Moduli Spaces of K3 Surfaces with Large Picard Number
... Let X be a K3 surface, then the K3 lattice is just the lattice given by the cup product form on H 2 (X, Z). This is a rank 22 lattice, and hence it is by no means simple, however, it is this complexity that makes K3 surfaces such a fertile place to study. In general, the moduli spaces of lattice pol ...
... Let X be a K3 surface, then the K3 lattice is just the lattice given by the cup product form on H 2 (X, Z). This is a rank 22 lattice, and hence it is by no means simple, however, it is this complexity that makes K3 surfaces such a fertile place to study. In general, the moduli spaces of lattice pol ...
Abelian Varieties
... Pic .A/ ' A_ .k/ and Pic0 .A_ / ' A.k/ (we shall define Pic0 in this context later). In the case of an elliptic curve, E _ D E. In general, A and A_ are isogenous, but they are not equal (and usually not even isomorphic). Appropriately interpreted, most of the statements in Silverman’s books on elli ...
... Pic .A/ ' A_ .k/ and Pic0 .A_ / ' A.k/ (we shall define Pic0 in this context later). In the case of an elliptic curve, E _ D E. In general, A and A_ are isogenous, but they are not equal (and usually not even isomorphic). Appropriately interpreted, most of the statements in Silverman’s books on elli ...
An Introduction to Topological Groups
... Since G → G, x 7→ yx is a homeomorphism, there exists a neighborhood W of e such that yW ⊆ V . Observe, that yW is a neighborhood of y ! By the continuity of the multiplication at (e, e), there is an open neighborhood U of e such that U · U ⊆ W . W.l.o.g. we may assume that U is symmetric (i.e. U = ...
... Since G → G, x 7→ yx is a homeomorphism, there exists a neighborhood W of e such that yW ⊆ V . Observe, that yW is a neighborhood of y ! By the continuity of the multiplication at (e, e), there is an open neighborhood U of e such that U · U ⊆ W . W.l.o.g. we may assume that U is symmetric (i.e. U = ...
670 notes - OSU Department of Mathematics
... (Z/nZ) As a set, define (Z/nZ) to be those congruence classes whose representatives are relatively prime to n. Here’s why we do this: Prop. An integer k has an inverse mod n k and n are relatively prime. Proof. One direction is easy. In the other direction, suppose that k and n are relatively pr ...
... (Z/nZ) As a set, define (Z/nZ) to be those congruence classes whose representatives are relatively prime to n. Here’s why we do this: Prop. An integer k has an inverse mod n k and n are relatively prime. Proof. One direction is easy. In the other direction, suppose that k and n are relatively pr ...
Computable Completely Decomposable Groups.
... completely decomposable groups can be found in [27, 13, 35]. There is a lot more hope to obtain a satisfactory description of higher computable categoricity in the special class of completely decomposable groups. Recently Downey and Melnikov [8] studied computable homogeneous completely decomposable ...
... completely decomposable groups can be found in [27, 13, 35]. There is a lot more hope to obtain a satisfactory description of higher computable categoricity in the special class of completely decomposable groups. Recently Downey and Melnikov [8] studied computable homogeneous completely decomposable ...
Genus three curves over finite fields
... Associated to C we have the Jacobian C 7→ Jac(C ) Properties of Jac(C ): i) Jac(C ) is a commutative projective group variety of dimension g . Such varieties are called Abelian. ii) Jac(C )(F̄q ) generated by formal differences x − y with x, y ∈ C (F̄q ). Modulo a certain relation: x1 + · · · + xn − ...
... Associated to C we have the Jacobian C 7→ Jac(C ) Properties of Jac(C ): i) Jac(C ) is a commutative projective group variety of dimension g . Such varieties are called Abelian. ii) Jac(C )(F̄q ) generated by formal differences x − y with x, y ∈ C (F̄q ). Modulo a certain relation: x1 + · · · + xn − ...
Algebra I: Section 6. The structure of groups. 6.1 Direct products of
... situation arises in linear algebra when we ask whether a family of vector subspaces V1 , . . . , Vr ⊆ V is “linearly independent.” This does not follow from the pairwise disjointness condition Vi ∩ Vj = (0), unless there are just two subspaces; P instead, we must require that each Vi have trivial in ...
