Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-28T07:11:02.627Z Has data issue: false hasContentIssue false
  • English
  • Français

Modular Abelian Varieties Over Number Fields

Published online by Cambridge University Press:  20 November 2018

Xavier Guitart  and
Jordi Quer
Xavier Guitart
Affiliation:
Max Planck Institute forMathematics, Vivatsgasse 7, 53111 Bonn, Germany and Departament de Matemàtica Aplicada II, Universitat Politécnica de Catalunya, C. Jordi Girona 1-3, 08034 Barcelona, Spain e-mail: [email protected]
Jordi Quer
Affiliation:
Departament de Matemàtica Aplicada II, Universitat Politécnica de Catalunya, C. Jordi Girona 1-3, 08034 Barcelona, Spain e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The main result of this paper is a characterization of the abelian varieties $B/K$ defined over Galois number fields with the property that the $L$-function $L\left( B/K;\,s \right)$ is a product of $L$-functions of non-$\text{CM}$ newforms over $Q$ for congruence subgroups of the form ${{\Gamma }_{1}}\,\left( N \right)$. The characterization involves the structure of End$\left( B \right)$, isogenies between the Galois conjugates of $B$, and a Galois cohomology class attached to $B/K$.

We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which the general results of the paper can be applied, we prove the strong modularity of some particular abelian surfaces belonging to that family, and we show how to find nontrivial examples of strongly modular varieties by twisting.

Keywords

11G1011G1811F11modular abelian varietiesGLsub2/sub-type varietiesmodular forms
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 1 , February 2014 , pp. 170 - 196
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Baba, S. and Granath, H., Genus 2 curves with quaternionic multiplication. Canad. J. Math. 60(2008), no. 4, 734–757. http://dx.doi.org/10.4153/CJM-2008-033-7 Google Scholar
[2] Breuil, C., Conrad, B., Diamond, F., and Taylor, R., On the modularity of elliptic curves over ℚ: wild 3-adic exercises. J. Amer. Math. Soc. 14(2001), no. 4, 843–939. http://dx.doi.org/10.1090/S0894-0347-01-00370-8 Google Scholar
[3] Cremona, J. E., Algorithms for modular elliptic curves. Second ed., Cambridge University Press, Cambridge, 1997.Google Scholar
[4] Dieulefait, L. and Urroz, J. J., Solving Fermat-type equations via modular ℚ-curves over polyquadratic fields. J. Reine Angew. Math. 633(2009), 183–195.Google Scholar
[5] Elkies, N., On elliptic K-curves. In: Modular curves and abelian varieties, Progr. Math., 224, Birkhäuser, Basel, 2004, pp. 81–91.Google Scholar
[6] González, J. and Lario, J.-C., Rational and elliptic parametrizations of ℚ-curves. J. Number Theory 72(1998), no. 1, 13–31. http://dx.doi.org/10.1006/jnth.1998.2259 Google Scholar
[7] Hashimoto, K.-I. and Murabayashi, N., Shimura curves as intersections of Humbert surfaces and defining equations of QM-curves of genus two. Tôhoku Math. J. 47(1995), no. 2, 271–296. http://dx.doi.org/10.2748/tmj/1178225596 Google Scholar
[8] Karpilovsky, G., Group representations. Vol. 2. North-Holland Mathematics Studies, 177, North-Holland Publishing Co., Amsterdam, 1993.Google Scholar
[9] Khare, C. and Wintenberger, J.-P., Serre's modularity conjecture. I. Invent. Math. 178(2009), no. 3, 485–504. http://dx.doi.org/10.1007/s00222-009-0205-7 Google Scholar
[10] Khare, C., Serre's modularity conjecture. II. Invent. Math. 178(2009), no. 3, 505–586. http://dx.doi.org/10.1007/s00222-009-0206-6 Google Scholar
[11] Milne, J. S., Complex multiplication. Course notes available at http://www.jmilne.org. Google Scholar
[12] Mumford, D., Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, 5, Oxford University Press, London, 1970.Google Scholar
[13] Pyle, E. E., Abelian varieties over ℚ with large endomorphism algebras and their simple components over In: Modular curves and abelian varieties, Progr. Math., 224, Birkhäuser, Basel, 2004, pp. 189–239.Google Scholar
[14] Quer, J., ℚ-curves and abelian varieties of GL2-type. Proc. London Math. Soc. 81(2000), no. 2, 285–317. http://dx.doi.org/10.1112/S0024611500012570 Google Scholar
[15] Quer, J., Embedding problems over abelian groups and an application to elliptic curves. J. Algebra 237(2001), no. 1, 186–202. http://dx.doi.org/10.1006/jabr.2000.8578 Google Scholar
[16] Quer, J., Fields of definition of building blocks. Math. Comp. 78(2009), no. 265, 537–554. http://dx.doi.org/10.1090/S0025-5718-08-02132-7 Google Scholar
[17] Ribet, K. A., Fields of definition of abelian varieties with real multiplication. In: Arithmetic geometry (Tempe, AZ, 1993), Contemp. Math., 174, American Mathematical Society, Providence, RI, 1994, pp. 107–118.Google Scholar
[18] Ribet, K. A., Abelian varieties over ℚ and modular forms. In: Modular curves and abelian varieties, Progr. Math., 224, Birkhäuser, Basel, 2004, pp. 241–261.Google Scholar
[19] Shimura, G., Introduction to the arithmetic theory of automorphic functions. Kanô Memorial Lectures, 1, Publications of the Mathematical Society of Japan, 11, Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, NJ, 1971.Google Scholar
[20] Stein, W.. Modular forms: A computational approach. With an appendix by Paul Gunnells. Graduate Studies in Mathematics, 79, American Mathematical Society, Providence, RI, 2007.Google Scholar
[21] Taylor, R. and Wiles, A., Ring-theoretic properties of certain Hecke algebras. Ann. of Math. 141(1995), no. 3, 553–572. http://dx.doi.org/10.2307/2118560 Google Scholar
[22] Wiles, A., Modular elliptic curves and Fermat's last theorem. Ann. of Math. 141(1995), no. 3, 443–551. http://dx.doi.org/10.2307/2118559 Google Scholar