambda = 0\) will be transcendental Liouville numbers, which seems unlikely. The result also applies if one replaces \(\lambda\) by a sufficiently continuous function without zeros, say \(a_2 = 24(1 + 2(k -12)^2).\) On the other hand, I don’t think these analytic methods will ever be enough to prove the conjectural bound, which is \(O(1).\)