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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)

##### MSC:
 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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