## Abelian surfaces over finite fields as Jacobians. With an appendix by Everett W. Howe.(English)Zbl 1101.14056

Summary: For any finite field $$k=\mathbb{F}_q$$, we explicitly describe the $$k$$-isogeny classes of abelian surfaces defined over $$k$$ and their behavior under finite field extension. In particular, we determine the absolutely simple abelian surfaces. Then, we analyze numerically what surfaces are $$k$$-isogenous to the Jacobian of a smooth projective curve of genus 2 defined over $$k$$. We prove some partial results suggested by these numerical data. For instance, we show that every absolutely simple abelian surface is $$k$$-isogenous to a Jacobian. Other facts suggested by these numerical computations are that the polynomials $$t^4+(1-2q)t^2+q^2$$ $$q$$ and $$t^4+(2-2q)t^2+q^2$$ (for $$q$$ odd) are never the characteristic polynomial of Frobenius of a Jacobian. These statements have been proved by E. Howe. The proof for the first polynomial is attached in an appendix.

### MSC:

 14K15 Arithmetic ground fields for abelian varieties 11G10 Abelian varieties of dimension $$> 1$$ 11G25 Varieties over finite and local fields
Full Text:

### References:

 [1] Adleman L. M., Primality testing and abelian varieties over finite fields (1992) · Zbl 0744.11065 [2] Cardona G., Curves of genus two over fields of even characteristic. · Zbl 1097.11033 [3] Guàrdia J., Geometria Aritmètica en una famìlia de corbes de gènere tres. (1998) [4] Howe E. W., Transactions of the American Mathematical Society 347 pp 2361– (1995) · Zbl 0859.14016 [5] Howe E. W., Journal of Algebraic Geometry 5 pp 583– (1996) [6] Howe E. W., Compositio Mathematical [7] Howe E. W., Journal of Number Theory 92 pp 139– (2002) · Zbl 0998.11031 [8] Igusa J.-I., Annals of Mathematics 72 pp 612– (1960) · Zbl 0122.39002 [9] Lachaud G., Journal of Number Theory 39 pp 18– (1991) · Zbl 0741.11048 [10] Lauter K., Proceedings of the American Mathematical Society 128 (2) pp 369– (2000) · Zbl 0983.11036 [11] López A., Finite Fields and Their Applications 8 pp 193– (2002) · Zbl 1036.14011 [12] Milne J. S., Arithmetic Geometry pp 167– (1986) [13] Mumford David, Abelian Varieties,, 2. ed. (1974) [14] Ruck H.-G., Compositio Mathematica 76 pp 351– (1990) [15] Stark H., Analytic Number Theory pp 285– (1973) [16] Tate J., Séminaire Bourbaki 1968/69, Exposé 352 pp 95– (1971) [17] van der Geer G., Mathematische Nachrichten 159 pp 73– (1992) · Zbl 0774.14045 [18] Waterhouse W. C., Annales Scientifiques de l’École Normale Supérieure (4) 2 pp 521– (1969) · Zbl 0188.53001 [19] Waterhouse, W. and Milne, J. ”Abelian varieties over finite fields.”. Proceedings of Symposia in Pure Mathematics. 1969 Number Theory Institute, Vol. XX, pp.53–64. Providence, RI: American Mathematical Society. [Waterhouse and Milne 69] · Zbl 0216.33102 [20] Weil A., Nachrichten der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse IIa pp 33– (1957) [21] Xing C. P., Archiv der Mathematik 63 pp 427– (1994) · Zbl 0813.14015 [22] Xing C. P., Finite Fields and Their Applications 2 pp 407– (1996) · Zbl 0923.11091
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.