Curves of given p-rank with trivial automorphism group
... The statements and proofs of our main results would be simpler if more were known about the geometry of Mg,f and Hg,f . For example, one could reduce to the case f = 0 if one knew that each irreducible component of Mg,f contained a component of Mg,0 . Even the number of irreducible components of Mg, ...
... The statements and proofs of our main results would be simpler if more were known about the geometry of Mg,f and Hg,f . For example, one could reduce to the case f = 0 if one knew that each irreducible component of Mg,f contained a component of Mg,0 . Even the number of irreducible components of Mg, ...
Variations on Belyi`s theorem - Universidad Autónoma de Madrid
... easy to apply (at least by us, the non experts) since, as often occurs in practice, it is difficult to check if its hypotheses are satisfied in a given problem (cf. [27]). On the contrary, our criterion (Criterion 1) is easy to handle. Although it is much less ambitious, in that it only attempts to det ...
... easy to apply (at least by us, the non experts) since, as often occurs in practice, it is difficult to check if its hypotheses are satisfied in a given problem (cf. [27]). On the contrary, our criterion (Criterion 1) is easy to handle. Although it is much less ambitious, in that it only attempts to det ...
GROUP ACTIONS 1. Introduction The symmetric groups S , alternating groups A
... The exponential notation xg in place of xg works well here, especially by writing the identity in the group as 1: x1 = x and (xg1 )g2 = xg1 g2 . The distinction between left and right actions is how a product gg 0 acts: in a left action g 0 acts first and g acts second, while in a right action g act ...
... The exponential notation xg in place of xg works well here, especially by writing the identity in the group as 1: x1 = x and (xg1 )g2 = xg1 g2 . The distinction between left and right actions is how a product gg 0 acts: in a left action g 0 acts first and g acts second, while in a right action g act ...
Algebra Qual Solutions September 12, 2009 UCLA ALGEBRA QUALIFYING EXAM Solutions
... nonzero elements in F24 , each spanning a line; but we over count by a factor of 4 − 1 = 3, the number of nonzero elements in each line. Now it is obvious that SL2 (F4 ) act by permuting the lines, thus we get a homomorphism SL2 (F4 ) −→ S5 . This is injective; indeed, a non-identity matrix of dimen ...
... nonzero elements in F24 , each spanning a line; but we over count by a factor of 4 − 1 = 3, the number of nonzero elements in each line. Now it is obvious that SL2 (F4 ) act by permuting the lines, thus we get a homomorphism SL2 (F4 ) −→ S5 . This is injective; indeed, a non-identity matrix of dimen ...
23. Dimension Dimension is intuitively obvious but - b
... rings the dimension does not seem to be such a useful concept: the Krull dimension is usually infinite and behaves in strange ways: for example the dimension of R[x] can be larger than the ”expected” value 1 + dim R. There seems to be no really satisfactory concept of dimension for non-Noetherian ri ...
... rings the dimension does not seem to be such a useful concept: the Krull dimension is usually infinite and behaves in strange ways: for example the dimension of R[x] can be larger than the ”expected” value 1 + dim R. There seems to be no really satisfactory concept of dimension for non-Noetherian ri ...
STRUCTURE OF LINEAR SETS
... Proof. If S^O, L is dense. If (g= 0, @' = 0, from which we conclude that if Az/= 0(L) and v^0(L), X has to be zero. This means that L is closed. The converse does not hold since an ideal is a dense set but may not be primary. However, every closed set is a prime linear set since every closed linear ...
... Proof. If S^O, L is dense. If (g= 0, @' = 0, from which we conclude that if Az/= 0(L) and v^0(L), X has to be zero. This means that L is closed. The converse does not hold since an ideal is a dense set but may not be primary. However, every closed set is a prime linear set since every closed linear ...
Genus three curves over finite fields
... Make an Abelian three fold A with a good trace And apply the Torelli theorem Proofs How to make A with a good trace for A = E 3 /G Trace Formula + Torelli + Deligne = Main result Conjecture for p = 3 ...
... Make an Abelian three fold A with a good trace And apply the Torelli theorem Proofs How to make A with a good trace for A = E 3 /G Trace Formula + Torelli + Deligne = Main result Conjecture for p = 3 ...
Algebraic Methods
... “A set of symbols all of them different, and such that the product of any two of them (no matter in what order), or the product of any one of them into itself, belongs to the set, is said to be a group. These symbols are not in general convertible [commutative], but are associative.” ...
... “A set of symbols all of them different, and such that the product of any two of them (no matter in what order), or the product of any one of them into itself, belongs to the set, is said to be a group. These symbols are not in general convertible [commutative], but are associative.” ...
