J.-P. Serre and J. Tate. Good reduction of abelian varieties. Ann. of Math. (2), 88:492--517, 1968.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Corollary 0.4 If dim 2, Conjecture 0.3 is true. To state more results on Conjecture 0.3, we recall the monodromy filtration and the weight monodromy conjecture. Let t # : I K Z # (1) be the canonical map sending I K to (#(# ) n Z # (1) By the monodromy theorem of Grothendieck [30] Appendix, there exist a nilpotent endomorphism N (X K , Q # ) 1) and an open subgroup J I K such that, for # J , the action of # on H (X K , Q # ) is given by exp(t # (#)N ) Let M . be the increasing filtration on H (X K , Q # ) characterized by the conditions (1) M s = ....

J.-P. Serre, J. Tate, Good reduction of abelian varieties, Ann. Math. 88 (1968), 492-517.

....T (N) be the subring of End(Pic ) generated by T # p and # . It turns out that T (N) is identi ed with T 1 (N) de ned earlier by [ Q ) T (Pic X 1 (N) an ) 2.2. 4) Now this gives our V (N) with a canonical continuous Galois action By N eron Ogg Shafarevich ([SeT68] Theorem 1) we have that this action is unrami ed at all p j N . We summarize our progress in this lemma: Lemma 2.2.1. Let T 1 (N) act on V (N) through either the ( # action on ( action. Then N; GQ Aut(V (N) GL 2 (Q T 1 (N) is a continuous representation, unrami ....

Serre, Jean-Pierre; Tate, John; Good reduction of Abelian Varieties, Ann. of Math. (2), 88, Issue. 3, 492-517, 1968.

....A(L) H1(L u =K u ; A(L u ) inflation to H 1 (L u =K; A( L u) followed by restriction) Corollary 2.2. Suppose that A has good reduction over L and let i be the exponent of the inertia group of L=K. Then the group Phi(k) is killed by i. Using the methods of Serre and Tate [ST] we deduce the following corollary. If n is a non zero integer with prime factorization n = p e1 1 p e2 2 Delta Delta Delta p er r we set L(n) p 2 Gamma 1 if n = 2p 2 , p 2 odd P r i=1 p e i Gamma1 i (p i Gamma 1) otherwise Theorem 2.3. Suppose that A has potentially good ....

....in H 1 (K; AK ) which are unramified and die on restriction to L. Proof of Corollary 2.2. By replacing K and L with suitable unramified extensions, we may assume that Phi(k) Phi(k) and that L=K is totally ramified. Hence i kills H 1 (L=K; A(L) Proof of Theorem 2.3. It follows from [ST] that the inertia group of L=K must be a finite subgroup of Gl(2; Z ) for all primes 6= p. Further, the prime to p part of the inertia group is cyclic, since it is the tame inertia group. Thus the theorem follows from the following lemma. Lemma 3.1. There is an element of order m in Gl(n; Z ....

J.-P. Serre and J. T. Tate, Good reduction of abelian varieties, Annals of Math. 68 (1968), 492-517.

.... . Recall that for primes p not dividing N , the Jacobian J has good reduction mod p, and the Eichler Shimura relation, theorem 1. 29, states that on J = p , we have T p = F hpiF 0 : For primes p not dividing N , we may identify T (J ) with the adic Tate module of the reduction (see [ST]) and consider the Frobenius endomorphism F on the free rank two module V. As a consequence of the Eichler Shimura relation, we nd: Theorem 1.41 For p not dividing N , the characteristic polynomial of F on the module V is X 2 T p X hpip: Proof: We are grateful to Brian Conrad for ....

....a p = p 1 N p as in section 1.1) If pjNE , then L p (Symm 2 E; s) 1 p s ) 1 . Let E = Z E( Neron E Neron E where Neron E is the N eron di erential de ned in section 1.1. Since E is modular by theorem 3. 48, a method of Shimura (see [Shi4] and the introduction of [St]) establishes the analytic continuation of L(Symm 2 E; s) to an entire function and shows that L(Symm 2 E; 2) is a non zero rational multiple of i E . We now explain how to deduce the following theorem from Wiles results and a formula of Hida, corollary 4.21. Theorem 3.53 Suppose that E is ....

J.-P. Serre, J.T. Tate, Good reduction of abelian varieties, Annals of Math. 88 (1968), 492-517.

....we de ne the global L function associated to T and the set by L (T; s) Y q = 2 L q (T; s) This function extends to an entire function on the complex plane, and satis es a functional equation interchanging s and 3 s. By using Rankin s method and some techniques of Shimura, Sturm [St] has shown that L (T; 2) is a rational multiple of a transcendental period = Z X(C) where is a N eron di erential attached to X. If x 2 Q, let [x] 5 = 5 ord 5 (x) denote the 5 part of x. Hida, using the fundamental work of Ribet and others, has succeeded in relating the special ....

Serre, J.-P., and Tate, J., Good reduction of abelian varieties, Ann. of Math. 88, 492-517 (1968) 62

....# 2, Conjecture 0.3 is true. To state more results on Conjecture 0.3, we recall the monodromy filtration and the weight monodromy conjecture. Let t # : I K # Z # (1) be the canonical map sending # # I K to (#(# 1 # n ) # 1 # n ) n # Z # (1) By the monodromy theorem of Grothendieck [18] Appendix, there exist a nilpotent endomorphism N # End(H r (X K , Q # ) 1) and an open subgroup J # I K such that, for # # J , the action of # on H r (X K , Q # ) is given by exp(t # (#)N ) Let M . be the increasing filtration on H r (X K , Q # ) characterized by the ....

