Skip to main content

The ℓ-parity conjecture over the constant quadratic extension


For a prime ℓ and an abelian variety A over a global field K, the ℓ-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton–Dyer, the ℤ-corank of the ℓ-Selmer group and the analytic rank agree modulo 2. Assuming that char K > 0, we prove that the ℓ-parity conjecture holds for the base change of A to the constant quadratic extension if ℓ is odd, coprime to char K, and does not divide the degree of every polarisation of A. The techniques involved in the proof include the étale cohomological interpretation of Selmer groups, the Grothendieck–Ogg–Shafarevich formula and the study of the behavior of local root numbers in unramified extensions.

Hide All
[Čes15] Česnavičius, K. Topology on cohomology of local fields. Forum Math. Sigma 3 (2015), e16, 55. DOI 10.1017/fms.2015.18.
[Čes16a] Česnavičius, K. Local factors valued in normal domains. Int. J. Number Theory 12 (2016), no. 1, 249272. DOI 10.1142/S1793042116500159. MR3455278.
[Čes16b] Česnavičius, K. Selmer groups as flat cohomology groups. J. Ramanujan Math. Soc. 31 (2016), no. 1, 3161, MR3476233.
[Čes16c] Česnavičius, K. The p-parity conjecture for elliptic curves with a p-isogeny. J. reine angew. Math. 719 (2016), 4573. DOI 10.1515/crelle-2014-0040. MR3552491.
[Čes17] Česnavičius, K. p-Selmer growth in extensions of degree p. J. Lond. Math. Soc. (2) 95 (2017), no. 3, 833852. DOI 10.1112/jlms.12038.
[CFKS10] Coates, J., Fukaya, T., Kato, K. and Sujatha, R. Root numbers, Selmer groups and non-commutative Iwasawa theory. J. Algebraic Geom. 19 (2010), no. 1, 1997. DOI 10.1090/S1056-3911-09-00504-9. MR2551757 (2011a:11127).
[Dav10] Davydov, A. Twisted automorphisms of group algebras. Noncommutative structures in mathematics and physics, Vlaam, K.. (Acad. Belgie Wet. Kunsten (KVAB), Brussels, 2010), pp. 131150, MR2742735 (2012b:16085).
[Del73] Deligne, P. Les constantes des équations fonctionnelles des fonctions L, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) (Springer: Berlin) (1973), pp. 501597. Lecture Notes in Math. vol. 349 (French). MR0349635 (50 #2128).
[DD08] Dokchitser, T. and Dokchitser, V. Parity of ranks for elliptic curves with a cyclic isogeny. J. Number Theory 128 (2008), no. 3, 662679. DOI 10.1016/j.jnt.2007.02.008. MR2389862 (2009c:11079).
[DD09a] Dokchitser, T. and Dokchitser, V. Regulator constants and the parity conjecture. Invent. Math. 178 (2009), no. 1, 2371. DOI 10.1007/s00222-009-0193-7. MR2534092 (2010j:11089).
[DD09b] Dokchitser, T. and Dokchitser, V. Self-duality of Selmer groups. Math. Proc. Camb. Phil. Soc. 146 (2009), no. 2, 257267. DOI 10.1017/S0305004108001989. MR2475965 (2010a:11219).
[DD10] Dokchitser, T. and Dokchitser, V. On the Birch–Swinnerton–Dyer quotients modulo squares. Ann. of Math. (2) 172 (2010), no. 1, 567596. DOI 10.4007/annals.2010.172.567. MR2680426 (2011h:11069).
[DD11] Dokchitser, T. and Dokchitser, V. Root numbers and parity of ranks of elliptic curves. J. reine angew. Math. 658 (2011), 3964. DOI 10.1515/CRELLE.2011.060. MR2831512.
[GA09] González-Avilés, C. D. Arithmetic duality theorems for 1-motives over function fields. J. reine angew. Math. 632 (2009), 203231. DOI 10.1515/CRELLE.2009.055. MR2544149 (2010i:11169).
[GH70] Gamst, J. and Hoechsmann, K. Products in sheaf-cohomology. Tôhoku Math. J. (2) 22 (1970), 143162. MR0289605 (44 #6793).
[GH71] Gamst, J. and Hoechsmann, K. Ext-products and edge-morphisms. Tôhoku Math. J. (2) 23 (1971), 581588. MR0302741 (46 #1884).
[GH81] Gross, B. H. and Harris, J. Real algebraic curves. Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 2, 157182. MR631748 (83a:14028).
[HR79] Hewitt, E. and Ross, K. A. Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] vol. 115, (Springer-Verlag: Berlin, 1979), Structure of topological groups, integration theory, group representations, MR551496 (81k:43001).
[HS05] Harari, D. and Szamuely, T. Arithmetic duality theorems for 1-motives. J. reine angew. Math. 578 (2005), 93128. DOI 10.1515/crll.2005.2005.578.93. MR2113891 (2006f:14053).
[HS05e] Harari, D. and Szamuely, T. Corrigenda for: Arithmetic duality theorems for 1-motives [MR 2113891]. J. reine angew. Math. 632 (2009), 233236. DOI 10.1515/CRELLE.2009.056, MR2544150 (2010i:14079).
[Kim07] Du Kim, B. The parity conjecture for elliptic curves at supersingular reduction primes. Compos. Math. 143 (2007), no. 1, 4772. DOI 10.1112/S0010437X06002569. MR2295194 (2007k:11091).
