An algebraic construction of an abelian variety with a given Weil number
|
|
- Blaze Fitzgerald
- 5 years ago
- Views:
Transcription
1 Submitted to Journal of the Mathematical Society of Japan An algebraic construction of an abelian variety with a given Weil number By Ching-Li Chai 1 & Frans Oort 2 Abstract. A classical theorem of Honda and Tate asserts that for every Weil q-number π, there exists an abelian variety over the finite field F q, unique up to F q-isogeny, whose q-frobenius is π. The standard proof (of the existence part in the Honda-Tate theory) uses the the fact that for a given CM field L and a given CM type Φ for L, there exists a CM abelian variety with CM type (L, Φ) over a field of characteristic 0. The usual proof of the last statement uses complex uniformization of (the set of C-points of) abelian varieties over C. In this short note we provide an algebraic proof of the existence of a CM abelian variety over an integral domain of characteristic 0 with a given CM type, resulting in an algebraic proof of the existence part of the Honda-Tate theorem which does not use complex uniformization. Dedicated to the memory of Taira Honda. Introduction. In this note we provide a new proof for a theorem already known for 45 years. We fix a prime number p and we consider q = p n with n Z >0. We say that π is a Weil q-number if π is an algebraic integer such that for every complex embedding ψ : Q(π) C we have ψ(π) = q. A result by A. Weil (which was the starting point of new developments in arithmetic algebraic geometry) states that for any abelian variety A over the finite field F q its associated q-frobenius morphism π A = Fr A,q : A A (q) = A 2010 Mathematics Subject Classification. Primary 11G10; Secondary 11G15, 14G15. Key Words and Phrases. abelian variety, complex multiplication, CM field, Weil number, CM lifting. 1 Partially supported by NSF grant DMS The Institute of Mathematics of Academia Sinica is gratefully acknowledged for excellent working conditions and for hospitality during a visit of the second-named author in November / December 2012.
2 2 Ching-Li Chai & Frans Oort is a Weil q-number, in the sense that π A is a root of a monic irreducible polynomial in Z[T ] all of whose roots are Weil q-numbers. See [20, p. 70], [19, p. 138] and [10, Th. 4, p. 206]. This enabled T. Honda and J. Tate to state their theorem: the map A π A defines a bijection {simple abelian variety over F q }/ {Weil q number}/ from the set of isogeny classes of simple abelian varieties over F q to the set of Weil q-numbers up to conjugacy. Here we say that π and π are conjugate if there exists an isomorphism Q(π ) = Q(π ) of fields which carries π into π. This map is well-defined because of the above theorem of Weil, and because isogenous abelian varieties have conjugate Frobenius endomorphisms. The injectivity was proved by Tate, and the surjectivity was proved by Honda and Tate. The surjectivity means: Theorem I. For any Weil q-number π there exists a simple abelian variety A over F q (unique up to F q -isogeny) such that π is conjugate to π A. See [6], Main Theorem; [18, Th. 1]. In the terminology of [18] we say that a Weil q-number is effective if it is conjugate to the q-frobenius of an abelian variety over F q. Theorem I asserts that every Weil number is effective. In the course of the proof of Theorem I we will show, in Theorem II in Step 5, that every CM type for a CM field L is realized by an abelian variety in characteristic zero of dimension [L : Q]/2 with complex multiplication by L. Proofs of these theorems were given by constructing a CM abelian variety over C (using complex uniformization and GAGA) with properties which ensure that the reduction modulo p of this CM abelian variety gives a Weil number which is a power of π A. We give a proof of both theorems without using complex uniformization. The remark in Step 8 gives this proof in the special case when g = 1; that proof is a guideline for the proof below for arbitrary g. In a sense, this answers a question posed in [14, 22,4]. (1) Step 1. A Weil q-number π has exactly one of the following three properties: (Q) It can happen that ψ(π) Q. In this case q = p n = p 2m and π = ± q = ±p m.
3 Abelian variety with a given Weil number 3 (R) It can happen that ψ(π) Q and ψ(π) R. In this case q = p n = p 2m+1 and π = ± q = ±p m p. In this case every embedding of Q(π) into C lands into R. ( R) If there is one embedding ψ : Q(π) C such that ψ (π) R then for every embedding ψ : Q(π) C we have ψ(π) R and in this case Q(π) is a CM field. The fact that L is a CM field means there is a subfield L 0 L such that L 0 is totally real (every embedding of L 0 into C lands into R) and [L : L 0 ] = 2 and L is totally complex (no embedding of L into C lands into R). We know from [18], page 97 Example (a) that every real Weil q-number comes from an abelian variety over F q, so the first two cases have been taken care of. In order to prove Theorem I we assume from now on that we are in the third case, i.e. ψ(π) R for all ψ : Q(π) C. We show in that case the properties of Q(π) mentioned above indeed hold. As ψ (π) = ψ (π) ψ (π) = q for some embedding ψ : Q(π) C we see that ψ (π) = q ( ψ (π) = q ) ψ. π As q/π π (because ψ (π) R) we see that for every embedding ψ the complex conjugate of ψ(π) equals ψ(q/π) which is different from ψ(π); hence π is totally complex. Moreover we see that β := π + q π is totally real and 2 q < ψ(β) < 2 q. for all ψ : Q(π) C. We note that π is a zero of the quadratic polynomial T 2 β T + q, and β 2 4q is totally negative. We have shown that Q(π) is a CM field. In [18], Th. 1 on page 96, we see a description of a finite dimensional central division algebra M over Q(π) determined by (p-adic) properties of π (denoted by E in [18]); these properties will be used below. We write 2g = [Q(π) : Q] [M : Q(π)]. If ever we would find an abelian variety A over F q with π A π then we would have
4 4 Ching-Li Chai & Frans Oort End 0 (A) = M and dim(a) = g. By [18], 3, Lemme 2 on page 100 we see that there exists a CM field L with Q(π) L M and [L : Q] = 2g. We fix notation q, and π a totally complex Weil q-number, and a finite dimensional central division algebra M over Q(π) determined by π and Q(π) L M with [L : Q] = 2g as above; Q(π) L M. We write L 0 for the maximal totally real subfield of L. (2) Step 2. Choosing a CM type for L. We follow [18], pp However our notation will be slightly different. A prime above p in Q(π) will be denoted by u. A prime in L 0 above p will be denoted by w and a prime in L above p will be denoted by v. We write ρ for the involution of the quadratic extension L/L 0 (which is the complex conjugation). Following Tate we write H v = Hom(L v, C p ), Hom(L, C p ) = v p H v, where C p is the p-adic completion of an algebraic closure of Q p. Let n v := v(π) v(q)) #(H v) N for each place v of L above p. Using properties of π we choose a suitable p-adic CM type for L, by choosing a subset v w Φ v v w H v for each place w of L 0 above p, as follows. [v = ρ(v)] For any v with v = ρ(v) the map ρ gives a fixed point free involution on H v ; in this case (once π and L are fixed and v is chosen) we choose a subset Φ v H v with #(Φ v ) = (1/2) #(H v ). Note that v(π) = (1/2)v(q) in this case and we have n v = (1/2) #(H v ) = (v(π)/v(q)) #(H v ). [v ρ(v)] For any pair v 1, v 2 above a place w of L 0 dividing p with v 1