... situation arises in linear algebra when we ask whether a family of vector subspaces V1 , . . . , Vr ⊆ V is “linearly independent.” This does not follow from the pairwise disjointness condition Vi ∩ Vj = (0), unless there are just two subspaces; P instead, we must require that each Vi have trivial in ...
M07/08
... h. This led to an early conjecture that C can be obtained by the method of group table modifications, as used in our earlier work [3], [4], [5], [13]. More precisely, we conjectured that there exists a group—we shall denote it again by G—such that C = (G, ∗), where x ∗ y ∈ {xy, xyh}, for a fixed cen ...
... h. This led to an early conjecture that C can be obtained by the method of group table modifications, as used in our earlier work [3], [4], [5], [13]. More precisely, we conjectured that there exists a group—we shall denote it again by G—such that C = (G, ∗), where x ∗ y ∈ {xy, xyh}, for a fixed cen ...
Finitely generated groups with automatic presentations
... As noted in [10], the main difference is that of substructures: the substructures of groups as structures (G, ◦) need only be subsemigroups, whereas, with (G, ◦, e, −1 ), they must be subgroups. For our purposes, we needn’t be too worried by this distinction. It is clear that, for the structure (G, ...
... As noted in [10], the main difference is that of substructures: the substructures of groups as structures (G, ◦) need only be subsemigroups, whereas, with (G, ◦, e, −1 ), they must be subgroups. For our purposes, we needn’t be too worried by this distinction. It is clear that, for the structure (G, ...
ABELIAN VARIETIES A canonical reference for the subject is
... Proof. Choose a nonempty open affine U ⊂ X and let D = X \ U with the reduced structure. We claim that D is a Cartier divisor (that is, its coherent ideal sheaf is invertible). Since X is regular, it is equivalent to say that all generic points of D have codimension 1. Mumford omits this explanation ...
... Proof. Choose a nonempty open affine U ⊂ X and let D = X \ U with the reduced structure. We claim that D is a Cartier divisor (that is, its coherent ideal sheaf is invertible). Since X is regular, it is equivalent to say that all generic points of D have codimension 1. Mumford omits this explanation ...
homework 1 - TTU Math Department
... Problem 4. Show that if the action of the group of deck transformations in one fiber is transitive, then its action in every fiber is transitive. Solution: Let p : E → B be the covering in question. We assume that the action of the group of deck transformations is transitive in some fiber p−1(b0) a ...
... Problem 4. Show that if the action of the group of deck transformations in one fiber is transitive, then its action in every fiber is transitive. Solution: Let p : E → B be the covering in question. We assume that the action of the group of deck transformations is transitive in some fiber p−1(b0) a ...
VISIBLE EVIDENCE FOR THE BIRCH AND SWINNERTON
... M2 (Γ0 (N )) ⊗ Q using Manin symbols, which are a finite set of generators for M2 (Γ0 (N )). In general, the easiest way we have found to compute M2 (Γ0 (N )) is to compute M2 (Γ0 (N ))⊗ Q and then to compute the Z-submodule of M2 (Γ0 (N ))⊗ Q generated by the Manin symbols. 3.2. Enumerating newform ...
... M2 (Γ0 (N )) ⊗ Q using Manin symbols, which are a finite set of generators for M2 (Γ0 (N )). In general, the easiest way we have found to compute M2 (Γ0 (N )) is to compute M2 (Γ0 (N ))⊗ Q and then to compute the Z-submodule of M2 (Γ0 (N ))⊗ Q generated by the Manin symbols. 3.2. Enumerating newform ...
Notes - Math Berkeley
... whence H̃ is finite and commutative. This implies that we can also view H̃ as a subgroup scheme of G(A,L) . Exercise 4.4. Show that if x 2 K(A,L) is an element, then there exists a level subgroup H̃ ⇢ G(A,L) whose image in K(A,L) contains x. 4.5. Let H̃ ⇢ G(A,L) be a level subgroup, and let H ⇢ K(A, ...
