By Shafarevich I.R.

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1. L^/L ^ K{Vj, hence N{aL)/aL ^ g{L) and g{L) is an algebraic Heisenberg group over C. PROOF: The homomorphism from N{aL)/aL to ^(L) already showed that L-^/L is contained in A'(L). If a G C^ is such that its image m X'j' j^ is in 38 TATA LECTURES ON THETA III /\ (L), then by applying an elementary lifting argument there is an automorphism {Ta, V') of the trivial line bundle on C^ (where Ta = translation by a) which normalizes the crL-action, and therefore centralizes it. Now putting ^ ( a , z) = {ah{z), z -\- a) where h is nowhere zero, the above condition can be read off as h(z — a)~^h(z — a — y) — exp 27ri*T/ia for all y G i^ and z £ C^.

Z) when the char- acteristic ofk is 0 and to ((AJ^))^^, (Z(^))2^) and {f\^f\i(P^) when the characteristic is p. 2. Let X(p^) = all points of X defined over k annihilated by some power of p. By definition V^(^) is the inverse limit of the chain of arrows: There is an isomorphism X{p^) = (Qp/Zp)^^. " (Qp/Zp)^^ — (Qp/Zp)^^ which by the commutative diagram above and the completeness of Q^^ in the p-adic topology is just Q^^. It is clear that Tp{X), given by the first member of the chain = 0, gets identified to Z^^.

1. L^/L ^ K{Vj, hence N{aL)/aL ^ g{L) and g{L) is an algebraic Heisenberg group over C. PROOF: The homomorphism from N{aL)/aL to ^(L) already showed that L-^/L is contained in A'(L). If a G C^ is such that its image m X'j' j^ is in 38 TATA LECTURES ON THETA III /\ (L), then by applying an elementary lifting argument there is an automorphism {Ta, V') of the trivial line bundle on C^ (where Ta = translation by a) which normalizes the crL-action, and therefore centralizes it. Now putting ^ ( a , z) = {ah{z), z -\- a) where h is nowhere zero, the above condition can be read off as h(z — a)~^h(z — a — y) — exp 27ri*T/ia for all y G i^ and z £ C^.