5 Abelian variety with a given Weil number 5 ρ(v 1 ) = v 2, the complex conjugation ρ defines a bijective map? ρ : H v1 H v2. We choose a subset Φ v1 H v1 with #(Φ v1 ) = n v1 and we define Φ v2 := H v2 Φ v1 ρ. Observe that indeed n vi + n ρ(vi) = [L v : Q p ] = #(H vi ) for i = 1, 2. We could as well have chosen first Φ v2 of the right size and then define Φ v1 as Φ v1 := H v1 Φ v2 ρ. Define a p-adic CM type Φ p Hom(L, C p ) = v p H v by Φ p := v p Φ v. By construction we have Φ p (Φ p ρ) =, Φ p (Φ p ρ) = Hom(L, C p ); i.e. Φ p is a p-adic CM type for the CM field L. Let j p : Q C p be the algebraic closure of Q in C p. The injection j p induces a bijection j p?: Hom(L, Q) Hom(L, C p ). The subset Φ := (j p?) 1 (Φ p ) Hom(L, Q) is a CM type in the usual sense, i.e. Φ (Φ ρ) = and Φ (Φ ρ) = Hom(L, Q). We fix the notation Φ p Hom(L, C p ) for the p-adic CM type constructed above, and the corresponding CM type Φ Hom(L, Q). (3) Step 3. Choosing a prime number r. Proposition A. For a given CM field L there exists a rational prime number r unramified in L such that r splits completely in L 0 and every place of L 0 above r is inert in L/L 0. Proof. Let N be the smallest Galois extension of Q containing L, and let G = Gal(N/Q). Note that the element ρ G induced by complex conjugation is a central element of order 2. By Chebotarev s theorem the set of rational primes unramified in N whose Frobenius conjugacy class in G is ρ has Dirichlet density 1/[G : 1] > 0; see [8, VIII.4, Th. 10]. Any prime number r in this subset satisfies the required properties. (4) Step 4. An abelian variety with an action by L.
6 6 Ching-Li Chai & Frans Oort We know that for every prime number (r in our case) there exists a supersingular elliptic curve E in characteristic r; e.g. for r > 2 we can consider the Legendre family Y 2 = X(X 1)(X λ) and we know from [4, 4.4.2] that there do exist values of λ in F r for which the corresponding elliptic curves over F r are supersingular; in characteristic 2 we can take the elliptic curve given by Y 2 + Y = X 3. Let E be a supersingular elliptic curve over the base field κ := F r ; we know that End(E) is non-commutative. Its endomorphism algebra End 0 (E) is the quaternion division algebra Q r, over Q in the notation of [2], which is ramified exactly at r and. Let B 1 := E g and let D := End 0 (B 1 ) = M g (Q r, ). Proposition B. Let L be a totally imaginary quadratic extension of a totally real number field L 0 such [L v : Q r ] is even for every place v of L above r. Let g = [L 0 : Q]. There exists a positive involution τ on the central simple algebra End Q (L 0) Q Q r, = Mg (Q r, ) over Q and a ring homomorphism ι: E End Q (L 0) Q Q r, such that ι(l ) is stable under the involution τ and τ induces the complex conjugation on L. Proof. Let End Q (L 0) = M g (Q) be the algebra of all endomorphisms of the Q- vector space underlying L 0. The trace form (x, y) Tr L 0 /Q(x y) for x, y L 0 is a positive definite quadratic form on (the Q-vector space underlying) L 0, so its associated involution τ 1 on End Q (L 0) is positive. Multiplication defines a natural embedding L 0 End Q (L 0), and every element of L 0 is fixed by τ 1. Let τ 2 be the canonical involution on Q r,. The involution τ 1 τ 2 on End Q (L 0) Q Q r, is clearly positive because τ 2 is. It is also clear that the subalgebra B := L 0 Q Q r, of End Q (L 0) Q Q r, is stable under τ. Moreover B is a positive definite quaternion division algebra over L 0, so the restriction to B of the positive involution τ is the canonical involution on B. The assumptions on L imply that there exists an L 0-linear embedding L B. From the elementary fact that every R-linear embedding of C in the Hamiltonian quaternions H is stable under the canonical involution on H, we deduce that the subalgebra L Q R B Q R is stable under the canonical involution of B Q R, which implies that L is stable under τ. Corollary C. (i) There exists a polarization µ 1 : B 1 B t 1 and an embedding L End 0 (B 1 ) = D such that the image of L in D = End 0 (B 1 ) is stable under the Rosati involution attached to µ 1. (ii) There exists an isogeny α : B 1 B 0 over F r such that the embedding L End 0 (B 1 ) = End 0 (B 0 ) factors through an action ι 0 : O L End(B 0 )
7 Abelian variety with a given Weil number 7 of O L on B 0, where O L is the ring of all algebraic integers in L. (iii) There exists a positive integer m such that the isogeny µ 0 := m (α t ) 1 µ 1 α 1 : B 0 B t 0 is a polarization on B 0 and the Rosati involution τ µ0 complex conjugation on the image of L in End 0 (B 0 ). attached to µ 0 induces the Proof. The statement (i) follows from Proposition B in view of the general structure of the Néron-Severi group NS(B) of B and the ample cone in NS(B) explained in [10, 21 pp ]. The statements (ii) and (iii) follow from (i). From now on we fix (L, Φ) as in Step 1, with r as in Proposition A, and (B 0, ι 0 : O L End(B 0 ), µ 0 : B 0 B t 0) as in Corollary C. We fix an algebraic closure Q r of Q r, an embedding j r : Q Q r, and an embedding i r,ur : W (F r )[1/p] Q r. We have bijections j p? Hom(L, C p ) Hom(L, Q) j r? i r? Hom(L, Q r ) Hom(L, W (F r )[1/r]) The last arrow Hom(L, Q r ) i r? Hom(L, W (F r )[1/r]) is a bijection because r is unramified in L. We regard the p-adic CM type Φ p as an r-adic CM type Φ r Hom(L, W (F r )[1/r]) via the bijection (j r?) (j p?) 1, i.e. Φ r := (j r?) (j p?) 1 (Φ p ) = (j r?)(φ). For each place w of L 0 above r, the w-adic completion L w := L L0 L 0,w of L is an unramified quadratic extension field of the w-adic completion L 0,w = Qr of L 0, and the intersection Φ w := Φ r Hom(L w, W (F r )[1/r]) is a singleton. (5) Step 5. Lifting to a CM abelian variety in characteristic zero. Theorem II. Let (B 0, ι 0 : O L End(B), µ 0 : B 0 B0) t be an ([L : Q]/2)- dimensional polarized supersingular abelian variety with an action by O L such that the subring O L End 0 (B 0 ) is stable under the Rosati involution τ µ0 as in Corollary C. There exists a lifting (B, ι, µ) of the triple (B, ι 0, µ 0 ) to the ring W (F r ) of