... whence H̃ is finite and commutative. This implies that we can also view H̃ as a subgroup scheme of G(A,L) . Exercise 4.4. Show that if x 2 K(A,L) is an element, then there exists a level subgroup H̃ ⇢ G(A,L) whose image in K(A,L) contains x. 4.5. Let H̃ ⇢ G(A,L) be a level subgroup, and let H ⇢ K(A, ...
A PROPERTY OF SMALL GROUPS A connected group of Morley
... Weakly small structures were introduced by Belegradek to give a common generalisation of small and minimal structures. Definition 3. (Poizat [17]) An infinite structure is d-minimal if any of its partitions has no more than d infinite definable subsets. Every d-minimal structure in a countable langu ...
... Weakly small structures were introduced by Belegradek to give a common generalisation of small and minimal structures. Definition 3. (Poizat [17]) An infinite structure is d-minimal if any of its partitions has no more than d infinite definable subsets. Every d-minimal structure in a countable langu ...
Degrees of curves in abelian varieties
... d ^ ^(A71/^!)1/71 ^ n. It is known (see [C], [R]) that d = n if and only if C is smooth and X is isomorphic to its Jacobian JC with its canonical principal polarization. What about the next cases? We get partial characterizations for d = n + 1 and d = n + 2, and we show (example 6.11) that all degre ...
... d ^ ^(A71/^!)1/71 ^ n. It is known (see [C], [R]) that d = n if and only if C is smooth and X is isomorphic to its Jacobian JC with its canonical principal polarization. What about the next cases? We get partial characterizations for d = n + 1 and d = n + 2, and we show (example 6.11) that all degre ...
Generalized Dihedral Groups - College of Arts and Sciences
... We are nearly ready to generalize the classical dihedral groups, but in order to do so we must discuss the role of automorphisms in semi-direct products. Definition 3.1. An automorphism of a group G is an isomorphism ϕ : G → G. The set of all such automorphisms is denoted Aut(G). One can check that ...
... We are nearly ready to generalize the classical dihedral groups, but in order to do so we must discuss the role of automorphisms in semi-direct products. Definition 3.1. An automorphism of a group G is an isomorphism ϕ : G → G. The set of all such automorphisms is denoted Aut(G). One can check that ...
structure of abelian quasi-groups
... is self-unit, this implies that all C„,r are equal, and since any element of § may then be chosen as factor set it follows that only one extension is possible in this case, namely the direct product. If § is not self-unit more possibilities occur, but a factor set for any extension can always be cho ...
... is self-unit, this implies that all C„,r are equal, and since any element of § may then be chosen as factor set it follows that only one extension is possible in this case, namely the direct product. If § is not self-unit more possibilities occur, but a factor set for any extension can always be cho ...
Solvable Groups
... Solvable Groups Mathematics 581, Fall 2012 In many ways, abstract algebra began with the work of Abel and Galois on the solvability of polynomial equations by radicals. The key idea Galois had was to transform questions about fields and polynomials into questions about finite groups. For the proof t ...
... Solvable Groups Mathematics 581, Fall 2012 In many ways, abstract algebra began with the work of Abel and Galois on the solvability of polynomial equations by radicals. The key idea Galois had was to transform questions about fields and polynomials into questions about finite groups. For the proof t ...
Projective p-adic representations of the K-rational geometric fundamental group (with G. Frey).
... matrix rings over Qp , i.e. Ep = EndRp (Vp (A)) is not split, contrary to hypothesis. Thus Rp ' Qep (for some e), and so all the characters of Rp are 1-dimensional, which means that Vp (A) has a decomposition of the form (7). Remark 3.2 a) As the above proof shows, we can also characterize the valid ...
... matrix rings over Qp , i.e. Ep = EndRp (Vp (A)) is not split, contrary to hypothesis. Thus Rp ' Qep (for some e), and so all the characters of Rp are 1-dimensional, which means that Vp (A) has a decomposition of the form (7). Remark 3.2 a) As the above proof shows, we can also characterize the valid ...
