## Local invariants of isogenous elliptic curves

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- by Tim Dokchitser and Vladimir Dokchitser PDF
- Trans. Amer. Math. Soc.
**367**(2015), 4339-4358 Request permission

## Abstract:

We investigate how various invariants of elliptic curves, such as the discriminant, Kodaira type, Tamagawa number and real and complex periods, change under an isogeny of prime degree $p$. For elliptic curves over $l$-adic fields, the classification is almost complete (the exception is wild potentially supersingular reduction when $l=p$), and is summarised in a table.## References

- B. J. Birch,
*Conjectures concerning elliptic curves*, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp.Â 106â€“112. MR**0174558** - K. ÄŚesnaviÄŤius,
*The $p$-parity conjecture for elliptic curves with a $p$-isogeny*, 2012, arxiv: 1207.0431, to appear in J. Reine Angew. Math. - John Coates,
*Elliptic curves with complex multiplication and Iwasawa theory*, Bull. London Math. Soc.**23**(1991), no.Â 4, 321â€“350. MR**1125859**, DOI 10.1112/blms/23.4.321 - David A. Cox,
*Primes of the form $x^2 + ny^2$*, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. Fermat, class field theory and complex multiplication. MR**1028322** - Fred Diamond and John Im,
*Modular forms and modular curves*, Seminar on Fermatâ€™s Last Theorem (Toronto, ON, 1993â€“1994) CMS Conf. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 1995, pp.Â 39â€“133. MR**1357209** - Tim Dokchitser and Vladimir Dokchitser,
*Parity of ranks for elliptic curves with a cyclic isogeny*, J. Number Theory**128**(2008), no.Â 3, 662â€“679. MR**2389862**, DOI 10.1016/j.jnt.2007.02.008 - Tim Dokchitser and Vladimir Dokchitser,
*On the Birch-Swinnerton-Dyer quotients modulo squares*, Ann. of Math. (2)**172**(2010), no.Â 1, 567â€“596. MR**2680426**, DOI 10.4007/annals.2010.172.567 - T. Dokchitser and V. Dokchitser,
*Growth of $\text {\eightcyr \cyracc {Sh}}$ in towers for isogenous curves*, preprint, 2013, arxiv:1301.4257. - Vladimir Dokchitser,
*Root numbers of non-abelian twists of elliptic curves*, Proc. London Math. Soc. (3)**91**(2005), no.Â 2, 300â€“324. With an appendix by Tom Fisher. MR**2167089**, DOI 10.1112/S0024611505015261 - Basil Gordon and Dale Sinor,
*Multiplicative properties of $\eta$-products*, Number theory, Madras 1987, Lecture Notes in Math., vol. 1395, Springer, Berlin, 1989, pp.Â 173â€“200. MR**1019331**, DOI 10.1007/BFb0086404 - Nicholas M. Katz,
*$p$-adic properties of modular schemes and modular forms*, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp.Â 69â€“190. MR**0447119** - Stefan Keil,
*Examples of non-simple abelian surfaces over the rationals with non-square order Tate-Shafarevich group*, J. Number Theory**144**(2014), 25â€“69. MR**3239151**, DOI 10.1016/j.jnt.2014.04.018 - A. A. Klyachko,
*Modular forms and representations of symmetric groups*, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)**116**(1982), 74â€“85, 162 (Russian). Integral lattices and finite linear groups. MR**687842** - K. Kramer and J. Tunnell,
*Elliptic curves and local $\varepsilon$-factors*, Compositio Math.**46**(1982), no.Â 3, 307â€“352. MR**664648** - Alain Kraus,
*Sur le dĂ©faut de semi-stabilitĂ© des courbes elliptiques Ă rĂ©duction additive*, Manuscripta Math.**69**(1990), no.Â 4, 353â€“385 (French, with English summary). MR**1080288**, DOI 10.1007/BF02567933 - Dino Lorenzini,
*Models of curves and wild ramification*, Pure Appl. Math. Q.**6**(2010), no.Â 1, Special Issue: In honor of John Tate., 41â€“82. MR**2591187**, DOI 10.4310/PAMQ.2010.v6.n1.a3 - Morris Newman,
*Construction and application of a class of modular functions. II*, Proc. London Math. Soc. (3)**9**(1959), 373â€“387. MR**107629**, DOI 10.1112/plms/s3-9.3.373 - Edward F. Schaefer,
*Class groups and Selmer groups*, J. Number Theory**56**(1996), no.Â 1, 79â€“114. MR**1370197**, DOI 10.1006/jnth.1996.0006 - Jean-Pierre Serre,
*Abelian $l$-adic representations and elliptic curves*, 2nd ed., Advanced Book Classics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. With the collaboration of Willem Kuyk and John Labute. MR**1043865** - Jean-Pierre Serre,
*PropriĂ©tĂ©s galoisiennes des points dâ€™ordre fini des courbes elliptiques*, Invent. Math.**15**(1972), no.Â 4, 259â€“331 (French). MR**387283**, DOI 10.1007/BF01405086 - Jean-Pierre Serre and John Tate,
*Good reduction of abelian varieties*, Ann. of Math. (2)**88**(1968), 492â€“517. MR**236190**, DOI 10.2307/1970722 - Joseph H. Silverman,
*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR**817210**, DOI 10.1007/978-1-4757-1920-8 - Joseph H. Silverman,
*Advanced topics in the arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR**1312368**, DOI 10.1007/978-1-4612-0851-8 - John Tate,
*On the conjectures of Birch and Swinnerton-Dyer and a geometric analog*, SĂ©minaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp.Â Exp. No. 306, 415â€“440. MR**1610977**

## Additional Information

**Tim Dokchitser**- Affiliation: Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom
- MR Author ID: 733080
- Email: tim.dokchitser@bristol.ac.uk
**Vladimir Dokchitser**- Affiliation: Department of Pure Mathematics and Mathematical Statistics, Emmanuel College, Cambridge CB2 3AP, United Kingdom
- Address at time of publication: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
- MR Author ID: 768165
- Email: v.dokchitser@dpmms.cam.ac.uk, v.dokchitser@warwick.ac.uk
- Received by editor(s): January 17, 2013
- Received by editor(s) in revised form: August 4, 2013
- Published electronically: October 10, 2014
- Additional Notes: The first author was supported by a Royal Society University Research Fellowship
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**367**(2015), 4339-4358 - MSC (2010): Primary 11G07; Secondary 11G05, 11G40
- DOI: https://doi.org/10.1090/S0002-9947-2014-06271-5
- MathSciNet review: 3324930