Root numbers of abelian varieties
HTML articles powered by AMS MathViewer
- by Maria Sabitova PDF
- Trans. Amer. Math. Soc. 359 (2007), 4259-4284 Request permission
Abstract:
We generalize a theorem of D. Rohrlich concerning root numbers of elliptic curves over number fields. Our result applies to arbitrary abelian varieties. Namely, under certain conditions which naturally extend the conditions used by D. Rohrlich, we show that the root number $W(A,\tau )$ associated to an abelian variety $A$ over a number field $F$ and a complex finite-dimensional irreducible representation $\tau$ of $\operatorname {Gal}(\overline {F}/F)$ with real-valued character is equal to $1$. We also show that our result is consistent with a refined version of the conjecture of Birch and Swinnerton-Dyer.References
- Claude Chevalley, Théorie des groupes de Lie. Tome III. Théorèmes généraux sur les algèbres de Lie, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1226, Hermann & Cie, Paris, 1955 (French). MR 0068552
- Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin, 1990. With an appendix by David Mumford. MR 1083353, DOI 10.1007/978-3-662-02632-8
- J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 103–150. MR 861974
- Michel Raynaud, 1-motifs et monodromie géométrique, Astérisque 223 (1994), 295–319 (French). Périodes $p$-adiques (Bures-sur-Yvette, 1988). MR 1293976
- David E. Rohrlich, The vanishing of certain Rankin-Selberg convolutions, Automorphic forms and analytic number theory (Montreal, PQ, 1989) Univ. Montréal, Montreal, QC, 1990, pp. 123–133. MR 1111015
- David E. Rohrlich, Elliptic curves and the Weil-Deligne group, Elliptic curves and related topics, CRM Proc. Lecture Notes, vol. 4, Amer. Math. Soc., Providence, RI, 1994, pp. 125–157. MR 1260960, DOI 10.1090/crmp/004/10
- David E. Rohrlich, Galois theory, elliptic curves, and root numbers, Compositio Math. 100 (1996), no. 3, 311–349. MR 1387669
- Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
- Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 236190, DOI 10.2307/1970722
- John Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144. MR 206004, DOI 10.1007/BF01404549
Additional Information
- Maria Sabitova
- Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
- Address at time of publication: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
- MR Author ID: 707297
- Email: sabitova@math.upenn.edu, sabitova@math.uiuc.edu
- Received by editor(s): May 6, 2005
- Received by editor(s) in revised form: July 21, 2005
- Published electronically: April 11, 2007
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 359 (2007), 4259-4284
- MSC (2000): Primary 11G10; Secondary 11F80, 11G40, 11R32
- DOI: https://doi.org/10.1090/S0002-9947-07-04148-7
- MathSciNet review: 2309184