CM newforms with rational coefficients. (English) Zbl 1226.11057
A motivation of this article comes from the complete determination of singular \(K3\) surfaces defined over \(\mathbb Q\). Here the term “singular” refers to the maximal Picard number 20 so that the transcendental lattice is of rank 2. A theorem of R. Livné [Isr. J. Math. 92, No. 1–3, 149–156 (1995; Zbl 0847.11035)] asserts that the two-dimensional Galois representation of the transcendental lattice of a singular \(K3\) surface over \(\mathbb Q\) is associated to a modular form of weight 3.
The first question dealt with in this article is to determine all CM newforms of weight 3 with rational coefficients. The result is the following
Theorem: Assume the extended Riemann hypotheses (ERH) for odd real Dirichlet characters. Then there are only finitely many CM newforms with rational coefficients for fixed weight up to twisting.
For weights \(2,3,4,5,6,2^r+1, 3\cdot 2^r+1\) \((r\in \mathbb N)\) this holds unconditionally.
The idea of proof is to translate the statement to Hecke characters, and classify Hecke characters. Tables for weight 3 (and 4) are produced, where finiteness holds unconditionally.
The next question is a geometric realization problem: how many of these CM newforms actually occur in association with singular K3 surfaces over \(\mathbb Q\)? This is answered in the following theorem with Elkies.
Theorem: Assuming ERH for odd real Dirichlet characters, every newform of weight 3 with rational coefficients is associated to a singular \(K3\) surface over \(\mathbb Q\).
The first question dealt with in this article is to determine all CM newforms of weight 3 with rational coefficients. The result is the following
Theorem: Assume the extended Riemann hypotheses (ERH) for odd real Dirichlet characters. Then there are only finitely many CM newforms with rational coefficients for fixed weight up to twisting.
For weights \(2,3,4,5,6,2^r+1, 3\cdot 2^r+1\) \((r\in \mathbb N)\) this holds unconditionally.
The idea of proof is to translate the statement to Hecke characters, and classify Hecke characters. Tables for weight 3 (and 4) are produced, where finiteness holds unconditionally.
The next question is a geometric realization problem: how many of these CM newforms actually occur in association with singular K3 surfaces over \(\mathbb Q\)? This is answered in the following theorem with Elkies.
Theorem: Assuming ERH for odd real Dirichlet characters, every newform of weight 3 with rational coefficients is associated to a singular \(K3\) surface over \(\mathbb Q\).
Reviewer: Noriko Yui (Kingston)
MSC:
| 11F11 | Holomorphic modular forms of integral weight |
| 11F30 | Fourier coefficients of automorphic forms |
| 11F23 | Relations with algebraic geometry and topology |
| 14J28 | \(K3\) surfaces and Enriques surfaces |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
Citations:
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