A non-archimedean Ax-Lindemann theorem - IMJ-PRG
... We shall use the fact that if Z is a rigid subanalytic subset of Cnp , then Z(F ) = Z ∩ Qnp is a subanalytic subset of Qnp . Indeed, Z can be defined by a quantifierfree formula of the above-mentioned variant of Lipshitz’s analytic language, and our claim follows from the very definition of this lan ...
... We shall use the fact that if Z is a rigid subanalytic subset of Cnp , then Z(F ) = Z ∩ Qnp is a subanalytic subset of Qnp . Indeed, Z can be defined by a quantifierfree formula of the above-mentioned variant of Lipshitz’s analytic language, and our claim follows from the very definition of this lan ...
Hopfian $\ell $-groups, MV-algebras and AF C $^* $
... investigated in Malcev’s book [29, p. 60], with the name “finite approximability”. The following result generalizes a group-theoretic theorem due to Malcev [28]: Theorem 1.1. [20, Theorem 1], [29, Lemma 6, p. 287] Every finitely generated residually finite algebra R is hopfian, meaning that every ho ...
... investigated in Malcev’s book [29, p. 60], with the name “finite approximability”. The following result generalizes a group-theoretic theorem due to Malcev [28]: Theorem 1.1. [20, Theorem 1], [29, Lemma 6, p. 287] Every finitely generated residually finite algebra R is hopfian, meaning that every ho ...
Towards a p-adic theory of harmonic weak Maass forms
... Let X be a nonsingular algebraic variety defined over a field k of characteristic 0 (by this we mean a smooth integral scheme of finite type over Spec(k)). If F is a sheaf on X and U ⊂ X is a Zariski open subset we denote by F(U ) the sections of F over U and by H i (X, −) the usual sheaf cohomology ...
... Let X be a nonsingular algebraic variety defined over a field k of characteristic 0 (by this we mean a smooth integral scheme of finite type over Spec(k)). If F is a sheaf on X and U ⊂ X is a Zariski open subset we denote by F(U ) the sections of F over U and by H i (X, −) the usual sheaf cohomology ...
Introductory Number Theory
... give us a kind of “Mickey Mouse” image of the integers (lumping together many integers into one residue class, thus losing information), with the advantage that the resulting structure Z/nZ has only finitely many elements. This means that all sorts of questions that are difficult to answer with resp ...
... give us a kind of “Mickey Mouse” image of the integers (lumping together many integers into one residue class, thus losing information), with the advantage that the resulting structure Z/nZ has only finitely many elements. This means that all sorts of questions that are difficult to answer with resp ...
upper half plane being filled with air and the lower... Math S21a: Multivariable calculus
... Two points P = (a, b, c) and Q = (x, y, z) in space define a vector !v = !x − a, y − b − z − c#. It points from P to Q and we write also !v = P!Q. The real numbers numbers p, q, r in a vector !v = !p, q, r# are called the components of !v . Vectors can be drawn everywhere in space but two vectors wi ...
... Two points P = (a, b, c) and Q = (x, y, z) in space define a vector !v = !x − a, y − b − z − c#. It points from P to Q and we write also !v = P!Q. The real numbers numbers p, q, r in a vector !v = !p, q, r# are called the components of !v . Vectors can be drawn everywhere in space but two vectors wi ...
MATH 436 Notes: Finitely generated Abelian groups.
... Example 2.10. We have previously seen that any nontrivial subgroup H of Z1 = Z is of the form dZ for a nonnegative integer d ≥ 1 and is hence itself free Abelian of rank 1 with basis {d}. This led to the classification of all one-generated Abelian groups as the cyclic groups Z or Z/dZ for d ≥ 1. No ...
... Example 2.10. We have previously seen that any nontrivial subgroup H of Z1 = Z is of the form dZ for a nonnegative integer d ≥ 1 and is hence itself free Abelian of rank 1 with basis {d}. This led to the classification of all one-generated Abelian groups as the cyclic groups Z or Z/dZ for d ≥ 1. No ...
