×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI EMIS
References:
[1] de Shalit E., Iwasawa theory of elliptic curves with complex multiplication; p-adic L functions (1987) · Zbl 0674.12004
[2] Deligne P., Hodge cycles, motives, and Shimura varieties (1982) · Zbl 0465.00010
[3] DOI: 10.1007/BF02940746 · Zbl 0025.02003 · doi:10.1007/BF02940746
[4] Elkies N. D., Invent. Math. 89 (3) pp 561– (1987) · Zbl 0631.14024 · doi:10.1007/BF01388985
[5] Elkies N. D., Journées Arithmétiques pp 127– (1991)
[6] Gonz’lez J., Tohoku Math. J. 49 (1997)
[7] Gonz’lez Rovira J., Ann. Inst. Fourier (Grenoble) 41 (4) pp 779– (1991) · Zbl 0758.14010 · doi:10.5802/aif.1273
[8] Hasse H., J. Reine angew. Math. 172 pp 77– (1934) · JFM 60.0101.03
[9] Hasse H., Monats. Math. Phys. 43 pp 477– (1936) · Zbl 0013.34102 · doi:10.1007/BF01707628
[10] Hecke E., Lectures on the theory of algebraic numbers (1981) · Zbl 0504.12001
[11] Huppert B., Endliche Gruppen. I (1967) · Zbl 0217.07201 · doi:10.1007/978-3-642-64981-3
[12] Lang S., Frobenius distributions in GL2-extensions (1976)
[13] Manin J. I., Izv. Akad. Nauk SSSR Ser. Mat. 25 pp 153– (1961)
[14] Manin J. I., Izv. Akad. Nauk SSSR Ser. Mat. 26 pp 281– (1962)
[15] Momose F., J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1) pp 89– (1981)
[16] Ribet K. A., Modular functions of one variable V pp 17– (1977)
[17] Ribet K. A., Math. Ann. 253 (1) pp 43– (1980) · Zbl 0421.14008 · doi:10.1007/BF01457819
[18] DOI: 10.1017/S0017089500006170 · Zbl 0596.10027 · doi:10.1017/S0017089500006170
[19] Ribet K. A., Algebra and topology pp 53– (1992)
[20] Ribet K. A., Arithmetic geometry pp 107– (1994) · doi:10.1090/conm/174/01854
[21] DOI: 10.1007/BF02698692 · Zbl 0496.12011 · doi:10.1007/BF02698692
[22] Serre J.-P., Annuaire du Collège de France pp 95– (1985)
[23] Serre J.-P., Abelian l-adic representations and elliptic curves, (1989)
[24] Shimura G., Introduction to the arithmetic theory of automorphic functions (1971) · Zbl 0221.10029
[25] Tate J. T., Arithmetical Algebraic Geometry pp 93– (1965)
[26] Tate J., Invent. Math. 2 pp 134– (1966) · Zbl 0147.20303 · doi:10.1007/BF01404549
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.