Restrictions on endomorphism rings of Jacobians and their minimal fields of definition

Goodman, Pip

doi:10.1090/tran/8347

Restrictions on endomorphism rings of Jacobians and their minimal fields of definition
HTML articles powered by AMS MathViewer

by Pip Goodman PDF

Trans. Amer. Math. Soc. 374 (2021), 4639-4654 Request permission

Abstract:

Zarhin has extensively studied restrictions placed on the endomorphism algebras of Jacobians $J$ for which the Galois group associated to their 2-torsion is insoluble and “large” (relative to the dimension of $J$). In this paper we examine what happens when this Galois group merely contains an element of “large” prime order.

References

Andrew R. Booker, Jeroen Sijsling, Andrew V. Sutherland, John Voight, and Dan Yasaki, A database of genus-2 curves over the rational numbers, LMS J. Comput. Math. 19 (2016), no. suppl. A, 235–254. MR 3540958, DOI 10.1112/S146115701600019X
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Timothy C. Burness and Michael Giudici, Classical groups, derangements and primes, Australian Mathematical Society Lecture Series, vol. 25, Cambridge University Press, Cambridge, 2016. MR 3443032, DOI 10.1017/CBO9781139059060
Benson Farb and R. Keith Dennis, Noncommutative algebra, Graduate Texts in Mathematics, vol. 144, Springer-Verlag, New York, 1993. MR 1233388, DOI 10.1007/978-1-4612-0889-1
Francesc Fité, Kiran S. Kedlaya, Víctor Rotger, and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, Compos. Math. 148 (2012), no. 5, 1390–1442. MR 2982436, DOI 10.1112/S0010437X12000279
Robert Guralnick and Kiran S. Kedlaya, Endomorphism fields of abelian varieties, Res. Number Theory 3 (2017), Paper No. 22, 10. MR 3720508, DOI 10.1007/s40993-017-0088-4
I. Martin Isaacs, Character theory of finite groups, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)]. MR 1280461
Christian U. Jensen, Arne Ledet, and Noriko Yui, Generic polynomials, Mathematical Sciences Research Institute Publications, vol. 45, Cambridge University Press, Cambridge, 2002. Constructive aspects of the inverse Galois problem. MR 1969648
The LMFDB Collaboration, The l-functions and modular forms database, 2018. [Online; accessed 10 December 2018].
Davide Lombardo, Computing the geometric endomorphism ring of a genus-2 Jacobian, Math. Comp. 88 (2019), no. 316, 889–929. MR 3882288, DOI 10.1090/mcom/3358
Pascal Molin and Christian Neurohr, Computing period matrices and the Abel-Jacobi map of superelliptic curves, Math. Comp. 88 (2019), no. 316, 847–888. MR 3882287, DOI 10.1090/mcom/3351
Shigefumi Mori, The endomorphism rings of some abelian varieties. II, Japan. J. Math. (N.S.) 3 (1977), no. 1, 105–109. MR 529440, DOI 10.4099/math1924.3.105
David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 0282985
Donald Passman, Permutation groups, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0237627
Gaël Rémond, Degré de définition des endomorphismes d’une variété abélienne, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 9, 3059–3099 (French, with English summary). MR 4127945, DOI 10.4171/jems/981
Kenneth A. Ribet, Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math. 98 (1976), no. 3, 751–804. MR 457455, DOI 10.2307/2373815
Murray M. Schacher, Subfields of division rings. I, J. Algebra 9 (1968), 451–477. MR 227224, DOI 10.1016/0021-8693(68)90015-X
A. Schwarz, M. Pohst, and F. Diaz y Diaz, A table of quintic number fields, Math. Comp. 63 (1994), no. 207, 361–376. MR 1219705, DOI 10.1090/S0025-5718-1994-1219705-3
Goro Shimura, On analytic families of polarized abelian varieties and automorphic functions, Ann. of Math. (2) 78 (1963), 149–192. MR 156001, DOI 10.2307/1970507
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original; Kanô Memorial Lectures, 1. MR 1291394
Goro Shimura, Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, vol. 46, Princeton University Press, Princeton, NJ, 1998. MR 1492449, DOI 10.1515/9781400883943
A. Silverberg, Fields of definition for homomorphisms of abelian varieties, J. Pure Appl. Algebra 77 (1992), no. 3, 253–262. MR 1154704, DOI 10.1016/0022-4049(92)90141-2
Yuri G. Zarhin, Hyperelliptic Jacobians without complex multiplication, Math. Res. Lett. 7 (2000), no. 1, 123–132. MR 1748293, DOI 10.4310/MRL.2000.v7.n1.a11
Yuri G. Zarhin, Hyperelliptic Jacobians and modular representations, Moduli of abelian varieties (Texel Island, 1999) Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 473–490. MR 1827030
Yuri G. Zarhin, Very simple representations: variations on a theme of Clifford, Progress in Galois theory, Dev. Math., vol. 12, Springer, New York, 2005, pp. 151–168. MR 2148465, DOI 10.1007/0-387-23534-5_{9}