This intriguing article presents some evidence for a conjecture attributed to Coleman on the endomorphism algebras associated to general abelian varieties, particularly in the setting of certain modular abelian varieties.
To state the general conjecture, let \(A\) be an abelian variety defined over a number field \(L\). We write \(\text{End}_L(A)\) to denote the ring of endomorphisms of \(A\) defined over \(L\), and \(\text{End}_L^0(A) = \text{End}_L(A) \otimes {\mathbb{Q}}\) its associated endomorphism algebra. Given integers \(n, m \geq 1\), it is conjectured that there exist ony finitely many \({\mathbb{Q}}\)-algebras \(M\) such that \(M \approx \text{End}_L^0(A)\) for some abelian variety \(A\) of dimension \(n\) defined over a number field \(L\) of degree \(m\).
Inspired by this conjecture, the author proves the following result. Let \(f\) be a normalized cuspidal newform of weight \(2\), level \(\Gamma_1(N)\), and trivial Nebentype. Let \[ f(z)= \sum_{n \geq 1} a_n(f) q^n, \qquad q = \exp(2 \pi i z) \] denote its Fourier series expansion, and \(L_f = {\mathbb{Q}}(a_v(f))\) the number field obtained by adjoining to \({\mathbb{Q}}\) the coefficients \(a_v(f)\) for any positive density set of primes \(v\). A classical construction associates to \(f\) an abelian variety \(A_f\) defined over \({\mathbb{Q}}\) of dimension \([L_f: {\mathbb{Q}}]\). This abelian variety occurs as a simple quotient over \({\mathbb{Q}}\) of the Jacobian of the modular curve \(X_1(N)\), and its endomorphism algebra \(\text{End}_L^0(A_f)\) is isomorphic to \(L_f\). Anyhow, for each integer \(n \geq 1\), consider the set \(\mathcal{S}_n\) of pairs of endomorphism algebras \[ \left( \text{End}^0_{\overline{\mathbb{Q}}}(A_f), \text{End}^0_{{\mathbb{Q}}}(A_f) \right), \] where \(f\) ranges over the set of normalized newforms of weight \(2\), level \(N\), and trivial Nebentype such that \(\dim(A_f)=n\). It is easy to see from the theory of complex multiplication for elliptic curves that \(\mathcal{S}_1 = 10\). The less straightforward task of determining the finiteness of the sets \(\mathcal{S}_n\) for \(n >1\) is the aim of this article, and of course an interesting test case for Coleman’s conjecture. Some partial results for abelian varieties with quaternionic multiplication have already been established by V. Rotger [“Which quaternion algebras act on a modular abelian variety?”, Math. Res. Lett. 15, No. 2–3, 251–263 (2008; Zbl 1226.11067)], and for surfaces by N. Bruin, E. V. Flynn, J. González and V. Rotger [“On finiteness conjectures for endomorphism algebras of abelian surfaces”, Math. Proc. Camb. Philos. Soc. 141, No. 3, 383–408 (2006; Zbl 1116.14042)].
The partial results presented here pertain to CM newforms \(f\) (see §2 of the paper under review), and the subsets of \(\mathcal{S}_n\) corresponding to such CM forms. In particular, these subsets are shown to be finite for all \(n \geq 1\), with the subset corresponding to \(n = 2\) to be of cardinality \(83\).