RIMS-1791 A pro-l version of the congruence subgroup problem for
... Curves: Let k be a field and (g, r) a pair of nonnegative integers. Then we shall say that a scheme X over k is a curve of type (g, r) over k if there exist a scheme X cpt which is of dimension 1, smooth, proper, geometrically connected over k of genus g and a closed subscheme D ⊆ X cpt which is fin ...
... Curves: Let k be a field and (g, r) a pair of nonnegative integers. Then we shall say that a scheme X over k is a curve of type (g, r) over k if there exist a scheme X cpt which is of dimension 1, smooth, proper, geometrically connected over k of genus g and a closed subscheme D ⊆ X cpt which is fin ...
Complex Analysis on Riemann Surfaces
... To see the first two surfaces are the same, we use the uniformization theorem b Now Z has symmetry group S4 , and the obvious rotation of to see Z ∼ = C. order 4 means we can take B equal to 0, ∞ and the 4th roots of unity. Then the uniqueness of branched coverings shows the result is isomorphic to ...
... To see the first two surfaces are the same, we use the uniformization theorem b Now Z has symmetry group S4 , and the obvious rotation of to see Z ∼ = C. order 4 means we can take B equal to 0, ∞ and the 4th roots of unity. Then the uniqueness of branched coverings shows the result is isomorphic to ...
Full text
... Since Nr(o)j) is an integer, fi must be a multiple of 8, which will turn out to be impossible unless all Cjh are even, a case already excluded. In fact, taking the congruence modulo 2 of the expression between square brackets, we find the condition CjX + Cj2 + Cj3 = 0, j = 1,2,3, where, for at least ...
... Since Nr(o)j) is an integer, fi must be a multiple of 8, which will turn out to be impossible unless all Cjh are even, a case already excluded. In fact, taking the congruence modulo 2 of the expression between square brackets, we find the condition CjX + Cj2 + Cj3 = 0, j = 1,2,3, where, for at least ...
Introduction to Algebraic Number Theory
... 5. Arithmetic geometry: This is a huge field that studies solutions to polynomial equations that lie in arithmetically interesting rings, such as the integers or number fields. A famous major triumph of arithmetic geometry is Faltings’s proof of Mordell’s Conjecture. Theorem 1.3.1 (Faltings). Let X ...
... 5. Arithmetic geometry: This is a huge field that studies solutions to polynomial equations that lie in arithmetically interesting rings, such as the integers or number fields. A famous major triumph of arithmetic geometry is Faltings’s proof of Mordell’s Conjecture. Theorem 1.3.1 (Faltings). Let X ...
7. Divisors Definition 7.1. We say that a scheme X is regular in
... for some quadratic polynomial g(x). If we assume that the characteristic is not three then we may complete the cube to get y 2 = x3 + ax + ab, for some a and b ∈ k. Now any two sets of three collinear points are linearly equivalent (since the equation of one line divided by another line is a rationa ...
... for some quadratic polynomial g(x). If we assume that the characteristic is not three then we may complete the cube to get y 2 = x3 + ax + ab, for some a and b ∈ k. Now any two sets of three collinear points are linearly equivalent (since the equation of one line divided by another line is a rationa ...
![A NOTE ON A THEOREM OF AX 1. Introduction In [1]](http://s1.studyres.com/store/data/002606018_1-a040f94f7e203e575591799a9dc26ca7-300x300.png)
A NOTE ON A THEOREM OF AX 1. Introduction In [1]
... way how to transfer them to positive characteristic. In our statement, the main character is not an algebraic group, but a formal map (which may be though of as an analytic map) between algebraic groups. In this interpretation Kirby’s and Bertrand’s statements are about the exponential map on the Li ...
... way how to transfer them to positive characteristic. In our statement, the main character is not an algebraic group, but a formal map (which may be though of as an analytic map) between algebraic groups. In this interpretation Kirby’s and Bertrand’s statements are about the exponential map on the Li ...
ARIZONA WINTER SCHOOL 2014 COURSE NOTES
... We will discuss heuristics (conjectural answers) for the questions we have considered so far, starting with the class group question. The Cohen-Lenstra heuristics start from the observation that structures often occur in nature with frequency inversely proportional to their number of automorphisms. ...
... We will discuss heuristics (conjectural answers) for the questions we have considered so far, starting with the class group question. The Cohen-Lenstra heuristics start from the observation that structures often occur in nature with frequency inversely proportional to their number of automorphisms. ...
1. Outline of Talk 1 2. The Kummer Exact Sequence 2 3
... Proposition 3.4. Let k be a quasi-algebraically closed field and let G be its absolute Galois group. Then, 1. H 2 (G, (k sep )× ) = 0. 2. For any r > 0, H r (G, M ) = 0 if M is a torsion discrete G-module. 3. For any r > 1, H r (G, M ) = 0 if M is a discrete G-module. The proof involves some difficu ...
... Proposition 3.4. Let k be a quasi-algebraically closed field and let G be its absolute Galois group. Then, 1. H 2 (G, (k sep )× ) = 0. 2. For any r > 0, H r (G, M ) = 0 if M is a torsion discrete G-module. 3. For any r > 1, H r (G, M ) = 0 if M is a discrete G-module. The proof involves some difficu ...
Ordinary forms and their local Galois representations
... is known about the converse except for some results in weight 2. However, it is possible to obtain some information about elliptic modular forms of higher weight, by descent, from Theorem 7. We have: Theorem 8 ([9]). Let p be an odd prime. Let S be the set of p-ordinary elliptic modular forms of wei ...
... is known about the converse except for some results in weight 2. However, it is possible to obtain some information about elliptic modular forms of higher weight, by descent, from Theorem 7. We have: Theorem 8 ([9]). Let p be an odd prime. Let S be the set of p-ordinary elliptic modular forms of wei ...
Publikationen - Mathematisches Institut
... Blanc, Jérémy; Hedén, Isac: The group of Cremona transformations generated by linear maps and the standard involution, in: Annales de l'institut Fourier 65, 2015, H. 6, S. 2641-2680.Blanc, Jérémy; Hedén, Isac: The group of Cremona transformations generated by linear maps and the standard involution, ...
... Blanc, Jérémy; Hedén, Isac: The group of Cremona transformations generated by linear maps and the standard involution, in: Annales de l'institut Fourier 65, 2015, H. 6, S. 2641-2680.Blanc, Jérémy; Hedén, Isac: The group of Cremona transformations generated by linear maps and the standard involution, ...
a theorem in finite protective geometry and some
... will be called identical if they contain the same integers. The order in which these integers are written is, of course, immaterial. Two perfect difference sets will be called equivalent if their reduced perfect difference sets are identical. Two perfect partitions will be called equivalent if their ...
... will be called identical if they contain the same integers. The order in which these integers are written is, of course, immaterial. Two perfect difference sets will be called equivalent if their reduced perfect difference sets are identical. Two perfect partitions will be called equivalent if their ...
Mathematics 6 - Phillips Exeter Academy
... The final blow to your intuition came when you encountered your first inflection point, where it is possible for a curve to lie locally on both sides of its tangent line. Now that you have begun to explore planes tangent to surfaces, you should be ready to meet some more counter-intuitive examples. ...
... The final blow to your intuition came when you encountered your first inflection point, where it is possible for a curve to lie locally on both sides of its tangent line. Now that you have begun to explore planes tangent to surfaces, you should be ready to meet some more counter-intuitive examples. ...
Concrete Algebra - the School of Mathematics, Applied Mathematics
... c. Determinacy of sign: Every integer a has precisely one of the following properties: a > 0, a < 0, a = 0. The law of well ordering: a. Any collection of positive integers has a least element; that is to say, an element a so that every element b satisfies a ≤ b. All of the other arithmetic laws we ...
... c. Determinacy of sign: Every integer a has precisely one of the following properties: a > 0, a < 0, a = 0. The law of well ordering: a. Any collection of positive integers has a least element; that is to say, an element a so that every element b satisfies a ≤ b. All of the other arithmetic laws we ...
RSA Cryptosystem and Factorization
... It was first invented by Pollard to factor integers of the special form n = re + s with small r and |s|. Lenstra, Lenstra, Jr., Manasse, Pollard, Buhler, Pomerance, et al. generalized it to factor any integers. The basic principle of number field sieve is the same for quadratic sieve, namely to find ...
... It was first invented by Pollard to factor integers of the special form n = re + s with small r and |s|. Lenstra, Lenstra, Jr., Manasse, Pollard, Buhler, Pomerance, et al. generalized it to factor any integers. The basic principle of number field sieve is the same for quadratic sieve, namely to find ...
Motivic interpretation of Milnor K
... under the canonical map G(Ow ) → G(F ); Then ∂v (g, h) is invariant under the action of Gal(F/k(v)), so that it belongs to G(k(v)). This definition of ∂v is independent of the choice of L and of the isomorphism from the torus to Gm ⊕n . 2.5 Let r ≥ 0 and s ≥ 0 be integers; let X1 , . . . , Xr be smo ...
... under the canonical map G(Ow ) → G(F ); Then ∂v (g, h) is invariant under the action of Gal(F/k(v)), so that it belongs to G(k(v)). This definition of ∂v is independent of the choice of L and of the isomorphism from the torus to Gm ⊕n . 2.5 Let r ≥ 0 and s ≥ 0 be integers; let X1 , . . . , Xr be smo ...
Exercises - Stanford University
... (51) Show that a curve γ ∈ C(F ) is p-typical if and only if F` γ = 0 for all primes ` 6= p. (52) Suppose R is torsion free and a formal group F over R is defined using a logarithm `F (X) ∈ Frac(R)[[X]]. Show that a curve γ(X) into F is p-typical if and only if all monomials appearing in `F (γ(X)) h ...
... (51) Show that a curve γ ∈ C(F ) is p-typical if and only if F` γ = 0 for all primes ` 6= p. (52) Suppose R is torsion free and a formal group F over R is defined using a logarithm `F (X) ∈ Frac(R)[[X]]. Show that a curve γ(X) into F is p-typical if and only if all monomials appearing in `F (γ(X)) h ...
What is Index Calculus?
... On the ”classical” index calculus Back to the ”classical case”: Let p be a prime number. We consider discrete logarithms in F∗p . Let N′ := {n ∈ N | p ∤ n}. Note that (N′ , ·) is a free abelian monoid on P − {p}. We have a surjective homomorphism of monoids N′ −→ F∗p , N 7→ [N ]p . ...
... On the ”classical” index calculus Back to the ”classical case”: Let p be a prime number. We consider discrete logarithms in F∗p . Let N′ := {n ∈ N | p ∤ n}. Note that (N′ , ·) is a free abelian monoid on P − {p}. We have a surjective homomorphism of monoids N′ −→ F∗p , N 7→ [N ]p . ...
NOTES ON FINITE LINEAR PROJECTIVE PLANES 1. Projective
... As an example, let F be a finite field with n = pk elements, which we denote by the integers 0, 1, . . . , n − 1. For each x 6= 0, let Lx be the n × n array with Lxij = xi + j. Then L1 , . . . , Ln−1 is a set of n − 1 mutually orthogonal latin squares. Note that L1 is the addition table for F, and t ...
... As an example, let F be a finite field with n = pk elements, which we denote by the integers 0, 1, . . . , n − 1. For each x 6= 0, let Lx be the n × n array with Lxij = xi + j. Then L1 , . . . , Ln−1 is a set of n − 1 mutually orthogonal latin squares. Note that L1 is the addition table for F, and t ...