8 8 Ching-Li Chai & Frans Oort r-adic Witt vectors with entries in F r such that the generic fiber B η is an abelian variety whose r-adic CM type is equal to Φ r. Proof. The prime number r was chosen so that for every place w of the totally real subfield L 0 L, the ring of local integers O L0,w of the w-adic completion of L 0 is Z p, and O L,w := O L OL0 O L0,w = W (F r 2). We have a product decomposition O L Z Z p = O L OL0 O L0,w = O L,w, w where w runs over the g places of L 0 above r. The g idempotents associated to the above decomposition of O L Z Z p define a decomposition w B 0 [r ] = w B 0 [w ] of the r-divisible group B 0 [r ] into a product of g factors, where each factor B 0 [w r ] is a height 2 r-divisible group with an action by O w. Similarly we have a decomposition B t 0[r ] = w B t 0[w ] of the r-divisible group attached to the dual B t 0 of B 0. The action of O L on B 0 induces an action of O L on B t 0 by y (ι 0 (ρ(y))) t for every y O L, so that the polarization µ 0 : B 0 B t 0 is O L -linear. The quasi-polarization µ 0 [r ] : B 0 [r ] B t 0[r ] decomposes into a product of quasi-polarizations µ 0 [w ] : B 0 [w ] B t 0[r ] on the O L,w -linear r-divisible groups B 0 [w ] of height 2. It suffices to show that for each place w of L 0 above r, the O L,w -linearly polarized r-divisible group (B 0 [w ], ι 0 [w ], µ 0 [w ]) over F r can be lifted to W (F r ) with r-adic CM type Φ w. For then the Serre-Tate theorem for deformation of abelian schemes tells us that (B 0, ι 0, µ 0 ) can be lifted over W (F r ) to a formal abelian scheme B with an action by O L and an O L -linear polarization whose r-adic CM type is Φ r ; see [7], [9, Ch. V, Th. 2.3] on page 166. Grothendieck s algebraization theorem implies that B comes from a unique abelian scheme B over W (F r ), see [3, III 1.5.4]. For any r-adic place w among the g places of L 0 above r, the existence of a CM lifting to W (F r ) of the O L,w -linear polarized r-divisible group (B 0 [w ], ι 0 [w ], µ 0 [w ]) of height 2 goes back to Deuring who proved that a supersingular elliptic curve with a given endomorphism can be lifted to characteristic zero, see [2, p. 259], and the proof on pp ; the case we need here is
9 Abelian variety with a given Weil number 9 [12, 14.7]. Below is a proof using Lubin-Tate formal groups. By [11, Th. 1], there exists a one-dimensional formal group X of height 2, over W (F r ) plus an action β : O L,w End(X) of O L,w on X whose r-adic CM type is Φ w. Let (X 0, β 0 : O L,w End(X 0 )) := (X, β) Spec(W (Fr)) Spec(F r) be the closed fiber of (X, β). It is well-known that (X 0, β 0 ) is isomorphic to (B 0 [w ], ι 0 [w ]); we choose and fix such an isomorphism and identify (B 0 [w ], ι 0 [w ]) with (X 0, β 0 ). The Serre dual X t of X, with the O L,w -action defined by γ : b (β(ρ(b))) t b O L,w, also has CM type Φ w. Let (X0, t γ 0 ) be the closed fiber of (X t, γ). The natural map ξ : Hom ( (X, β), (X t, γ) ) Hom ( (X 0, β 0 ), (X t 0, γ 0 ) ) defined by reduction modulo r is a bijection: [11, Th. 1] implies that (X t, γ) is isomorphic to (X, β), and after identifying them via a chosen isomorphism both the source and the target of ξ are isomorphic to O L,w so that ξ is an O L,w -linear isomorphism. Under our identification of (X 0, β 0 ) with (B 0 [w ], ι 0 [w ]) above, the quasipolarization µ 0 [w ] on B 0 [w ] is identified with a quasi-polarization ν 0 on X 0. The quasi-polarization ν 0 : X 0 X0 t extends over W (κ L,w ) to a quasipolarization ν : X X t because ξ is a bijection. We have shown that the triple (B 0 [w ], ι 0 [w ], µ 0 [w ]) can be lifted over W (F r ). Remark. One can also prove the existence of a lifting of (B 0 [w ], ι 0 [w ], µ 0 [w ]) to W (F r ) using the Grothendieck-Messing deformation theory for abelian schemes. The point is that the deformation functor for (B 0 [w ], ι 0 [w ]) is represented by Spf(W (F r )) because O L,w is unramified over Z p. We fix the generic fiber (B η, µ, ι) of a lifting as in Theorem II over the fraction field W (F r )[1/r] of W (F r ) with an O L -linear action ι : O L End(B η ), whose r-adic CM type is Φ r (6) Step 6. We change to a number field and we reduce modulo p. We have arrived at a situation where we have an abelian variety B η over a field of characteristic zero with an action O L End(B η ) by O L, whose r-adic CM type with respect to an embedding of the base field in Q r is equal to the r-adic CM type Φ r constructed at the end of Step 4. We know that any CM abelian variety in characteristic 0 can be defined over a number field K, see e.g. [16, Prop. 26, p. 109] or [1, Prop ]. By [15, Th. 6]
10 10 Ching-Li Chai & Frans Oort we may assume, after passing to a suitable finite extension of K, that this CM abelian variety has good reduction at every place of K above p. Again we may pass to a finite extension of K, if necessary, to ensure that K has a place with residue field δ of characteristic p with F q δ. We have arrive at the following situation. We have a CM abelian variety (C, L End 0 (C)) of dimension g = [L : Q]/2 over a number field K, of p-adic CM type Φ p with respect to an embedding K C p such that C has good reduction C 0 at a p-adic place of K induced by the embedding K C p and the residue class field of that place contains F q (7) Step 7. Some power of π is effective. Let i Z >0 such that δ = F q i. We have C 0 over δ and π i, π C0 L. We know that π i and π C0 are units at all places of L not dividing p. We know that these two algebraic numbers have the same absolute value under every embedding into C. By the construction of Φ in Step 2 and by [18], Lemme 5 on page 103, we know that π i and π C0 have the same valuation at every place above p. As remarked in [18, pp. 103/104], the essence of this step is the factorization of a Frobenius endomorphism into a product of prime ideals in [16]. This shows that π i /π C0 is a unit locally everywhere and has absolute value equal to one at all infinite places. This implies, by standard finiteness properties for algebraic number fields, that π i /π C0 is a root of unity in O L. See for instance [5, 34 Hilfsatz a)]; the main point is that the set {y O L ψ(y) 1 complex embedding ψ : L C} is finite because the image of O L in ν L ν is a lattice, where the index ν runs through all archimedean places of L, hence π i /π C0 = y O L is a root of unity. We conclude that there exists a positive integer j Z >0 such that π ij = (π C0 ) j. (8) Step 8. End of the proof. The previous step shows that π ij is effective, because it is (conjugate to) the q ij - Frobenius of the base change of C 0 to F q ij. By [18, Lemma 1, p. 100] this implies that π is effective, and this ends the proof of the theorem in the introduction.