Part III. Homomorphisms and Factor Groups
... Note. By Theorem 13.12 part (4), we know that for K < G0 , where K = {e0}, we have φ−1 [K] < G. This subgroup φ−1 [K] includes all elements of G mapped under φ to e0 . You encountered a similar idea in linear algebra when considering an m × n matrix A as a linear transformation from Rn to Rm. Rn and ...
... Note. By Theorem 13.12 part (4), we know that for K < G0 , where K = {e0}, we have φ−1 [K] < G. This subgroup φ−1 [K] includes all elements of G mapped under φ to e0 . You encountered a similar idea in linear algebra when considering an m × n matrix A as a linear transformation from Rn to Rm. Rn and ...
groups with no free subsemigroups
... since we shall show that G/Fitt(G) has bounded exponent and the nilpotency class of Fitt(C7) is bounded. The bound on Fitt(G) is obtained from Proposition 5. By Theorem 1, G is nilpotent-by-finite. Let A be a normal abelian subgroup of G. Then for any a £ A and g £ G consider the ordered pair (g, fl ...
... since we shall show that G/Fitt(G) has bounded exponent and the nilpotency class of Fitt(C7) is bounded. The bound on Fitt(G) is obtained from Proposition 5. By Theorem 1, G is nilpotent-by-finite. Let A be a normal abelian subgroup of G. Then for any a £ A and g £ G consider the ordered pair (g, fl ...
SECTION 2: UNIVERSAL COEFFICIENT THEOREM IN SINGULAR
... Let now B be a further abelian group and let F ⊗ B ∈ Ch≥0 (Z) be the levelwise tensor product, still concentrated in dimensions 0, 1. Then we make the following definitions, Tork,F (A, B) := Hk (F ⊗ B), k = 0, 1. ∼ A ⊗ B. In fact, Tor0,F (A, B) is defined Let us observe that we have an isomorphism T ...
... Let now B be a further abelian group and let F ⊗ B ∈ Ch≥0 (Z) be the levelwise tensor product, still concentrated in dimensions 0, 1. Then we make the following definitions, Tork,F (A, B) := Hk (F ⊗ B), k = 0, 1. ∼ A ⊗ B. In fact, Tor0,F (A, B) is defined Let us observe that we have an isomorphism T ...
The expected number of random elements to generate a finite
... larger is the value of k, the closer Corollary 4 is to the truth for these numbers. Remarks. The argument at the start of the proof of the theorem is well known. In fact a more general result is known: a subset of a finite group G generates G if and only if the projection of the subset in G/Φ(G) gen ...
... larger is the value of k, the closer Corollary 4 is to the truth for these numbers. Remarks. The argument at the start of the proof of the theorem is well known. In fact a more general result is known: a subset of a finite group G generates G if and only if the projection of the subset in G/Φ(G) gen ...
On the field of definition of superspecial polarized
... (A, 0), where A is as above, and some arithmetic nature on quaternion hermitian lattices, for example, Hecke operators, type numbers and so on. When dim A = 1, that is, when A is a supersingular elliptic curve, Deuring [3] has shown that A has a model defined over the finite field Fp2 and the number ...
... (A, 0), where A is as above, and some arithmetic nature on quaternion hermitian lattices, for example, Hecke operators, type numbers and so on. When dim A = 1, that is, when A is a supersingular elliptic curve, Deuring [3] has shown that A has a model defined over the finite field Fp2 and the number ...
Introduction for the seminar on complex multiplication
... • A 1-dimensional abelian variety is exactly the same as an elliptic curve. • Every abelian variety over k = C is (as a complex manifold) isomorphic to a complex torus, i.e. a manifold of the form Cg /Λ, for a lattice Λ of rank 2g in Cg . A complex torus is an abelian variety if and only if it has a ...
... • A 1-dimensional abelian variety is exactly the same as an elliptic curve. • Every abelian variety over k = C is (as a complex manifold) isomorphic to a complex torus, i.e. a manifold of the form Cg /Λ, for a lattice Λ of rank 2g in Cg . A complex torus is an abelian variety if and only if it has a ...