[Kis04] Kisilevsky, H. Rank determines semi-stable conductor. J. Number Theory 104 (2004), no. 2, 279286. DOI 10.1016/S0022-314X(03)00157-4. MR2029506 (2005h:11137).
[KMR13] Klagsbrun, Z., Mazur, B. and Rubin, K. Disparity in Selmer ranks of quadratic twists of elliptic curves. Ann. of Math. (2) 178 (2013), no. 1, 287320. DOI 10.4007/annals.2013.178.1.5. MR3043582.
[McC86] McCallum, W. G. Duality theorems for Néron models. Duke Math. J. 53 (1986), no. 4, 10931124. DOI 10.1215/S0012-7094-86-05354-8. MR874683 (88c:14062).
[Mil80] Milne, J. S. Étale cohomology. Princeton Mathematical Series, vol. 33. Princeton University Press: Princeton, N.J., 1980. MR559531 (81j:14002).
[Mil06] Milne, J. S. Arithmetic Duality Theorems, 2nd. ed., (BookSurge, LLC, Charleston, SC, 2006), MR2261462 (2007e:14029).
[Mon96] Monsky, P. Generalizing the Birch-Stephens theorem. I. Modular curves. Math. Z. 221 (1996), no. 3, 415420. DOI 10.1007/PL00004518. MR1381589 (97a:11103).
[Mor15] Morgan, A. 2-Selmer parity for hyperelliptic curves in quadratic extensions. Preprint, Available at (2015).
[MR07] Mazur, B. and Rubin, K. Finding large Selmer rank via an arithmetic theory of local constants. Ann. of Math. (2) 166 (2007), no. 2, 579612. DOI 10.4007/annals.2007.166.579. MR2373150 (2009a:11127).
[Nek06] Nekovář, J. Selmer complexes. Astérisque, no. 310 (2006), viii+559 (English, with English and French summaries). MR2333680 (2009c:11176).
[Nek13] Nekovář, J. Some consequences of a formula of Mazur and Rubin for arithmetic local constants. Algebra Number Theory 7 (2013), no. 5, 11011120. DOI 10.2140/ant.2013.7.1101. MR3101073.
[Nek15] Nekovář, J. Compatibility of arithmetic and algebraic local constants (the case ℓ ≠ p). Compos. Math. 151 (2015), no. 9, 16261646. DOI 10.1112/S0010437X14008069. MR3406439.
[Nek16] Nekovář, J. Compatibility of arithmetic and algebraic local constants II. The tame abelian potentially Barsotti-Tate case. Preprint, (2016). Available at
[Oda69] Oda, T. The first de Rham cohomology group and Dieudonné modules. Ann. Sci. École Norm. Sup. (4) 2 (1969), 63135. MR0241435 (39 #2775).
[PR12] Poonen, B. and Rains, E. Random maximal isotropic subspaces and Selmer groups. J. Amer. Math. Soc. 25 (2012), no. 1, 245269. DOI 10.1090/S0894-0347-2011-00710-8. MR2833483.
[PS99] Poonen, B. and Stoll, M. The Cassels–Tate pairing on polarised abelian varieties. Ann. of Math. (2) 150 (1999), no. 3, 11091149. DOI 10.2307/121064. MR1740984 (2000m:11048).
[Ray65] Raynaud, M. Caractéristique d'Euler-Poincaré d'un faisceau et cohomologie des variétés abéliennes. Séminaire Bourbaki, Vol. 9 (Soc. Math. France, Paris, 1995), pp. Exp. No. 286, 129–147 (French). MR1608794.
[Sab07] Sabitova, M. Root numbers of abelian varieties. Trans. Amer. Math. Soc. 359 (2007), no. 9, 4259–4284 (electronic). DOI 10.1090/S0002-9947-07-04148-7. MR2309184 (2008c:11090).
[Ser67] Serre, J.-P.. Local class field theory. Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), (Thompson, Washington, D.C., 1967), pp. 128161, MR0220701 (36 #3753).
[Ser70] Serre, J.-P.. Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures), Séminaire Delange-Pisot-Poitou. 11e année: 1969/70. Théorie des nombres. Fasc. 1: Exposés 1 à 15; Fasc. 2: Exposés 16 à 24, Secrétariat Mathématique, 1970, pp. 19-01–19-15. MR0401396 (53 #5224).
[Ser77] Serre, J.-P.. Linear Representations of Finite Groups (Springer-Verlag: New York, 1977). Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42, MR0450380 (56 #8675).
[Sha64] Shatz, S. St.. Cohomology of artinian group schemes over local fields. Ann. of Math. (2) 79 (1964), 411449. MR0193093 (33 #1314).
[ST68] Serre, J.-P. and Tate, J.. Good reduction of abelian varieties. Ann. of Math. (2) 88 (1968), 492517. MR0236190 (38 #4488).
[TW11] Trihan, F. and Wuthrich, C. Parity conjectures for elliptic curves over global fields of positive characteristic. Compos. Math. 147 (2011), no. 4, 11051128. DOI 10.1112/S0010437X1100532X. MR2822863 (2012j:11134).
[TY14] Trihan, F. and Yasuda, S. The ℓ -parity conjecture for abelian varieties over function fields of characteristic p > 0. Compos. Math. 150 (2014), no. 4, 507522. DOI 10.1112/S0010437X13007501. MR3200666.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 17 *
Loading metrics...

Abstract views

Total abstract views: 95 *
Loading metrics...

* Views captured on Cambridge Core between 7th August 2017 - 23rd March 2018. This data will be updated every 24 hours.