11 Abelian variety with a given Weil number 11 Remark. When g = 1 the proof of Theorem I is easier. This simple proof, sketched below, was the starting point of this note. Suppose that π is a Weil q-number and L = Q(π) is an imaginary quadratic field such that the positive integer g defined by p-adic properties of π is equal to 1. This means (the first case) either that there is an i Z >0 with π i Q, or (the second case) that for every i we have L = Q(π i ), with p split in L/Q and at one place v above p in L we have v(π)/v(q) = 1 while at the other place v above p we have v (π)/v (q) = 0. If π i Q we know that π is the q-frobenius of a supersingular elliptic curve over F q, see Step 1, and π is effective. If the second case occurs, we choose a prime number r which is inert in L/Q, then choose a supersingular elliptic curve in characteristic r, lift it to characteristic zero together with an action of (an order in) L; the reduction modulo p (over some extension of F p ) gives an elliptic curves whose Frobenius is a power of π; by [18, Lemme 1] on page 100 we conclude π is effective. This scheme of the proof of the general case is the same as the simple proof described in the previous paragraph when g = 1, except that (as we do in steps 2, 4 and 5) we specify the CM type in order to keep control of the p-adic properties of the Weil number eventually constructed. Note that the CM lifting problem considered in Step 5 reduces to the g = 1 case by the Serre-Tate theorem and by the Grothendieck algebraization result (as we explained in the proof of Theorem II). References [1] C.-L. Chai, B. Conrad & F. Oort CM liftings. [To appear] Available from [2] M. Deuring Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Hamburg, 14 (1941), [3] A. Grothendieck & J. Dieudonné Élements de géométrie algébriue, III1. Étude cohomologique des faisceaux cohérents. Publ. Math. IHES 11, [4] R. Hartshorne Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer- Verlag, New York-Heidelberg, [5] E. Hecke Vorlesungen über die Theorie der algebraischen Zahlen. Akademische Verlagsgesellschaft, Leipzig, [6] T. Honda Isogeny classes of abelian varieties over finite fields. J. Math. Soc. Japan, 20 (1968), [7] N. Katz Appendix to Exposé V: Cristaux ordinaires et coordonnées canoniques. Algebraic surfaces (Orsay, ), Lecture Notes in Mathematics, vol. 868, Springer, Berlin, 1981, Appendix to an article of P. Deligne, pp [8] S. Lang Algebraic number theory. Addison-Wesley Publishing Co., Inc., Reading, Mass.- London-Don Mills, Ont [9] W. Messing The crystals associated to Barsotti-Tate groups: with applications to abelian schemes. Lecture Notes in Mathematics, vol. 264, Springer-Verlag, Berlin, [10] D. Mumford Abelian varieties. Tata Institute of Fundamental Research Studies in Math. 5, Oxford University Press, London, 1970.
12 12 Ching-Li Chai & Frans Oort [11] J. Lubin & J. Tate Formal complex multiplication Ann. of Math. 81 (1965), [12] F. Oort Lifting algebraic curves, abelian varieties, and their endomorphisms to characteristic zero. Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., 46, Part 2, pp , Amer. Math. Soc., Providence, RI, [13] F. Oort Moduli of abelian varieties in mixed and in positive characteristic. The Handbook of Moduli (G. Farkas, I, Morrison, editors), Vol. III, pp [To appear in 2012.] [14] F. Oort Abelian varieties over finite fields. Summer School on Varieties over finite fields, Göttingen, 25-VI 6-VII Higher-dimensional geometry over finite fields. Proceedings of the NATO Advanced Study Institute 2007 (Editors: Dmitry Kaledin, Yuri Tschinkel). IOS Press, 2008, [15] J-P. Serre & J. Tate Good reduction of abelian varieties. Ann. of Math. 88 (1968) [16] G. Shimura & Y. Taniyama Complex multiplication of abelian varieties and its applications to number theory. Publications of the Mathematical Society of Japan, Vol. 6. The Mathematical Society of Japan, Tokyo [17] J. Tate Endomorphisms of abelian varieties over finite fields. Invent. Math. 2 (1966), [18] J. Tate Class d isogenie des variétés abéliennes sur un corps fini (d après T. Honda). Séminaire Bourbaki, 1968/69, no Lecture Notes Math. 179, Springer-Verlag, 1971, [19] W. Waterhouse & J. Milne Abelian varieties over finite fields Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), [20] A. Weil Sur les courbes algébriques et les variétés qui s en déduisent. Act. Sci. Ind., no. 1041, Publ. Inst. Math. Univ. Strasbourg 7 (1945). Hermann et Cie., Paris, 1948.
13 Abelian variety with a given Weil number 13 Ching-Li Chai Institute of Mathematics, Academia Sinica 6F Astronomy-Mathematics Building No. 1, Sec. 4, Roosevelt Rd. Taipei 10617, Taiwan (R.O.C.) chai@ math.sinica.edu.tw and Department of Mathematics University of Pennsylvania 209 S. 33rd St Philadelphia, PA 19104, U.S.A. chai@ math.upenn.edu Frans Oort Mathematisch Instituut Utrecht University P.O. Box NL TA Utrecht The Netherlands f.oort@ uu.nl
An algebraic construction of an abelian variety with a given Weil number
An algebraic construction of an abelian variety with a given Weil number Ching-Li Chai a & Frans Oort b version 10, April 20, 2015 Abstract A classical theorem of Honda and Tate asserts that for every
More informationHONDA-TATE THEOREM FOR ELLIPTIC CURVES
HONDA-TATE THEOREM FOR ELLIPTIC CURVES MIHRAN PAPIKIAN 1. Introduction These are the notes from a reading seminar for graduate students that I organised at Penn State during the 2011-12 academic year.
More informationFORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS
Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ
More informationSurjectivity in Honda-Tate
Surjectivity in Honda-Tate Brian Lawrence May 5, 2014 1 Introduction Let F q be a finite field with q = p a elements, p prime. Given any simple Abelian variety A over F q, we have seen that the characteristic
More informationDieudonné Modules and p-divisible Groups
Dieudonné Modules and p-divisible Groups Brian Lawrence September 26, 2014 The notion of l-adic Tate modules, for primes l away from the characteristic of the ground field, is incredibly useful. The analogous
More informationψ l : T l (A) T l (B) denotes the corresponding morphism of Tate modules 1
1. An isogeny class of supersingular elliptic curves Let p be a prime number, and k a finite field with p 2 elements. The Honda Tate theory of abelian varieties over finite fields guarantees the existence
More informationLocal root numbers of elliptic curves over dyadic fields
Local root numbers of elliptic curves over dyadic fields Naoki Imai Abstract We consider an elliptic curve over a dyadic field with additive, potentially good reduction. We study the finite Galois extension
More informationComplex multiplication and lifting problems. Ching-Li Chai Brian Conrad Frans Oort
Complex multiplication and lifting problems Ching-Li Chai Brian Conrad Frans Oort Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104 E-mail address: chai@math.upenn.edu Institute
More informationTHE THEOREM OF HONDA AND TATE
THE THEOREM OF HONDA AND TATE KIRSTEN EISENTRÄGER 1. Motivation Consider the following theorem of Tate (proved in [Mum70, Theorems 2 3, Appendix I]): Theorem 1.1. Let A and B be abelian varieties over
More informationAn Introduction to Supersingular Elliptic Curves and Supersingular Primes
An Introduction to Supersingular Elliptic Curves and Supersingular Primes Anh Huynh Abstract In this article, we introduce supersingular elliptic curves over a finite field and relevant concepts, such
More informationChing-Li Chai 1 & Frans Oort
1. Introduction CM-lifting of p-divisible groups Ching-Li Chai 1 & Frans Oort Let p be a prime number fixed throughout this article. An abelian variety A over the algebraic closure F of F p is said to
More informationNotes on p-divisible Groups
Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More informationThe Splitting of Primes in Division Fields of Elliptic Curves
The Splitting of Primes in Division Fields of Elliptic Curves W.Duke and Á. Tóth Introduction Dedicated to the memory of Petr Cižek Given a Galois extension L/K of number fields with Galois group G, a
More informationKleine AG: Travaux de Shimura
Kleine AG: Travaux de Shimura Sommer 2018 Programmvorschlag: Felix Gora, Andreas Mihatsch Synopsis This Kleine AG grew from the wish to understand some aspects of Deligne s axiomatic definition of Shimura
More informationMASS FORMULA FOR SUPERSINGULAR ABELIAN SURFACES
MASS FORMULA FOR SUPERSINGULAR ABELIAN SURFACES CHIA-FU YU AND JENG-DAW YU Abstract. We show a mass formula for arbitrary supersingular abelian surfaces in characteristic p.. Introduction In [] Chai studied
More informationFIELDS OF DEFINITION OF RATIONAL POINTS ON VARIETIES
FIELDS OF DEFINITION OF RATIONAL POINTS ON VARIETIES JORDAN RIZOV Abstract. Let X be a scheme over a field K and let M X be the intersection of all subfields L of K such that X has a L-valued point. In
More informationGalois Representations
9 Galois Representations This book has explained the idea that all elliptic curves over Q arise from modular forms. Chapters 1 and introduced elliptic curves and modular curves as Riemann surfaces, and
More informationSHIMURA VARIETIES AND TAF
SHIMURA VARIETIES AND TAF PAUL VANKOUGHNETT 1. Introduction The primary source is chapter 6 of [?]. We ve spent a long time learning generalities about abelian varieties. In this talk (or two), we ll assemble
More informationModuli of abelian varieties in mixed and in positive characteristic
Moduli of abelian varieties in mixed and in positive characteristic Frans Oort Abstract. We start with a discussion of CM abelian varieties in characteristic zero, and in positive characteristic. An abelian
More informationMODULI SPACES OF CURVES
MODULI SPACES OF CURVES SCOTT NOLLET Abstract. My goal is to introduce vocabulary and present examples that will help graduate students to better follow lectures at TAGS 2018. Assuming some background
More informationIN POSITIVE CHARACTERISTICS: 3. Modular varieties with Hecke symmetries. 7. Foliation and a conjecture of Oort
FINE STRUCTURES OF MODULI SPACES IN POSITIVE CHARACTERISTICS: HECKE SYMMETRIES AND OORT FOLIATION 1. Elliptic curves and their moduli 2. Moduli of abelian varieties 3. Modular varieties with Hecke symmetries
More informationTAMAGAWA NUMBERS OF ELLIPTIC CURVES WITH C 13 TORSION OVER QUADRATIC FIELDS
TAMAGAWA NUMBERS OF ELLIPTIC CURVES WITH C 13 TORSION OVER QUADRATIC FIELDS FILIP NAJMAN Abstract. Let E be an elliptic curve over a number field K c v the Tamagawa number of E at v and let c E = v cv.
More informationVARIETIES WITHOUT EXTRA AUTOMORPHISMS II: HYPERELLIPTIC CURVES
VARIETIES WITHOUT EXTRA AUTOMORPHISMS II: HYPERELLIPTIC CURVES BJORN POONEN Abstract. For any field k and integer g 2, we construct a hyperelliptic curve X over k of genus g such that #(Aut X) = 2. We
More informationGenus 2 Curves of p-rank 1 via CM method
School of Mathematical Sciences University College Dublin Ireland and Claude Shannon Institute April 2009, GeoCrypt Joint work with Laura Hitt, Michael Naehrig, Marco Streng Introduction This talk is about
More informationCHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998
CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic
More informationAbelian varieties with l-adic Galois representation of Mumford s type
Abelian varieties with l-adic Galois representation of Mumford s type Rutger Noot Abstract This paper is devoted to the study of 4-dimensional abelian varieties over number fields with the property that
More informationCM lifting of abelian varieties
CM lifting of abelian varieties Ching-Li Chai Brian Conrad Frans Oort Dept. of Math Dept. of Math Dept. of Math Univ. of Pennsylvania Stanford Univ. Univ. of Utrecht Philadelphia, PA 19104 Stanford, CA
More informationRESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES RIMS A Note on an Anabelian Open Basis for a Smooth Variety. Yuichiro HOSHI.
