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On the Hasse-Witt invariants of modular curves. (English) Zbl 0923.11090
Let \(\overline A\) be an abelian variety over a field of characteristic \(p>0\). Then \(r(\overline A):= \dim_{\mathbb{F}_p} \overline A[p]\), the dimension of the space of \(p\)-torsion points on \(\overline A\), is called the Hasse-Witt invariant of \(\overline A\). It is an integer which satisfies \(0\leq r(\overline A)\leq\dim \overline A\).
Now let \(A\) be an abelian variety over \(\mathbb{Q}\) and consider for each prime number \(p\) the reduction \(\overline A=A\bmod p\) of \(A\). It is a natural question to ask for the distribution of the corresponding Hasse-Witt invariants. In particular it is interesting to know the density of those primes \(p\), where \(r(A\bmod p) =\dim A\), \(<\dim A\) or \(=0\) respectively. If \(A\) has complex multiplication, then the density of these sets is equal to the density of primes with given decomposition law on the endomorphism algebra. If \(A\) does not admit complex multiplication only little is known.
In the case of an elliptic curve \(E\) there are partial results of Elkies, Elkies and Murty and a conjecture of Lang and Trotter on the density of supersingular primes, i.e. primes with \(r(E\bmod p)=0\).
The authors generalize the conjecture of Lang and Trotter to the case of abelian varieties \(A/ \mathbb{Q}\) of dimension \(d\geq 2\) which are factors of the Jacobians of modular curves \(X_0(N)\). It is shown that these conjectures are based on certain probabilistic models. Numerical examples at the end of the paper give some support for these ideas.
Reviewer: H.-G.Rück (Essen)

11G10 Abelian varieties of dimension \(> 1\)
14K10 Algebraic moduli of abelian varieties, classification
14G35 Modular and Shimura varieties
11G18 Arithmetic aspects of modular and Shimura varieties
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