Representation theory and applications in classical quantum
... H. If End(H) is equipped with the operator norm, then v 7→ Pv defines a continuous map H \ {0} → End(H), which factors to a continuous embedding P(H) ,→ End(H). Show that the topology of P(H) coincides with the restriction topology associated with the embedding P(H) ,→ End(H). We shall now investiga ...
... H. If End(H) is equipped with the operator norm, then v 7→ Pv defines a continuous map H \ {0} → End(H), which factors to a continuous embedding P(H) ,→ End(H). Show that the topology of P(H) coincides with the restriction topology associated with the embedding P(H) ,→ End(H). We shall now investiga ...
Another Look at Square Roots and Traces (and Quadratic Equations
... odd degree d to yield a low weight x. In particular, we give examples of pentanomials and heptanomials, but in at least one case, that of F2233 , that can be defined by trinomials, we show how one can perform square root computations even faster than in [19]. Since the main motivation comes from ell ...
... odd degree d to yield a low weight x. In particular, we give examples of pentanomials and heptanomials, but in at least one case, that of F2233 , that can be defined by trinomials, we show how one can perform square root computations even faster than in [19]. Since the main motivation comes from ell ...
Full Text (PDF format)
... whose exponent is a power of a prime p. Must the dimension of H be a power of the same prime? This is a special case of Question 5.1, but we still do not know the answer, except for the case exp(H) = 2, when the answer is trivially positive. For group algebras, the statement is equivalent to the wel ...
... whose exponent is a power of a prime p. Must the dimension of H be a power of the same prime? This is a special case of Question 5.1, but we still do not know the answer, except for the case exp(H) = 2, when the answer is trivially positive. For group algebras, the statement is equivalent to the wel ...
diagram algebras, hecke algebras and decomposition numbers at
... applies when R is an algebraically closed field whose characteristic is either 0 or p > 0 such that pe > n, where e is the multiplicative order of q 2 . na (q) be the extended affine Hecke algebra of type A ...
... applies when R is an algebraically closed field whose characteristic is either 0 or p > 0 such that pe > n, where e is the multiplicative order of q 2 . na (q) be the extended affine Hecke algebra of type A ...
ON SOME DIFFERENTIALS IN THE MOTIVIC COHOMOLOGY
... The strategy of the proof is the following: First, I show, using Adams operations, that the first non-trivial differential may appear only in Ep -term (Proposition 1.2). Then, computing the motivic Steenrod algebra in the corresponding degree, it is possible to show that the differential in question ...
... The strategy of the proof is the following: First, I show, using Adams operations, that the first non-trivial differential may appear only in Ep -term (Proposition 1.2). Then, computing the motivic Steenrod algebra in the corresponding degree, it is possible to show that the differential in question ...
HOW TO DO A p-DESCENT ON AN ELLIPTIC CURVE
... We can simplify the computation of the p-Selmer group in the case that p splits in the endomorphism ring of E. In section 8 we give an algorithm for this special case. When the elliptic curve has a rational p-isogeny h : E → E 0 , we can use Selmer groups related to h and the dual isogeny instead of ...
... We can simplify the computation of the p-Selmer group in the case that p splits in the endomorphism ring of E. In section 8 we give an algorithm for this special case. When the elliptic curve has a rational p-isogeny h : E → E 0 , we can use Selmer groups related to h and the dual isogeny instead of ...
Derived splinters in positive characteristic
... one works ‘up to finite covers’. This idea has been pursued in much more depth in the recent work [BST11], where ‘up to finite cover’ analogues of the Nadel vanishing have been established. Moreover, we can also prove a weak mixed-characteristic analogue of Theorem 1.5 (see [Bha11b]), and this analo ...
... one works ‘up to finite covers’. This idea has been pursued in much more depth in the recent work [BST11], where ‘up to finite cover’ analogues of the Nadel vanishing have been established. Moreover, we can also prove a weak mixed-characteristic analogue of Theorem 1.5 (see [Bha11b]), and this analo ...
Ideals (prime and maximal)
... disjoint from S if the ideal generated by I and any element not in I contains some element of S. Definition. An ideal I ⊂ A is said to be prime if its complement contains 1 and is closed under multiplication. Fact. I is maximal if and only if A/I is a field. I is prime if and only if A/I is a domain ...
... disjoint from S if the ideal generated by I and any element not in I contains some element of S. Definition. An ideal I ⊂ A is said to be prime if its complement contains 1 and is closed under multiplication. Fact. I is maximal if and only if A/I is a field. I is prime if and only if A/I is a domain ...