RIMS-1898 A Note on an Anabelian Open Basis for a Smooth Variety By Yuichiro HOSHI January 2019 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan A Note on an Anabelian Open Basis
More informationCM p-divisible Groups over Finite Fields
CM p-divisible Groups over Finite Fields by Xinyun Sun A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan
More informationCONSTRUCTING SUPERSINGULAR ELLIPTIC CURVES. Reinier Bröker
CONSTRUCTING SUPERSINGULAR ELLIPTIC CURVES Reinier Bröker Abstract. We give an algorithm that constructs, on input of a prime power q and an integer t, a supersingular elliptic curve over F q with trace
More informationwith many rational points
Algebraic curves over F 2 with many rational points, René Schoof version May 7, 1991 Algebraic curves over F 2 with many rational points René Schoof Dipartimento di Matematica Università degli Studi di
More information2,3,5, LEGENDRE: ±TRACE RATIOS IN FAMILIES OF ELLIPTIC CURVES
2,3,5, LEGENDRE: ±TRACE RATIOS IN FAMILIES OF ELLIPTIC CURVES NICHOLAS M. KATZ 1. Introduction The Legendre family of elliptic curves over the λ-line, E λ : y 2 = x(x 1)(x λ), is one of the most familiar,
More informationarxiv: v2 [math.ac] 7 May 2018
INFINITE DIMENSIONAL EXCELLENT RINGS arxiv:1706.02491v2 [math.ac] 7 May 2018 HIROMU TANAKA Abstract. In this note, we prove that there exist infinite dimensional excellent rings. Contents 1. Introduction
More informationOn some congruence properties of elliptic curves
arxiv:0803.2809v5 [math.nt] 19 Jun 2009 On some congruence properties of elliptic curves Derong Qiu (School of Mathematical Sciences, Institute of Mathematics and Interdisciplinary Science, Capital Normal
More informationTranscendence theory in positive characteristic
Prof. Dr. Gebhard Böckle, Dr. Patrik Hubschmid Working group seminar WS 2012/13 Transcendence theory in positive characteristic Wednesdays from 9:15 to 10:45, INF 368, room 248 In this seminar we will
More informationOn Hrushovski s proof of the Manin-Mumford conjecture
On Hrushovski s proof of the Manin-Mumford conjecture Richard PINK and Damian ROESSLER May 16, 2006 Abstract The Manin-Mumford conjecture in characteristic zero was first proved by Raynaud. Later, Hrushovski
More informationON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS
ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS LISA CARBONE Abstract. We outline the classification of K rank 1 groups over non archimedean local fields K up to strict isogeny,
More informationA p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1
A p-adic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 ALEXANDER G.M. PAULIN Abstract. The (de Rham) geometric Langlands correspondence for GL n asserts that to an irreducible rank n integrable connection
More informationIntroduction to Elliptic Curves
IAS/Park City Mathematics Series Volume XX, XXXX Introduction to Elliptic Curves Alice Silverberg Introduction Why study elliptic curves? Solving equations is a classical problem with a long history. Starting
More informationLifting Galois Representations, and a Conjecture of Fontaine and Mazur
Documenta Math. 419 Lifting Galois Representations, and a Conjecture of Fontaine and Mazur Rutger Noot 1 Received: May 11, 2001 Revised: November 16, 2001 Communicated by Don Blasius Abstract. Mumford
More informationOn the equality case of the Ramanujan Conjecture for Hilbert modular forms
On the equality case of the Ramanujan Conjecture for Hilbert modular forms Liubomir Chiriac Abstract The generalized Ramanujan Conjecture for unitary cuspidal automorphic representations π on GL 2 posits
More informationNUNO FREITAS AND ALAIN KRAUS
ON THE DEGREE OF THE p-torsion FIELD OF ELLIPTIC CURVES OVER Q l FOR l p NUNO FREITAS AND ALAIN KRAUS Abstract. Let l and p be distinct prime numbers with p 3. Let E/Q l be an elliptic curve with p-torsion
More informationAlgebraic Hecke Characters
Algebraic Hecke Characters Motivation This motivation was inspired by the excellent article [Serre-Tate, 7]. Our goal is to prove the main theorem of complex multiplication. The Galois theoretic formulation
More informationAUTOMORPHISMS OF X(11) OVER CHARACTERISTIC 3, AND THE MATHIEU GROUP M 11
AUTOMORPHISMS OF X(11) OVER CHARACTERISTIC 3, AND THE MATHIEU GROUP M 11 C. S. RAJAN Abstract. We show that the automorphism group of the curve X(11) is the Mathieu group M 11, over a field of characteristic
More informationOn Mumford s families of abelian varieties
On Mumford s families of abelian varieties Rutger Noot Abstract In [Mum69], Mumford constructs families of abelian varieties which are parametrized by Shimura varieties but which are not of PEL type. In
More informationEndomorphism Rings of Abelian Varieties and their Representations
Endomorphism Rings of Abelian Varieties and their Representations Chloe Martindale 30 October 2013 These notes are based on the notes written by Peter Bruin for his talks in the Complex Multiplication
More informationETA-QUOTIENTS AND ELLIPTIC CURVES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 11, November 1997, Pages 3169 3176 S 0002-9939(97)03928-2 ETA-QUOTIENTS AND ELLIPTIC CURVES YVES MARTIN AND KEN ONO (Communicated by
More informationFiniteness of the Moderate Rational Points of Once-punctured Elliptic Curves. Yuichiro Hoshi
Hokkaido Mathematical Journal ol. 45 (2016) p. 271 291 Finiteness of the Moderate Rational Points of Once-punctured Elliptic Curves uichiro Hoshi (Received February 28, 2014; Revised June 12, 2014) Abstract.
More informationModularity of Abelian Varieties
1 Modularity of Abelian Varieties This is page 1 Printer: Opaque this 1.1 Modularity Over Q Definition 1.1.1 (Modular Abelian Variety). Let A be an abelian variety over Q. Then A is modular if there exists
More informationCLASS FIELD THEORY WEEK Motivation
CLASS FIELD THEORY WEEK 1 JAVIER FRESÁN 1. Motivation In a 1640 letter to Mersenne, Fermat proved the following: Theorem 1.1 (Fermat). A prime number p distinct from 2 is a sum of two squares if and only
More informationA Version of the Grothendieck Conjecture for p-adic Local Fields
A Version of the Grothendieck Conjecture for p-adic Local Fields by Shinichi MOCHIZUKI* Section 0: Introduction The purpose of this paper is to prove an absolute version of the Grothendieck Conjecture
More informationTitchmarsh divisor problem for abelian varieties of type I, II, III, and IV
Titchmarsh divisor problem for abelian varieties of type I, II, III, and IV Cristian Virdol Department of Mathematics Yonsei University cristian.virdol@gmail.com September 8, 05 Abstract We study Titchmarsh
More informationRiemann Forms. Ching-Li Chai. 1. From Riemann matrices to Riemann forms
Riemann Forms Ching-Li Chai The notion of Riemann forms began as a coordinate-free reformulation of the Riemann period relations for abelian functions discovered by Riemann. This concept has evolved with
More information1.5.4 Every abelian variety is a quotient of a Jacobian
16 1. Abelian Varieties: 10/10/03 notes by W. Stein 1.5.4 Every abelian variety is a quotient of a Jacobian Over an infinite field, every abelin variety can be obtained as a quotient of a Jacobian variety.