J.P.Serre and J.Tate, Good reduction of abelian varieties, Ann. Math. 88 (1968), 492-517.

....C = a 2 d Gamma 4bd c 2 : Making the substitution Z= 3Y b, and substituting from (1.2) the result follows. This result was probably known to Cayley, but we were unable to find any reference. The next result is a weak form of the criterion of Neron Ogg Shafarevich (see Serre Tate [10]) but follows directly from Proposition 1.3. Proposition 1.4. If an elliptic curve E is defined over a number field K, then the extension of K by the X coordinates of the points of order 3 on E is ramified only at primes dividing 3 and Delta. 2. Extension by a subgroup of order 3. In this ....

J.-P. Serre and J. Tate, Good reduction of Abelian varieties, Ann. Maths. 88 (1968), 492-527.

....say that an abelian variety h as stable reduction at v, if its N eron minimal model X at v has a special fibre whose connected component X 0 v is an extension of an abelian variety by an algebraic torus. As Grothendieck showed: III) eigenvalues of algebraic monodromy are roots of unity (see [30], appendix, page 515; see [22] 1.3) see [23] 7.4) Conclusion: there exists a finite extension K ae L, and a discrete valuation w of L over v such that: the eigenvalues of ae X; are roots of unity = the eigenvalues of ae XL ; are equal to 1 = XL has stable reduction at w = take X = ....

J-P. Serre & J. Tate, Good reduction of abelian varieties. Ann. Math. 88 (1968), 492-517 [Serre OEII, 79].

....of E. We will see that not only are torsion points abelian over F , in fact they are almost abelian over K, so that (using class eld theory) we can describe the action of GF on torsion points in terms of an action of the ideles of K. The reference for this section is [Sh] Chapter 5; see also [ST]. We continue to suppose that E has complex multiplication by the maximal order of K. 17 De nition 5.10. Let A K denote the group of ideles of K. There is a natural map from A K to the group of fractional ideals of K, and if x 2 A K and a is a fractional ideal of K we will write xa ....

Serre, J-P., Tate, J., Good reduction of abelian varieties, Ann. of Math. 88 (1968) 492-517.

....there is a constant c such that the ramification degree e v (K(A[p] K) c, for infinitly many primes p. Proof. This is well known, but for convenience of the reader we present the proof. i) ii) This follows from the criterion of N eron Ogg Shafarevich for good reduction; cf. Serre Tate[ST], Theorem 2. ii) iii) Trivial. iii) i) By replacing K by a suitable finite extension K 0 , we may assume without loss of generality that A has semi stable reduction at v. For example, we could take K 0 = K(A[p] for any prime p 3, p 6= char( v) cf. Grothendieck[Gro] Prop. 4.7) ....

....field. If L denotes the compositum of the K(A[p] s for the primes in question, then the hypothesis (and tame ramification) implies that L is a finite (cyclic) extension of K. By construction, L(A[p] L for infinitely many p s, so by the N eron Ogg Shafarevich criterion for good reduction (cf. [ST], Theorem 1) A has good reduction over L and hence potentially good reduction over K. Let us now come back to Theorem 5.7. By imposing further restrictions on the m i s, we can conclude that all of J C has potentially good reduction: Corollary 5.9 Suppose that C=K is as above and satifies in ....

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J.-P. Serre and J. Tate, Good reduction of abelian varieties. Ann. of Math. 88 (1968), 492--517.

....by A the extension of scalars of A into K, a nonramified closure of the completion of A with respect to the valuation. But then I is isomorphic to the whole Galois group Gal( L= K) and hence jIj = L : K] kills the group OE A . 2 The following proposition was originally proved in [14] in the case that the residue field is finite; a proof in the general case can be find in [13] see also [17] Proposition 13 (Serre Tate, Oort) Let A be an abelian variety over a discrete valuation field K with potentially good reduction. Then, if 2g 1 is prime different from p, A ....

J-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. 88 (1968) 492--517.

....Thus Proposition 4.4 (b) implies that Y s is smooth, and the corollary is proved when g(Y ) 2. Let us now present a different argument that also applies to the case where g(Y ) 1. Let J(X) resp. J(Y ) be the Jacobian of X (resp. of Y ) Then J(X) has good reduction, and so does J(Y ) [S T], x1, Cor. 2) Since X is smooth, it has a section over some etale extension of OK ( EGA] IV.17.16.3 (ii) Let Y be the minimal regular model of Y over OK . Then by composition Y also has an etale quasi section. So the identity component of the N eron model of J(Y ) over OK is isomorphic to ....

J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. Math. 88 (1968), 492-517.

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J.-P. Serre and J. Tate. Good reduction of abelian varieties. Ann. of Math. (2), 88:492--517, 1968.

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J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. 88 (1968), 492-517.

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J.P.Serre and J.Tate, Good reduction of abelian varieties, Ann. Math. 88 (1968), 492--517.

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