AI{D RELATED SPACES
... clopen. Every clopen subset of a topological space is C-embedded and every C*embedded subspace of a strongly O-dimensional space is strongly 0-dimensional. Thus each component of X is connected and strongly O-dimensional, hence ultrapseudocompact, by Theorem 2. (5)"=+(6) is trivial. (6)+(l): By (6), ...
... clopen. Every clopen subset of a topological space is C-embedded and every C*embedded subspace of a strongly O-dimensional space is strongly 0-dimensional. Thus each component of X is connected and strongly O-dimensional, hence ultrapseudocompact, by Theorem 2. (5)"=+(6) is trivial. (6)+(l): By (6), ...
(pdf)
... and natural transformations introduced by Eilenberg and MacLane in 1945. It was perhaps inevitable that some such language would have appeared eventually. It was certainly not inevitable that such an early systematization would have proven so remarkably durable and appropriate. I talked then about a ...
... and natural transformations introduced by Eilenberg and MacLane in 1945. It was perhaps inevitable that some such language would have appeared eventually. It was certainly not inevitable that such an early systematization would have proven so remarkably durable and appropriate. I talked then about a ...
Classical Period Domains - Stony Brook Mathematics
... symmetric domain. This can be done in terms of standard Lie theory (see [Viv13, §2.1] and the references therein). However, we shall answer this question from the viewpoint of Shimura data. Specifically, we shall replace the Lie group H by an algebraic group G, replace cosets of K by certain homomor ...
... symmetric domain. This can be done in terms of standard Lie theory (see [Viv13, §2.1] and the references therein). However, we shall answer this question from the viewpoint of Shimura data. Specifically, we shall replace the Lie group H by an algebraic group G, replace cosets of K by certain homomor ...
Math 850 Algebra - San Francisco State University
... 10. Find applications of Zorn’s lemma and the Axiom of Choice to group and ring theory and linear algebra. 11. Is the existence of maximal ideals equivalent to Zorn’s lemma? 12. Define the functors Exti and/or T ori and present the basic properties. 13. What is a Gröbner basis for an ideal in a poly ...
... 10. Find applications of Zorn’s lemma and the Axiom of Choice to group and ring theory and linear algebra. 11. Is the existence of maximal ideals equivalent to Zorn’s lemma? 12. Define the functors Exti and/or T ori and present the basic properties. 13. What is a Gröbner basis for an ideal in a poly ...
Automorphisms of 2--dimensional right
... about the automorphism groups of general right-angled Artin groups. In this paper we begin a systematic study of automorphism groups of right-angled Artin groups. We restrict our attention to the case that the defining graph is connected and triangle-free or, equivalently, A is freely indecomposa ...
... about the automorphism groups of general right-angled Artin groups. In this paper we begin a systematic study of automorphism groups of right-angled Artin groups. We restrict our attention to the case that the defining graph is connected and triangle-free or, equivalently, A is freely indecomposa ...
An Introduction to K-theory
... We very briefly summarize the content of each of these six lectures. Lecture 1 introduces low dimensional K-theory, with emphasis on K0 (X), the Grothendieck group of finitely generated projective R-modules for a (commutative) ring R if Spec R = X, of topological vector vector bundles over X if X is ...
... We very briefly summarize the content of each of these six lectures. Lecture 1 introduces low dimensional K-theory, with emphasis on K0 (X), the Grothendieck group of finitely generated projective R-modules for a (commutative) ring R if Spec R = X, of topological vector vector bundles over X if X is ...
Notes on Galois Theory
... For simple extensions, the converse to Lemma 2 is true. In fact, we can say much more. Lemma 3: Let α be an element in an overfield L of a field K. Then: K(α)/K is algebraic ⇔ α is algebraic over K ⇔ K[α] = K(α) ⇔ [K(α) : K] < ∞. Moreover, if α is algebraic over K and f (X) =Irr(α, K), then there ex ...
... For simple extensions, the converse to Lemma 2 is true. In fact, we can say much more. Lemma 3: Let α be an element in an overfield L of a field K. Then: K(α)/K is algebraic ⇔ α is algebraic over K ⇔ K[α] = K(α) ⇔ [K(α) : K] < ∞. Moreover, if α is algebraic over K and f (X) =Irr(α, K), then there ex ...
Polynomial Rings
... common divisor. So the last pair has the same greatest common divisor as the first pair — but the last pair consists of 0 and the last nonzero remainder, so the last nonzero remainder is the greatest common divisor. This process is called the Euclidean algorithm, just as in the case of the integers. ...
... common divisor. So the last pair has the same greatest common divisor as the first pair — but the last pair consists of 0 and the last nonzero remainder, so the last nonzero remainder is the greatest common divisor. This process is called the Euclidean algorithm, just as in the case of the integers. ...