More informationTwisted L-Functions and Complex Multiplication
Journal of umber Theory 88, 104113 (2001) doi:10.1006jnth.2000.2613, available online at http:www.idealibrary.com on Twisted L-Functions and Complex Multiplication Abdellah Sebbar Department of Mathematics
More information2-ADIC ARITHMETIC-GEOMETRIC MEAN AND ELLIPTIC CURVES
-ADIC ARITHMETIC-GEOMETRIC MEAN AND ELLIPTIC CURVES KENSAKU KINJO, YUKEN MIYASAKA AND TAKAO YAMAZAKI 1. The arithmetic-geometric mean over R and elliptic curves We begin with a review of a relation between
More informationFields of definition of abelian varieties with real multiplication
Contemporary Mathematics Volume 174, 1994 Fields of definition of abelian varieties with real multiplication KENNETH A. RIBET 1. Introduction Let K be a field, and let K be a separable closure of K. Let
More informationANNIHILATING POLYNOMIALS, TRACE FORMS AND THE GALOIS NUMBER. Seán McGarraghy
ANNIHILATING POLYNOMIALS, TRACE FORMS AND THE GALOIS NUMBER Seán McGarraghy Abstract. We construct examples where an annihilating polynomial produced by considering étale algebras improves on the annihilating
More informationOn elliptic curves in characteristic 2 with wild additive reduction
ACTA ARITHMETICA XCI.2 (1999) On elliptic curves in characteristic 2 with wild additive reduction by Andreas Schweizer (Montreal) Introduction. In [Ge1] Gekeler classified all elliptic curves over F 2
More informationMaximal Class Numbers of CM Number Fields
Maximal Class Numbers of CM Number Fields R. C. Daileda R. Krishnamoorthy A. Malyshev Abstract Fix a totally real number field F of degree at least 2. Under the assumptions of the generalized Riemann hypothesis
More informationMULTIPLICITIES OF GALOIS REPRESENTATIONS IN JACOBIANS OF SHIMURA CURVES
MULTIPLICITIES OF GALOIS REPRESENTATIONS IN JACOBIANS OF SHIMURA CURVES BY KENNETH A. RIBET Mathematics Department, University of California, Berkeley CA 94720 USA Let p and q be distinct primes. The new
More information1. Introduction. Mathematisches Institut, Seminars, (Y. Tschinkel, ed.), p Universität Göttingen,
Mathematisches Institut, Seminars, (Y. Tschinkel, ed.), p. 203 209 Universität Göttingen, 2004-05 THE DIOPHANTINE EQUATION x 4 + 2y 4 = z 4 + 4w 4 A.-S. Elsenhans J. Jahnel Mathematisches Institut, Bunsenstr.
More informationThe Grothendieck-Katz Conjecture for certain locally symmetric varieties
The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-
More informationarxiv: v1 [math.ag] 23 Oct 2007
ON THE HODGE-NEWTON FILTRATION FOR p-divisible O-MODULES arxiv:0710.4194v1 [math.ag] 23 Oct 2007 ELENA MANTOVAN AND EVA VIEHMANN Abstract. The notions Hodge-Newton decomposition and Hodge-Newton filtration
More information1.6.1 What are Néron Models?
18 1. Abelian Varieties: 10/20/03 notes by W. Stein 1.6.1 What are Néron Models? Suppose E is an elliptic curve over Q. If is the minimal discriminant of E, then E has good reduction at p for all p, in
More informationIsogeny invariance of the BSD conjecture
Isogeny invariance of the BSD conjecture Akshay Venkatesh October 30, 2015 1 Examples The BSD conjecture predicts that for an elliptic curve E over Q with E(Q) of rank r 0, where L (r) (1, E) r! = ( p
More informationBoundary of Cohen-Macaulay cone and asymptotic behavior of system of ideals
Boundary of Cohen-Macaulay cone and asymptotic behavior of system of ideals Kazuhiko Kurano Meiji University 1 Introduction On a smooth projective variety, we can define the intersection number for a given
More informationp-divisible Groups: Definitions and Examples
p-divisible Groups: Definitions and Examples Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 18, 2013 Connected vs. étale
More informationNéron models of abelian varieties
Néron models of abelian varieties Matthieu Romagny Summer School on SGA3, September 3, 2011 Abstract : We present a survey of the construction of Néron models of abelian varieties, as an application of
More informationAlgebraic Number Theory Notes: Local Fields
Algebraic Number Theory Notes: Local Fields Sam Mundy These notes are meant to serve as quick introduction to local fields, in a way which does not pass through general global fields. Here all topological
More informationNEWTON POLYGONS OF CYCLIC COVERS OF THE PROJECTIVE LINE BRANCHED AT THREE POINTS
NEWTON POLYGONS OF CYCLIC COVERS OF THE PROJECTIVE LINE BRANCHED AT THREE POINTS WANLIN LI, ELENA MANTOVAN, RACHEL PRIES, AND YUNQING TANG ABSTRACT. We review the Shimura-Taniyama method for computing
More informationAn analogue of the Weierstrass ζ-function in characteristic p. José Felipe Voloch
An analogue of the Weierstrass ζ-function in characteristic p José Felipe Voloch To J.W.S. Cassels on the occasion of his 75th birthday. 0. Introduction Cassels, in [C], has noticed a remarkable analogy
More informationSets of Completely Decomposed Primes in Extensions of Number Fields
Sets of Completely Decomposed Primes in Extensions of Number Fields by Kay Wingberg June 15, 2014 Let p be a prime number and let k(p) be the maximal p-extension of a number field k. If T is a set of primes
More informationSome remarks on Frobenius and Lefschetz in étale cohomology
Some remarks on obenius and Lefschetz in étale cohomology Gabriel Chênevert January 5, 2004 In this lecture I will discuss some more or less related issues revolving around the main idea relating (étale)
More informationThe Néron Ogg Shafarevich criterion Erik Visse
The Néron Ogg Shafarevich criterion Erik Visse February 17, 2017 These are notes from the seminar on abelian varieties and good reductions held in Amsterdam late 2016 and early 2017. The website for the
More informationMath 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille
Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is
More informationON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF
ON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF MATTHEW H. BAKER AND JÁNOS A. CSIRIK Abstract. We give a new proof of the isomorphism between the dualizing sheaf and the canonical
