×

zbMATH — the first resource for mathematics

On the Sato-Tate conjecture for non-generic abelian surfaces. (English) Zbl 1428.11100
Summary: We prove many non-generic cases of the Sato-Tate conjecture for abelian surfaces as formulated by F. FitĂ© et al. [Compos. Math. 148, No. 5, 1390–1442 (2012; Zbl 1269.11094)], using the potential automorphy theorems of Barnet-Lamb, Gee, Geraghty and Taylor.

MSC:
11F80 Galois representations
11G10 Abelian varieties of dimension \(> 1\)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Arthur, James; Clozel, Laurent, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies 120, xiv+230 pp., (1989), Princeton University Press, Princeton, NJ · Zbl 0682.10022
[2] Artin, Emil; Tate, John, Class field theory, Advanced Book Classics, xxxviii+259 pp., (1990), Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA · Zbl 0681.12003
[3] Banaszak, Grzegorz; Kedlaya, Kiran S., An algebraic Sato-Tate group and Sato-Tate conjecture, Indiana Univ. Math. J., 64, 1, 245-274, (2015) · Zbl 1392.11041
[4] Barnet-Lamb, Thomas; Gee, Toby; Geraghty, David, The Sato-Tate conjecture for Hilbert modular forms, J. Amer. Math. Soc., 24, 2, 411-469, (2011) · Zbl 1269.11045
[5] Barnet-Lamb, Thomas; Gee, Toby; Geraghty, David; Taylor, Richard, Potential automorphy and change of weight, Ann. of Math. (2), 179, 2, 501-609, (2014) · Zbl 1310.11060
[6] Barnet-Lamb, Tom; Geraghty, David; Harris, Michael; Taylor, Richard, A family of Calabi-Yau varieties and potential automorphy II, Publ. Res. Inst. Math. Sci., 47, 1, 29-98, (2011) · Zbl 1264.11044
[7] Cardona, Gabriel, \(\mathbb{Q}\)-curves and abelian varieties of \(\rm GL_2\)-type from dihedral genus 2 curves. Modular curves and abelian varieties, Progr. Math. 224, 45-52, (2004), Birkh\"auser, Basel · Zbl 1080.11045
[8] Cardona, Gabriel; Quer, Jordi, Curves of genus 2 with group of automorphisms isomorphic to \(D_8\) or \(D_{12}\), Trans. Amer. Math. Soc., 359, 6, 2831-2849 (electronic), (2007) · Zbl 1192.11038
[9] [Far]Far L. Fargues, Motives and automorphic forms: The (potentially) abelian case, available at http://www.math.jussieu.fr/\textasciitilde fargues/Motifs_abeliens.pdf.
[10] Fit\'e, Francesc; Kedlaya, Kiran S.; Rotger, V\'\i ctor; Sutherland, Andrew V., Sato-Tate distributions and Galois endomorphism modules in genus 2, Compos. Math., 148, 5, 1390-1442, (2012) · Zbl 1269.11094
[11] [FS]FS Francesc Fit\'e and Andrew Sutherland, Sato-Tate distributions of twists of \(y^2=x^5-x\) and \(y^2=x^6+1\), preprint, available at http://arxiv.org/abs/1203.1476. · Zbl 1303.14051
[12] Harris, Michael, Potential automorphy of odd-dimensional symmetric powers of elliptic curves and applications. Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, Progr. Math. 270, 1-21, (2009), Birkh\"auser Boston, Inc., Boston, MA · Zbl 1234.11068
[13] Harris, Michael; Shepherd-Barron, Nick; Taylor, Richard, A family of Calabi-Yau varieties and potential automorphy, Ann. of Math. (2), 171, 2, 779-813, (2010) · Zbl 1263.11061
[14] Kedlaya, Kiran S.; Sutherland, Andrew V., Computing \(L\)-series of hyperelliptic curves. Algorithmic number theory, Lecture Notes in Comput. Sci. 5011, 312-326, (2008), Springer, Berlin · Zbl 1232.11078
[15] Neukirch, J\"urgen; Schmidt, Alexander; Wingberg, Kay, Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 323, xvi+699 pp., (2000), Springer-Verlag, Berlin · Zbl 0948.11001
[16] Ribet, Kenneth A., Abelian varieties over \(\textbf{Q}\) and modular forms. Algebra and topology 1992 (Taej\u on), 53-79, (1992), Korea Adv. Inst. Sci. Tech., Taej\u on
[17] Serre, Jean-Pierre, Abelian \(l\)-adic representations and elliptic curves, McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute, xvi+177 pp. (not consecutively paged) pp., (1968), W. A. Benjamin, Inc., New York-Amsterdam
[18] \bibSer2book label=Ser2, author=Serre, Jean-Pierre, title=Lectures on \(N_X (p)\), series=Chapman & Hall/CRC Research Notes in Mathematics, volume=11, pages=x+163, publisher=CRC Press, Boca Raton, FL, date=2012, isbn=978-1-4665-0192-8, review=\MR 2920749,
[19] Tate, J., Number theoretic background. Automorphic forms, representations and \(L\)-functions, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977, Proc. Sympos. Pure Math., XXXIII, 3-26, (1979), Amer. Math. Soc., Providence, R.I.
[20] Tautz, Walter; Top, Jaap; Verberkmoes, Alain, Explicit hyperelliptic curves with real multiplication and permutation polynomials, Canad. J. Math., 43, 5, 1055-1064, (1991) · Zbl 0793.14022
[21] Taylor, Richard, Remarks on a conjecture of Fontaine and Mazur, J. Inst. Math. Jussieu, 1, 1, 125-143, (2002) · Zbl 1047.11051
[22] Weil, Andr\'e, Sur la th\'eorie du corps de classes, J. Math. Soc. Japan, 3, 1-35, (1951) · Zbl 0044.02901
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.