Equidistribution of cusp forms in the level aspect. (English) Zbl 1273.11069
Summary: Let \(f\) traverse a sequence of classical holomorphic newforms of fixed weight and increasing square-free level \(q\to\infty\). We prove that the pushforward of the mass of \(f\) to the modular curve of level 1 equidistributes with respect to the Poincaré measure.
Our result answers affirmatively the square-free level case of a conjecture spelled out in 2002 by E. Kowalski, P. Michel and J. Vanderkam [Duke Math. J. 114, 123–191 (2002; Zbl 1035.11018)] in the spirit of a conjecture that Z. Rudnick and P. Sarnak [Commun. Math. Phys. 161, No. 1, 195–213 (1994; Zbl 0836.58043)] made in 1994.
Our proof follows the strategy of R. Holowinsky and K. Soundararajan [Ann. Math. (2) 172, No. 2, 1499–1516 (2010; Zbl 1214.11054); ibid., 1469–1498 (2010; Zbl 1234.11066); ibid., 1517–1528 (2010; Zbl 1211.11050), who showed that newforms of level 1 and large weight have equidistributed mass. The new ingredients required to treat forms of fixed weight and large level are an adaptation of Holowinsky’s reduction of the problem to one of bounding shifted sums of Fourier coefficients, a refinement of his bounds for shifted sums, an evaluation of the \(p\)-adic integral needed to extend Watson’s formula to the case of three newforms of not necessarily equal square-free levels, and some additional technical work in the problematic case that the level has many small prime factors.
Our result answers affirmatively the square-free level case of a conjecture spelled out in 2002 by E. Kowalski, P. Michel and J. Vanderkam [Duke Math. J. 114, 123–191 (2002; Zbl 1035.11018)] in the spirit of a conjecture that Z. Rudnick and P. Sarnak [Commun. Math. Phys. 161, No. 1, 195–213 (1994; Zbl 0836.58043)] made in 1994.
Our proof follows the strategy of R. Holowinsky and K. Soundararajan [Ann. Math. (2) 172, No. 2, 1499–1516 (2010; Zbl 1214.11054); ibid., 1469–1498 (2010; Zbl 1234.11066); ibid., 1517–1528 (2010; Zbl 1211.11050), who showed that newforms of level 1 and large weight have equidistributed mass. The new ingredients required to treat forms of fixed weight and large level are an adaptation of Holowinsky’s reduction of the problem to one of bounding shifted sums of Fourier coefficients, a refinement of his bounds for shifted sums, an evaluation of the \(p\)-adic integral needed to extend Watson’s formula to the case of three newforms of not necessarily equal square-free levels, and some additional technical work in the problematic case that the level has many small prime factors.
MSC:
| 11F11 | Holomorphic modular forms of integral weight |
| 11M36 | Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) |
| 58J51 | Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity |
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