Tate module


Let AA be an abelian group, let pp be a prime number.

The pp-adic Tate Module T p(A)T_p(A) of AA is defined to limit

T p(A):=lim nA[p n]:=lim nker(Ap nA)T_p(A):=lim_n A[p^n]:=lim_n ker(A\stackrel{\cdot p^n }{\to}A)

of the directed diagram given by kernels of the endomorphism of AA defined by multiplication with p np^n and transition maps d i:=(p):A[p n+1]A[p n]d_i:=(\cdot p):A[p^{n+1}]\to A[p^n].

The Tate module can equivalently be described in terms of the endorphism ring of the Prüfer group.


  • Faltings, Gerd (1983), “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern”, Inventiones Mathematicae 73 (3): 349–366, doi:10.1007/BF01388432

  • Hazewinkel, Michiel, ed. (2001), “Tate module”, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104

  • Murty, V. Kumar (2000), Introduction to abelian varieties, CRM Monograph Series, 3, American Mathematical Society, ISBN 978-0-8218-1179-5

  • Section 13 of Rohrlich, David (1994), “Elliptic curves and the Weil–Deligne group”, in Kisilevsky, Hershey; Murty, M. Ram, Elliptic curves and related topics, CRM Proceedings and Lecture Notes, 4, American Mathematical Society, ISBN 978-0-8218-6994-9

  • Tate, John (1966), “Endomorphisms of abelian varieties over finite fields”, Inventiones Mathematicae 2: 134–144, MR 0206004John Tate, endomorphisms of abelian varieties over finite fields, web

  • Tate module. Encyclopedia of Mathematics. web

Revised on June 12, 2012 15:45:57 by Stephan Alexander Spahn (