More informationMATH 8253 ALGEBRAIC GEOMETRY WEEK 12
MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f
More informationGROSS-ZAGIER REVISITED
GROSS-ZAGIER REVISITED BRIAN CONRAD Contents 1. Introduction 1 2. Some properties of abelian schemes and modular curves 3 3. The Serre-Tate theorem and the Grothendieck existence theorem 7 4. Computing
More informationRAMIFIED PRIMES IN THE FIELD OF DEFINITION FOR THE MORDELL-WEIL GROUP OF AN ELLIPTIC SURFACE
PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 116. Number 4, December 1992 RAMIFIED PRIMES IN THE FIELD OF DEFINITION FOR THE MORDELL-WEIL GROUP OF AN ELLIPTIC SURFACE MASATO KUWATA (Communicated
More informationLecture 2: Elliptic curves
Lecture 2: Elliptic curves This lecture covers the basics of elliptic curves. I begin with a brief review of algebraic curves. I then define elliptic curves, and talk about their group structure and defining
More informationarxiv: v1 [math.ag] 30 Mar 2016
ON PRIMES OF ORDINARY AND HODGE-WITT REDUCTION KIRTI JOSHI arxiv:1603.09404v1 [math.ag] 30 Mar 2016 Abstract. Jean-Pierre Serre has conjectured, in the context of abelian varieties, that there are infinitely
More information15 Lecture 15: Points and lft maps
15 Lecture 15: Points and lft maps 15.1 A noetherian property Let A be an affinoid algebraic over a non-archimedean field k and X = Spa(A, A 0 ). For any x X, the stalk O X,x is the limit of the directed
More informationREDUCTION OF ELLIPTIC CURVES OVER CERTAIN REAL QUADRATIC NUMBER FIELDS
MATHEMATICS OF COMPUTATION Volume 68, Number 228, Pages 1679 1685 S 0025-5718(99)01129-1 Article electronically published on May 21, 1999 REDUCTION OF ELLIPTIC CURVES OVER CERTAIN REAL QUADRATIC NUMBER
More informationLecture 4: Abelian varieties (algebraic theory)
Lecture 4: Abelian varieties (algebraic theory) This lecture covers the basic theory of abelian varieties over arbitrary fields. I begin with the basic results such as commutativity and the structure of
More informationINTEGER VALUED POLYNOMIALS AND LUBIN-TATE FORMAL GROUPS
INTEGER VALUED POLYNOMIALS AND LUBIN-TATE FORMAL GROUPS EHUD DE SHALIT AND ERAN ICELAND Abstract. If R is an integral domain and K is its field of fractions, we let Int(R) stand for the subring of K[x]
More informationGross Zagier Revisited
Heegner Points and Rankin L-Series MSRI Publications Volume 49, 2004 Gross Zagier Revisited BRIAN CONRAD WITH AN APPENDIX BY W. R. MANN Contents 1. Introduction 67 2. Some Properties of Abelian Schemes
More informationRigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture
Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture Benson Farb and Mark Kisin May 8, 2009 Abstract Using Margulis s results on lattices in semisimple Lie groups, we prove the Grothendieck-
More informationINJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA
INJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA These notes are intended to give the reader an idea what injective modules are, where they show up, and, to
More information14 Ordinary and supersingular elliptic curves
18.783 Elliptic Curves Spring 2015 Lecture #14 03/31/2015 14 Ordinary and supersingular elliptic curves Let E/k be an elliptic curve over a field of positive characteristic p. In Lecture 7 we proved that
More informationDensity of rational points on Enriques surfaces
Density of rational points on Enriques surfaces F. A. Bogomolov Courant Institute of Mathematical Sciences, N.Y.U. 251 Mercer str. New York, (NY) 10012, U.S.A. e-mail: bogomolo@cims.nyu.edu and Yu. Tschinkel
More informationHigher Class Field Theory
Higher Class Field Theory Oberseminar im Sommersemester 2011 Content The aim of higher class field theory is to describe the abelian fundamental group of an arithmetic scheme using a suitable, arithmetically
More informationElliptic Curves Spring 2015 Lecture #23 05/05/2015
18.783 Elliptic Curves Spring 2015 Lecture #23 05/05/2015 23 Isogeny volcanoes We now want to shift our focus away from elliptic curves over C and consider elliptic curves E/k defined over any field k;
More informationTHE MODULI SPACE OF CURVES
THE MODULI SPACE OF CURVES In this section, we will give a sketch of the construction of the moduli space M g of curves of genus g and the closely related moduli space M g,n of n-pointed curves of genus
More informationLemma 1.1. The field K embeds as a subfield of Q(ζ D ).
Math 248A. Quadratic characters associated to quadratic fields The aim of this handout is to describe the quadratic Dirichlet character naturally associated to a quadratic field, and to express it in terms
More informationDESCENT FOR SHIMURA VARIETIES
DESCENT FOR SHIMURA VARIETIES J.S. MILNE Abstract. This note proves that the descent maps provided by Langlands s Conjugacy Conjecture do satisfy the continuity condition necessary for them to be effective.
More informationSERRE S CONJECTURE AND BASE CHANGE FOR GL(2)
SERRE S CONJECTURE AND BASE CHANGE OR GL(2) HARUZO HIDA 1. Quaternion class sets A quaternion algebra B over a field is a simple algebra of dimension 4 central over a field. A prototypical example is the
More information1. Artin s original conjecture
A possible generalization of Artin s conjecture for primitive root 1. Artin s original conjecture Ching-Li Chai February 13, 2004 (1.1) Conjecture (Artin, 1927) Let a be an integer, a 0, ±1, and a is not
More informationHamburger Beiträge zur Mathematik
Hamburger Beiträge zur Mathematik Nr. 712, November 2017 Remarks on the Polynomial Decomposition Law by Ernst Kleinert Remarks on the Polynomial Decomposition Law Abstract: we first discuss in some detail
More informationCLASS FIELD THEORY AND COMPLEX MULTIPLICATION FOR ELLIPTIC CURVES
CLASS FIELD THEORY AND COMPLEX MULTIPLICATION FOR ELLIPTIC CURVES FRANK GOUNELAS 1. Class Field Theory We ll begin by motivating some of the constructions of the CM (complex multiplication) theory for
More information