Jacobians with automorphisms of prime order
Yuri Zarhin1
1Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
Mathematics Research Reports, Volume 7 (2026), pp. 1-26

In this paper we study principally polarized complex abelian varieties $(X,\lambda )$ that admit an automorphism $\delta $ of prime order $p>2$. It turns out that certain natural conditions on the multiplicities of the action of $\delta $ on $\Omega ^1(X)$ do guarantee that those polarized varieties are not canonically polarized jacobians of curves.

Received:
Revised:
Published online:
DOI: 10.5802/mrr.24
Classification: 14J45, 14K30
Keywords: Jacobians, polarizations, endomorphisms rings of abelian varieties

Yuri Zarhin  1

1 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{MRR_2026__7__1_0,
     author = {Yuri Zarhin},
     title = {Jacobians with automorphisms of prime order},
     journal = {Mathematics Research Reports},
     pages = {1--26},
     year = {2026},
     publisher = {MathOA foundation},
     volume = {7},
     doi = {10.5802/mrr.24},
     language = {en},
     url = {https://mrr.centre-mersenne.org/articles/10.5802/mrr.24/}
}
TY  - JOUR
AU  - Yuri Zarhin
TI  - Jacobians with automorphisms of prime order
JO  - Mathematics Research Reports
PY  - 2026
SP  - 1
EP  - 26
VL  - 7
PB  - MathOA foundation
UR  - https://mrr.centre-mersenne.org/articles/10.5802/mrr.24/
DO  - 10.5802/mrr.24
LA  - en
ID  - MRR_2026__7__1_0
ER  - 
%0 Journal Article
%A Yuri Zarhin
%T Jacobians with automorphisms of prime order
%J Mathematics Research Reports
%D 2026
%P 1-26
%V 7
%I MathOA foundation
%U https://mrr.centre-mersenne.org/articles/10.5802/mrr.24/
%R 10.5802/mrr.24
%G en
%F MRR_2026__7__1_0
Yuri Zarhin. Jacobians with automorphisms of prime order. Mathematics Research Reports, Volume 7 (2026), pp. 1-26. doi: 10.5802/mrr.24

[1] Jeffrey D. Achter; Rachel Pries The integral monodromy of hyperelliptic and trielliptic curves, Math. Ann., Volume 338 (2007) no. 1, pp. 187-206 | DOI | Zbl | MR

[2] George E. Andrews; Kimmo Eriksson Integer partitions, Cambridge University Press, Cambridge, 2004, x+141 pages | DOI | Zbl | MR

[3] M. F. Atiyah; R. Bott A Lefschetz fixed point formula for elliptic differential operators, Bull. Amer. Math. Soc., Volume 72 (1966), pp. 245-250 | DOI | Zbl | MR

[4] Christina Birkenhake; Herbert Lange Complex abelian varieties, Grundlehren der mathematischen Wissenschaften, 302, Springer-Verlag, Berlin, 2004, xii+635 pages | DOI | Zbl | MR

[5] C. Herbert Clemens; Phillip A. Griffiths The intermediate Jacobian of the cubic threefold, Ann. of Math. (2), Volume 95 (1972), pp. 281-356 | DOI | Zbl | MR

[6] Phillip Griffiths; Joseph Harris Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978, xii+813 pages | Zbl | MR

[7] Tsuyoshi Hayashida; Mieo Nishi Existence of curves of genus two on a product of two elliptic curves, J. Math. Soc. Japan, Volume 17 (1965), pp. 1-16 | Zbl | MR

[8] Kenneth Ireland; Michael Rosen A classical introduction to modern number theory, Graduate Texts in Mathematics, 84, Springer-Verlag, New York, 1998, xiv+389 pages | Zbl | MR

[9] Serge Lang Algebra, Graduate Texts in Mathematics, 211, Springer-Verlag, New York, 2002, xvi+914 pages | DOI

[10] John Milnor Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999, viii+257 pages | Zbl | MR

[11] Hugh L. Montgomery; Robert C. Vaughan Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, Cambridge, 2007, xviii+552 pages | Zbl | MR

[12] Ben Moonen; Yuri G. Zarhin Weil classes on abelian varieties, J. Reine Angew. Math., Volume 496 (1998), pp. 83-92 (Erratum https://www.math.ru.nl/ bmoonen/Papers/ErratumCrelle98.pdf) | DOI | Zbl | MR

[13] David Mumford Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Tata Institute of Fundamental Research, Bombay; by Oxford University Press, London, 1974, xii+279 pages | Zbl | MR

[14] Bjorn Poonen; Edward F. Schaefer Explicit descent for Jacobians of cyclic covers of the projective line, J. Reine Angew. Math., Volume 488 (1997), pp. 141-188 | Zbl | MR

[15] I. Reiner Maximal orders, London Mathematical Society Monographs. New Series, 28, The Clarendon Press, Oxford University Press, Oxford, 2003, xiv+395 pages (Corrected reprint of the 1975 original, With a foreword by M. J. Taylor) | DOI | Zbl | MR

[16] Kenneth A. Ribet Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math., Volume 98 (1976) no. 3, pp. 751-804 | DOI | Zbl | MR

[17] Goro Shimura Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, 46, Princeton University Press, Princeton, NJ, 1997, xvi+218 pages | MR

[18] André Weil Zum Beweis des Torellischen Satzes, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa., Volume 1957 (1957), pp. 33-53 (Œuvres, vol. III, [1957a]) | Zbl | MR

[19] André Weil Sur le théorème de Torelli, Séminaire Bourbaki (Mai 1957) no. 151 | Zbl

[20] Yuri G. Zarhin The endomorphism rings of Jacobians of cyclic covers of the projective line, Math. Proc. Cambridge Philos. Soc., Volume 136 (2004) no. 2, pp. 257-267 | DOI | Zbl | MR

[21] Yuri G. Zarhin Cubic surfaces and cubic threefolds, Jacobians and intermediate Jacobians, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II (Progr. Math.), Volume 270, Birkhäuser Boston, Boston, MA, 2009, pp. 687-691 | DOI | Zbl

[22] Yuri G. Zarhin Endomorphism algebras of abelian varieties with special reference to superelliptic Jacobians, Geometry, algebra, number theory, and their information technology applications (Springer Proc. Math. Stat.), Volume 251, Springer, Cham, 2018, pp. 477-528 | DOI | Zbl

[23] Yuri G. Zarhin Superelliptic jacobians and central simple representations, Arithmetic, Geometry, Cryptography and Coding Theory (Contemporary Mathematics), Volume 832, American Mathematical Society, Providence, RI, 2026 (To appear; arXiv:2305.12022 [math.NT]) | DOI | Zbl

Cited by Sources: