Quadratic reciprocity (after Weil) 1. Standard set-up and Poisson summation
|
|
- Vernon Davis
- 5 years ago
- Views:
Transcription
1 (December 19, 010 Quadratic reciprocity (after Weil Paul Garrett garrett/ I show that over global fields k (characteristic not the quadratic norm residue symbol is a Hecke character, that is, a k -invariant continuous character on the ideles of k. From this reciprocity law one directly obtains the traditional reciprocity laws for quadratic Hilbert symbols, and for quadratic symbols. Striking is the role of certain quadratic exponential functions, as tempered distributions. The archimedean prototype is S x (z = e πix z (x R and z C This argument is suggested by, and essential to, a careful treatment of Weil representations, and proof that theta series are automorphic forms. 1. Standard set-up and Poisson summation. Weil s quadratic exponential distributions 3. Quadratic norm residue symbols and local integrals 4. The reciprocity law for quadratic norm residue symbols 5. Quadratic Hilbert-symbol reciprocity 6. Quadratic reciprocity 7. The simplest examples 1. Standard set-up and Poisson summation This section reviews standard items: measures, convolutions, characters, bilinear forms. We are concerned not with L Fourier inversion, but rather with a Schwartz-Bruhat Fourier inversion, which is easier. Let k be a global field of characteristic not. On each completion k v of k fix a non-trivial additive unitary character ψ v. For global applications, make these choices so that, for all but finitely-many v, ψ v (xy = 1 for all y o v x o v (o v the local ring of integers Further, choose the family of characters so that the global character ψ = v ψ v is trivial on k k A. For a character and (additive Haar measure on k v there is a local Fourier transform F f(x = ˆf(x = Take the Haar measure so that Fourier inversion holds: k v f (x = f( x f(y ψ v (xy dy The aggregate of local measures should make the total measure of the quotient A k /k be 1. Let K be either a quadratic field extension of k or isomorphic to k k. In either case let σ be the non-trivial k-algebra automorphism of K. Define a k-valued k-bilinear form, on K by α, β = tr K/k (αβ σ (with tr K/k (α = α + α σ 1
2 Paul Garrett: Quadratic reciprocity (after Weil (December 19, 010 Extend this k v -linearly to a k v -valued k v -bilinear form, on = K k k v Give the spaces additive compatible Haar measures, such that Fourier Inversion holds locally everywhere, with respect to the pairing a b ψ a, b Locally and globally the convolution on Schwartz-Bruhat functions is (f ϕ(a = f(a b ϕ(b db The groups are abelian, so convolution is commutative: f ϕ = ϕ f For a Schwartz-Bruhat function f on K A, Poisson summation is f(x = f(x x K x K Poisson summation is equivalent to the assertion that the tempered distribution u defined by u(f = f(x x K is its own Fourier transform. This may be proven by proving the one-dimensionality of the space of distributions supported on K and annihilated by multiplication by all the functions x ψ ξ, x (for ξ K Fourier transform exchanges these conditions, from which follows the Poisson formula up to a constant. Our normalization of measure makes the constant be 1. [1.0.1] Lemma: (p-adic version For Schwartz-Bruhat f on and smooth ϕ on, F (ϕf = F ϕ F f where F ϕ is the Fourier transform of the tempered distribution ϕ. [1.0.] Note: In the lemma, smooth means locally constant. The analogue of the lemma in the archimedean case is more delicate, since in that case Schwartz-Bruhat and test functions differ, and the notion of moderate growth is needed in addition to smoothness. Proof: (p-adic case The proof is a reduction to the corresponding property for Schwartz-Bruhat functions. Let L b be the (left regular representation operator on Schwartz-Bruhat functions by L b f(a = f(a b For a function h on, let θh(x = h( x
3 Paul Garrett: Quadratic reciprocity (after Weil (December 19, 010 The convolution of a distribution u and a test function f is [1] (u f(a = u(l a θf When the distribution u is integration against a function S, convolution is the usual convolution of functions. WIth ch be the characteristic function of a (large compact open subgroup of ch ϕ ϕ (in the topology of tempered distributions Indeed, for large enough depending upon the Schwartz-Bruhat function f, ϕ(a f(a da = ϕ(a f(a da since in the p-adic case f has compact support. Likewise, since the Fourier transform is a topological automorphism of Schwartz-Bruhat functions, lim F (ch u = F u Thus, (F ϕ F f(a = (F ϕ(l a θf f = lim (F (ch ϕ(l a θf f = lim (F (ch ϕ F f(a = lim F (ch ϕf(a = F (ϕf(a using the identity for Schwartz-Bruhat functions. F (α β = F α F β. Weil s quadratic exponential distributions Weil s quadratic exponential functions are introduced, as tempered distributions. contain the germ of the reciprocity law. The first two lemmas For x k v define S x (a = ψ v ( x a, a View this as a tempered distribution, as usual, by identifying it with the integration-against functional f S x (a f(a da [.0.1] Lemma: (p-adic case where F S x γ(x = lim = γ(x S x 1 ψ v ( x a, a da as ranges over larger and larger compact open subgroups of. In fact, there is a large-enough compact open subgroup Y of K so that the limit is reached for any Y. [1] Expression of convolution in terms of left and right regular representations proceeds similarly on any unimodular topological group, by changing variables in the integrals, using the invariance of the measure. For that matter, the precise relationship is often less significant than the fact that there is a relationship. 3
4 Paul Garrett: Quadratic reciprocity (after Weil (December 19, 010 [.0.] Lemma: Let f be a Schwartz-Bruhat function on. For x k v and a (S x f(a = S x (a F (S x f(xa Proof: (of first lemma In the course of the proof, we show that the limit exists in the stronger sense indicated. By the usual definition of Fourier transform of a tempered distribution, F S x (f = S x (F f = S x (a F f(a da v Since F f is also a Schwartz-Bruhat function, the characteristic function ch of any sufficiently large compact open set can be inserted into the integral without affecting its value. Thus, F S x (f = S x (a ch (a F f(a da Then S x ch is itself a Schwartz-Bruhat function, so apply the identity K f 1 (a F f (a da = F f 1 (a f (a da Thus, Since generally given a, = F (S x ch (a = F S x (f = F (S x ch (a f(a da ψ v a, b = ψ v ( a, b σ = ψ v b, a S x (b ψ v a, b db = ψ v ( x b, b 1 a, b 1 b, a db = ψ v ( x b, b a, b db ψ v ( x b x 1 a, b x 1 a x 1 a, a db For large enough (depending upon a, replace b by b + x 1 a to obtain F (S x ch (a = S x 1 a, a ψ v ( x b, b db Thus, as claimed. F S x = S x 1 lim ψ v ( x b, b db Proof: (of second lemma From the definitions, and from we have (f S x (a = (S x f(a = = K This is all we want here. K ψ a, b = ψ b, a S x (a b f(b db = S x (a ψ xa, b S x (b f(b db = S x (a K K ψ ( x a, a x a, b x b, a + x b, b f(b db ψ xa, b S x (b f(b db = S x (a F (S x f(xa 4
5 Paul Garrett: Quadratic reciprocity (after Weil (December 19, Quadratic norm residue symbols and local integrals One more bit of preparation is required. We define the local norm residue symbol ν v : k v {±1} attached to the separable quadratic extension /k v as follows. If = K k k v is not a field, then just put ν v (x = 1 for all x k v. If is a field, put ν v (x = 1 if x is a norm from, otherwise ν v (x = 1. It is standard but non-trivial that the norms from are of index two in k v for a separable quadratic field extension of k v. We invoke this know that ν v is a group homomorphism. As in the previous section, let γ(x = γ v (x = lim S x (a da where ranges over larger and larger compact open subgroups of, and x k v. [3.0.1] Lemma: (p-adic case γ v (x = ν v (x x 1 k v γ v (1 (for x k v Proof: From the definition, γ(x = lim ψ(xaa σ da and the limit is reached for sufficiently large. For x k v of the form x = bb σ, replacing a by ab 1 in the integral gives γ(x = b 1 lim ψ(aa σ da We are using the local norms making the product formula hold, so b x kv = bb σ kv = b Kv Thus, we have the desired formula when x is a local norm. When x is not a local norm, then it must be that is a field, otherwise the local norm map is onto. Let Θ be the subgroup of of elements of norm 1; it is compact. Letting vary over Θ-stable compact open subgroups, γ(x = lim ψ(xaa σ da = lim ψ(xaθθ σ a σ dθ da = lim ψ(xaa σ da Θ\ Θ Θ\ where we give Θ total measure 1. Taking the quotient of K v by the kernel Θ of the norm, is an isomorphism to its image. Note that ϕ : Θ\K v k v (by α αα σ d α = α 1 dα d y = y 1 k v dy are multiplicative Haar measures on and k v, respectively. The (topological isomorphism just above yields an identity γ(x = lim ψ(xaa σ da = lim ψ(xαα σ α Kv d α = lim ψ(xy y kv d y Y Θ\ Θ\ 5 Y
6 Paul Garrett: Quadratic reciprocity (after Weil (December 19, 010 where y = αα σ, = 0, and Y is the image of under the norm map. (Here we choose some compatible normalizations of the measures: it doesn t matter which. Since in this quadratic field extension the norms are of index, lim Y Y ψ(xy d y = lim ψ(xy 1 (1 + ord v(y y kv d y where runs over larger and larger compact open additive subgroups of k v (ignoring the point 0 k v. A typical elementary cancellation argument shows Then lim ψ(xy y kv d y = 0 (for x 0 γ(x = lim ψ(xy 1 ord v(y y kv d y Replace y by yx 1 to obtain the desired identity. 4. Reciprocity law for quadratic norm residue symbols The first reciprocity law is that (quadratic global norm residue symbols x ν K/k (x = Π v ν v (x (with x an idele of k are Hecke characters, that is, are trivial on k. Continuity is clear. Proof: This global assertion needs a global source: Poisson summation. For f an adelic Schwartz-Bruhat function, x k, and an adele a = {a v }, write S x (a = Π v S v x(a v where now S v x(a = ψ v ( x a va σ v Since S x is 1 on K, By Poisson summation, F (S x f(a = f(a = S x (a f(a (F S x F f(a = γ(x (S x 1 F f(a by the first lemma, which computed the Fourier transform of S x as tempered distribution. By the second lemma, which computed S x f, this is γ(x S x 1(a F (S x 1F f(xa = γ(x F (S x 1F f(xa since S x 1 = 1 on K. Change variables in the sum, replacing a by ax 1, to obtain (so far f(a = γ(x F (S x 1F f(a = γ(x S x 1(a F f(a 6
7 Paul Garrett: Quadratic reciprocity (after Weil (December 19, 010 the latter by Poisson summation, and this is = γ(x F f(a = γ(x f(a since S x 1(a = 1, and again applying Poisson summation. Taking any f so that f(a 0 necessarily γ(x = 1 (for all x k Then 1 = γ(x = Π v γ v (x = Π v x 1 k v ν v (xγ v (x = Π v ν v (xγ v (1 = ν(xγ(1 from the product formula and from the earlier result that γ v (x = ν v (xγ v (1 Thus, ν is a Hecke character. 5. Quadratic Hilbert-symbol reciprocity Recall the definition of quadratic Hilbert symbols, and obtain the their reciprocity law from the fact that the norm residue symbol is a Hecke character. For a, b k v the (quadratic Hilbert symbol is (a, b v = 1 1 (when ax + by = z has non-trivial solution in k v (otherwise [5.0.1] Theorem: For a, b k Π v (a, b v = 1 Proof: We prove this from the fact that the quadratic norm residue symbol is a Hecke character. When b (or a is a square in k, the equation ax + by = z has a solution over the global field k, with x = 0. Then there is a solution over k v for all v, so all the Hilbert symbols are all 1. Thus, the reciprocity assertion holds in this case. Suppose that b is not a square in k. Rewrite the equation as where K = k( b is a quadratic field extension of k. ax = z by = Norm K/k (z + y b At a prime v of k split in K, the local extension K k k v is not a field, and the norm is a surjection, so ν v 1 in that case. 7
8 Paul Garrett: Quadratic reciprocity (after Weil (December 19, 010 At a prime v of k not split in K, the local extension K k k v is a field, so ax = z by can have no (non-trivial solution x, y, z even in k v unless x 0. In that case, divide by x and find that a is a norm if and only if this equation has a solution. In summary, the values (a, b v of the Hilbert symbol coincide with the values of the local norm residue symbol ν v (a attached to the local field extension k( b/k. Thus, the reciprocity law for the norm residue symbol gives the corresponding result for the Hilbert symbol. In the proof, we proved a local statement stronger than that required to obtain the reciprocity law: we showed that the quadratic Hilbert symbol (a, b v is the quadratic local norm residue symbol ν v (a attached to the local extension k v ( b, for any non-zero a, b k v. When b is a square in k we must interpret this extension as k v [] mod b 6. Quadratic reciprocity Now we obtain the most traditional quadratic reciprocity law from the reciprocity law for the quadratic Hilbert symbol. We only derive what is often called the main part, refering to non-archimedean odd primes. Inspection of the relation (indicated in the proof of the quadratic symbols to Hilbert symbols will make clear how to obtain the auxiliary parts of quadratic reciprocity. Fix a ring of integers o inside k: for number fields k take the integral closure of in k, and for a function field which is a separable extension of F q ( take the integral closure of F q [] in k. For a (non-archimedean prime v of o, and for x o define the quadratic symbol ( x v = 1 (forx a non-zero square mod v 0 (for x = 0 mod v 1 (for x a non-square mod v For π o generating a prime ideal v, also write ( x π = ( x v A prime v is odd when the cardinality of its residue class field is odd. In the number field case, a prime is infinite when it lies over the real prime of Q. In the function field case, the prime at infinity in F q (T is given by the valuation : P q deg P A prime (i.e., valuation v of a finite separable extension k of F q (T, lying over, is an infinite prime of k. The reciprocity law here is: Quadratic Reciprocity ( main part : Let π and ϖ be two elements of o generating distinct odd prime ideals. Then ( ϖ ( π = Π v (π, ϖ v π ϖ where v runs over all even or infinite primes, and (, v is the (quadratic Hilbert symbol. 8
9 Paul Garrett: Quadratic reciprocity (after Weil (December 19, 010 Quadratic Reciprocity ( supplementary part : Let π o generate an odd prime ideal. For any other element α of o which is a πo-unit, ( α = Π v (π, α v π where v runs over all even or infinite primes and over all primes at which α is not a local unit. The proof of the main part illustrates well-enough the connection, so we omit explicit proof of the supplementary part. Proof: (of main part We claim that, since πo and ϖo are odd primes, (π, ϖ v = ( ϖ π for v = πo ( π ϖ for v = ϖo 1 for v odd and v πo, ϖo Let v = πo. Suppose that there is a solution x, y, z in k v to πx + ϖy = z Then (via the ultrametric property ord v y and ord v z must be identical, and less than ord v x, since ϖ is a v-unit and ord v πx is odd. Then multiply through by π n so that π n y and π n z are v-units. Then we see that ϖ must be a square modulo v. On the other hand, if ϖ is a square modulo v, then we can use Hensel s lemma to infer that ϖ is a square in k v. Then ϖy = z certainly has a non-trivial solution. Further, for v is an odd prime distinct from both πo and ϖo, then both π and ϖ are v-units. If ϖ is a square in k v, then ϖ = z certainly has a solution, so the Hilbert symbol is 1. Suppose ϖ is not a square in k v. Then, k v ( ϖ is an unramified field extension of k v, since v is odd. Thus, the norm map is surjective to units in k v. Thus, there are y, z k v so that π = Norm(z + y ϖ = z ϖy Thus, all but the even prime and infinite prime quadratic Hilbert symbols have interpretations in terms of quadratic symbols. 7. The simplest examples First, let s recover quadratic reciprocity for two (positive odd prime numbers p, q in. We wish to recover the assertion ( ( q p = ( 1 (p 1(q 1/4 p q We have proven so far that ( q p ( p q = (p, q (p, q where (p, q is the -adic Hilbert symbol and (p, q is the Q R Hilbert symbol. 9
10 Paul Garrett: Quadratic reciprocity (after Weil (December 19, 010 Since both p, q are positive, the equation px + qy = z has non-trivial real solutions x, y, z. That is, the real Hilbert symbol (p, q for the archimedean completion of Q has the value 1. Therefore, only the -adic Hilbert symbol contributes to the right-hand side of Gauss formula: so far ( ( q p = (p, q p q Hensel s lemma shows that the solvability of the equation above (for p, q both -adic units depends only upon their residue classes mod 8. The usual formula is but one way of interpolating the -adic Hilbert symbol by elementary-looking formulas. For contrast, let us derive the analogue for F q [T ] with q odd: for distinct monic irreducible polynomials π, ϖ in F q [T ], we have ( ϖ ( π ( (deg π(deg ϖ 1 = π ϖ F q ( 1 where F q is ± depending upon whether 1 is a square in F q or not. So far, from the general assertion of the previous section, where is the prime (valuation ( ϖ π ( π ϖ P q deg P = (π, ϖ This valuation has valuation ring consisting of all rational functions in T which can be written as power series in the local parameter t = T 1. Then π = t deg π (1 + t (... where (1 + t (... is some power series in t. A similar assertion holds for ϖ. Thus, if either degree is even, then one of π, ϖ is a local square, so the Hilbert symbol is +1. If t deg π (1 + t (... is a non-square, then deg π is odd. Nevertheless, any expression of the form is a local square (by Hensel s lemma. contemplating the equation 1 + t (... Thus, without loss of generality for local purposes, we are t (x + y = z The t -order of the right-hand side is even. If there is no 1 in F q, then the left-hand side is t -times a norm from the unramified extension F q ( 1(T = F q (T ( 1 so has odd order. This is impossible. On the other hand if there is a 1 in F q then the equation has non-trivial solutions. Thus, if neither π nor ϖ is a local square (i.e., both are of odd degree, then the Hilbert symbol is 1 if and only if there is a 1 in F q. The formula given above is an elementary interpolation of this assertion (much as was done for the case k = Q. 10
Quadratic reciprocity (after Weil) 1. Standard set-up and Poisson summation
(September 17, 010) Quadratic reciprocity (after Weil) Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ I show that over global fields (characteristic not ) the quadratic norm residue
More informationChapter 8. P-adic numbers. 8.1 Absolute values
Chapter 8 P-adic numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.
More informationMath 676. A compactness theorem for the idele group. and by the product formula it lies in the kernel (A K )1 of the continuous idelic norm
Math 676. A compactness theorem for the idele group 1. Introduction Let K be a global field, so K is naturally a discrete subgroup of the idele group A K and by the product formula it lies in the kernel
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 informationPublic-key Cryptography: Theory and Practice
Public-key Cryptography Theory and Practice Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Chapter 2: Mathematical Concepts Divisibility Congruence Quadratic Residues
More informationAbsolute Values and Completions
Absolute Values and Completions B.Sury This article is in the nature of a survey of the theory of complete fields. It is not exhaustive but serves the purpose of familiarising the readers with the basic
More informationFactorization of unitary representations of adele groups Paul Garrett garrett/
(February 19, 2005) Factorization of unitary representations of adele groups Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ The result sketched here is of fundamental importance in
More informationA BRIEF INTRODUCTION TO LOCAL FIELDS
A BRIEF INTRODUCTION TO LOCAL FIELDS TOM WESTON The purpose of these notes is to give a survey of the basic Galois theory of local fields and number fields. We cover much of the same material as [2, Chapters
More informationp-adic fields Chapter 7
Chapter 7 p-adic fields In this chapter, we study completions of number fields, and their ramification (in particular in the Galois case). We then look at extensions of the p-adic numbers Q p and classify
More informationthen it is called a non-archimedean absolute value. If condition (1.1.2) fails for some x, y F, then is called an archimedean absolute value.
CHAPTER I ADELES OVER Q 1.1 Absolute values Definition 1.1.1 (Absolute value) An absolute value on a field F is a nonnegative real valued function on F which satisfies the conditions: (i) x = 0 if and
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 informationProof of a simple case of the Siegel-Weil formula. 1. Weil/oscillator representations
(March 6, 2014) Proof of a simple case of the Siegel-Weil formula Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ First, I confess I never understood Siegel s arguments for his mass
More information= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2
8. p-adic numbers 8.1. Motivation: Solving x 2 a (mod p n ). Take an odd prime p, and ( an) integer a coprime to p. Then, as we know, x 2 a (mod p) has a solution x Z iff = 1. In this case we can suppose
More informationNOTES ON DIOPHANTINE APPROXIMATION
NOTES ON DIOPHANTINE APPROXIMATION Jan-Hendrik Evertse January 29, 200 9 p-adic Numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics
More information(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d
The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers
More informationLECTURE 2. Hilbert Symbols
LECTURE 2 Hilbert Symbols Let be a local field over Q p (though any local field suffices) with char() 2. Note that this includes fields over Q 2, since it is the characteristic of the field, and not the
More informationa = a i 2 i a = All such series are automatically convergent with respect to the standard norm, but note that this representation is not unique: i<0
p-adic Numbers K. Sutner v0.4 1 Modular Arithmetic rings integral domains integers gcd, extended Euclidean algorithm factorization modular numbers add Lemma 1.1 (Chinese Remainder Theorem) Let a b. Then
More informationTheorem 5.3. Let E/F, E = F (u), be a simple field extension. Then u is algebraic if and only if E/F is finite. In this case, [E : F ] = deg f u.
5. Fields 5.1. Field extensions. Let F E be a subfield of the field E. We also describe this situation by saying that E is an extension field of F, and we write E/F to express this fact. If E/F is a field
More informationAbelian topological groups and (A/k) k. 1. Compact-discrete duality
(December 21, 2010) Abelian topological groups and (A/k) k Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ 1. Compact-discrete duality 2. (A/k) k 3. Appendix: compact-open topology
More information1. Statement of the theorem
[Draft] (August 9, 2005) The Siegel-Weil formula in the convergent range Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ We give a very simple, mostly local, argument for the equality
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
More informationMA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2018 57 5. p-adic Numbers 5.1. Motivating examples. We all know that 2 is irrational, so that 2 is not a square in the rational field Q, but that we can
More informationLecture Notes: Tate s thesis
Lecture Notes: Tate s thesis September 9, 2 Motivation To prove the analytic continuation of the Riemann zeta function (85), we start with the Gamma function: Γ(s) Substitute: Γ(s/2) And now put πn 2 x
More informationφ(xy) = (xy) n = x n y n = φ(x)φ(y)
Groups 1. (Algebra Comp S03) Let A, B and C be normal subgroups of a group G with A B. If A C = B C and AC = BC then prove that A = B. Let b B. Since b = b1 BC = AC, there are a A and c C such that b =
More informationMath 120 HW 9 Solutions
Math 120 HW 9 Solutions June 8, 2018 Question 1 Write down a ring homomorphism (no proof required) f from R = Z[ 11] = {a + b 11 a, b Z} to S = Z/35Z. The main difficulty is to find an element x Z/35Z
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #24 12/03/2013
18.78 Introduction to Arithmetic Geometry Fall 013 Lecture #4 1/03/013 4.1 Isogenies of elliptic curves Definition 4.1. Let E 1 /k and E /k be elliptic curves with distinguished rational points O 1 and
More informationx = π m (a 0 + a 1 π + a 2 π ) where a i R, a 0 = 0, m Z.
ALGEBRAIC NUMBER THEORY LECTURE 7 NOTES Material covered: Local fields, Hensel s lemma. Remark. The non-archimedean topology: Recall that if K is a field with a valuation, then it also is a metric space
More informationarxiv:math/ v1 [math.nt] 21 Sep 2004
arxiv:math/0409377v1 [math.nt] 21 Sep 2004 ON THE GCD OF AN INFINITE NUMBER OF INTEGERS T. N. VENKATARAMANA Introduction In this paper, we consider the greatest common divisor (to be abbreviated gcd in
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 information15 Elliptic curves and Fermat s last theorem
15 Elliptic curves and Fermat s last theorem Let q > 3 be a prime (and later p will be a prime which has no relation which q). Suppose that there exists a non-trivial integral solution to the Diophantine
More information08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms
(February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops
More informationOn The Weights of Binary Irreducible Cyclic Codes
On The Weights of Binary Irreducible Cyclic Codes Yves Aubry and Philippe Langevin Université du Sud Toulon-Var, Laboratoire GRIM F-83270 La Garde, France, {langevin,yaubry}@univ-tln.fr, WWW home page:
More informationElliptic Curves over p-adic Fields
Elliptic Curves over p-adic Fields Robert L. Benedetto Amherst College University of Connecticut Saturday, May 17, 2014 Quick review of p-adic numbers The p-adic absolute value p on Q has 0 p = 0 and r
More informationFILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.
FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0
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 informationIntroduction to L-functions I: Tate s Thesis
Introduction to L-functions I: Tate s Thesis References: - J. Tate, Fourier analysis in number fields and Hecke s zeta functions, in Algebraic Number Theory, edited by Cassels and Frohlich. - S. Kudla,
More informationFebruary 1, 2005 INTRODUCTION TO p-adic NUMBERS. 1. p-adic Expansions
February 1, 2005 INTRODUCTION TO p-adic NUMBERS JASON PRESZLER 1. p-adic Expansions The study of p-adic numbers originated in the work of Kummer, but Hensel was the first to truly begin developing the
More informationSelected exercises from Abstract Algebra by Dummit and Foote (3rd edition).
Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition). Bryan Félix Abril 12, 2017 Section 14.2 Exercise 3. Determine the Galois group of (x 2 2)(x 2 3)(x 2 5). Determine all the subfields
More information2 Garrett: `A Good Spectral Theorem' 1. von Neumann algebras, density theorem The commutant of a subring S of a ring R is S 0 = fr 2 R : rs = sr; 8s 2
1 A Good Spectral Theorem c1996, Paul Garrett, garrett@math.umn.edu version February 12, 1996 1 Measurable Hilbert bundles Measurable Banach bundles Direct integrals of Hilbert spaces Trivializing Hilbert
More informationTC10 / 3. Finite fields S. Xambó
TC10 / 3. Finite fields S. Xambó The ring Construction of finite fields The Frobenius automorphism Splitting field of a polynomial Structure of the multiplicative group of a finite field Structure of the
More informationEXTENSIONS OF ABSOLUTE VALUES
CHAPTER III EXTENSIONS OF ABSOLUTE VALUES 1. Norm and Trace Let k be a field and E a vector space of dimension N over k. We write End k (E) for the ring of k linear endomorphisms of E and Aut k (E) = End
More informationDefinability in fields Lecture 2: Defining valuations
Definability in fields Lecture 2: Defining valuations 6 February 2007 Model Theory and Computable Model Theory Gainesville, Florida Defining Z in Q Theorem (J. Robinson) Th(Q) is undecidable. Defining
More informationPart 1. For any A-module, let M[x] denote the set of all polynomials in x with coefficients in M, that is to say expressions of the form
Commutative Algebra Homework 3 David Nichols Part 1 Exercise 2.6 For any A-module, let M[x] denote the set of all polynomials in x with coefficients in M, that is to say expressions of the form m 0 + m
More informationHARTSHORNE EXERCISES
HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing
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 informationMath 210B. Artin Rees and completions
Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an A-module. In class we defined the I-adic completion of M to be M = lim M/I n M. We will soon show
More information9. Integral Ring Extensions
80 Andreas Gathmann 9. Integral ing Extensions In this chapter we want to discuss a concept in commutative algebra that has its original motivation in algebra, but turns out to have surprisingly many applications
More information1 Absolute values and discrete valuations
18.785 Number theory I Lecture #1 Fall 2015 09/10/2015 1 Absolute values and discrete valuations 1.1 Introduction At its core, number theory is the study of the ring Z and its fraction field Q. Many questions
More informationGalois theory (Part II)( ) Example Sheet 1
Galois theory (Part II)(2015 2016) Example Sheet 1 c.birkar@dpmms.cam.ac.uk (1) Find the minimal polynomial of 2 + 3 over Q. (2) Let K L be a finite field extension such that [L : K] is prime. Show that
More informationc ij x i x j c ij x i y j
Math 48A. Class groups for imaginary quadratic fields In general it is a very difficult problem to determine the class number of a number field, let alone the structure of its class group. However, in
More informationCounting points on elliptic curves: Hasse s theorem and recent developments
Counting points on elliptic curves: Hasse s theorem and recent developments Igor Tolkov June 3, 009 Abstract We introduce the the elliptic curve and the problem of counting the number of points on the
More information1 Adeles over Q. 1.1 Absolute values
1 Adeles over Q 1.1 Absolute values Definition 1.1.1 (Absolute value) An absolute value on a field F is a nonnegative real valued function on F which satisfies the conditions: (i) x = 0 if and only if
More informationAHAHA: Preliminary results on p-adic groups and their representations.
AHAHA: Preliminary results on p-adic groups and their representations. Nate Harman September 16, 2014 1 Introduction and motivation Let k be a locally compact non-discrete field with non-archimedean valuation
More information1. Pseudo-Eisenstein series
(January 4, 202) Spectral Theory for SL 2 (Z)\SL 2 (R)/SO 2 (R) Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ Pseudo-Eisenstein series Fourier-Laplace-Mellin transforms Recollection
More informationProjects on elliptic curves and modular forms
Projects on elliptic curves and modular forms Math 480, Spring 2010 In the following are 11 projects for this course. Some of the projects are rather ambitious and may very well be the topic of a master
More informationWhat is the Langlands program all about?
What is the Langlands program all about? Laurent Lafforgue November 13, 2013 Hua Loo-Keng Distinguished Lecture Academy of Mathematics and Systems Science, Chinese Academy of Sciences This talk is mainly
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 informationORAL QUALIFYING EXAM QUESTIONS. 1. Algebra
ORAL QUALIFYING EXAM QUESTIONS JOHN VOIGHT Below are some questions that I have asked on oral qualifying exams (starting in fall 2015). 1.1. Core questions. 1. Algebra (1) Let R be a noetherian (commutative)
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 informationLecture 8: The Field B dr
Lecture 8: The Field B dr October 29, 2018 Throughout this lecture, we fix a perfectoid field C of characteristic p, with valuation ring O C. Fix an element π C with 0 < π C < 1, and let B denote the completion
More informationAUTOMORPHIC FORMS NOTES, PART I
AUTOMORPHIC FORMS NOTES, PART I DANIEL LITT The goal of these notes are to take the classical theory of modular/automorphic forms on the upper half plane and reinterpret them, first in terms L 2 (Γ \ SL(2,
More informationMore Model Theory Notes
More Model Theory Notes Miscellaneous information, loosely organized. 1. Kinds of Models A countable homogeneous model M is one such that, for any partial elementary map f : A M with A M finite, and any
More information4. Noether normalisation
4. Noether normalisation We shall say that a ring R is an affine ring (or affine k-algebra) if R is isomorphic to a polynomial ring over a field k with finitely many indeterminates modulo an ideal, i.e.,
More informationCDM. Finite Fields. Klaus Sutner Carnegie Mellon University. Fall 2018
CDM Finite Fields Klaus Sutner Carnegie Mellon University Fall 2018 1 Ideals The Structure theorem Where Are We? 3 We know that every finite field carries two apparently separate structures: additive and
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 informationEXERCISES IN MODULAR FORMS I (MATH 726) (2) Prove that a lattice L is integral if and only if its Gram matrix has integer coefficients.
EXERCISES IN MODULAR FORMS I (MATH 726) EYAL GOREN, MCGILL UNIVERSITY, FALL 2007 (1) We define a (full) lattice L in R n to be a discrete subgroup of R n that contains a basis for R n. Prove that L is
More information(3) Let Y be a totally bounded subset of a metric space X. Then the closure Y of Y
() Consider A = { q Q : q 2 2} as a subset of the metric space (Q, d), where d(x, y) = x y. Then A is A) closed but not open in Q B) open but not closed in Q C) neither open nor closed in Q D) both open
More informationON GALOIS GROUPS OF ABELIAN EXTENSIONS OVER MAXIMAL CYCLOTOMIC FIELDS. Mamoru Asada. Introduction
ON GALOIS GROUPS OF ABELIAN ETENSIONS OVER MAIMAL CYCLOTOMIC FIELDS Mamoru Asada Introduction Let k 0 be a finite algebraic number field in a fixed algebraic closure Ω and ζ n denote a primitive n-th root
More informationNumber Theory Fall 2016 Problem Set #6
18.785 Number Theory Fall 2016 Problem Set #6 Description These problems are related to the material covered in Lectures 10-12. Your solutions are to be written up in latex (you can use the latex source
More informationA GENERALIZATION OF DIRICHLET S S-UNIT THEOREM
A GENERALIZATION OF DIRICHLET -UNIT THEOREM PAUL FILI AND ZACHARY MINER Abstract. We generalize Dirichlet s -unit theorem from the usual group of -units of a number field K to the infinite rank group of
More informationA field F is a set of numbers that includes the two numbers 0 and 1 and satisfies the properties:
Byte multiplication 1 Field arithmetic A field F is a set of numbers that includes the two numbers 0 and 1 and satisfies the properties: F is an abelian group under addition, meaning - F is closed under
More informationGarrett Z p. Few explicit parametrizations of algebraic closures of fields are known: not Q, for sure. But we do also know
Garrett 0-2-20 Examples (cont d): Function fields in one variable... as algebraic parallels to Z and Q. Theorem: All finite field extensions of C((X z)) are by adjoining solutions to Y e = X z for e =
More informationLecture 4: Examples of automorphic forms on the unitary group U(3)
Lecture 4: Examples of automorphic forms on the unitary group U(3) Lassina Dembélé Department of Mathematics University of Calgary August 9, 2006 Motivation The main goal of this talk is to show how one
More informationALGEBRA PH.D. QUALIFYING EXAM September 27, 2008
ALGEBRA PH.D. QUALIFYING EXAM September 27, 2008 A passing paper consists of four problems solved completely plus significant progress on two other problems; moreover, the set of problems solved completely
More informationQuaternion algebras over local fields
Chapter 11 Quaternion algebras over local fields In this chapter, we classify quaternion algebras over local fields; this generalizes the classification of quaternion algebras over R. 11.1 Local quaternion
More informationReal Analysis Prelim Questions Day 1 August 27, 2013
Real Analysis Prelim Questions Day 1 August 27, 2013 are 5 questions. TIME LIMIT: 3 hours Instructions: Measure and measurable refer to Lebesgue measure µ n on R n, and M(R n ) is the collection of measurable
More informationFACTORIZATION OF IDEALS
FACTORIZATION OF IDEALS 1. General strategy Recall the statement of unique factorization of ideals in Dedekind domains: Theorem 1.1. Let A be a Dedekind domain and I a nonzero ideal of A. Then there are
More informationMath 2070BC Term 2 Weeks 1 13 Lecture Notes
Math 2070BC 2017 18 Term 2 Weeks 1 13 Lecture Notes Keywords: group operation multiplication associative identity element inverse commutative abelian group Special Linear Group order infinite order cyclic
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationLocal Fields. Chapter Absolute Values and Discrete Valuations Definitions and Comments
Chapter 9 Local Fields The definition of global field varies in the literature, but all definitions include our primary source of examples, number fields. The other fields that are of interest in algebraic
More informationSolutions 13 Paul Garrett garrett/
(March 1, 005) Solutions 13 Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [13.1] Determine the degree of Q( 65 + 56i) over Q, where i = 1. We show that 65 + 56i is not a suare in
More informationRepresentations of moderate growth Paul Garrett 1. Constructing norms on groups
(December 31, 2004) Representations of moderate growth Paul Garrett Representations of reductive real Lie groups on Banach spaces, and on the smooth vectors in Banach space representations,
More information1. If 1, ω, ω 2, -----, ω 9 are the 10 th roots of unity, then (1 + ω) (1 + ω 2 ) (1 + ω 9 ) is A) 1 B) 1 C) 10 D) 0
4 INUTES. If, ω, ω, -----, ω 9 are the th roots of unity, then ( + ω) ( + ω ) ----- ( + ω 9 ) is B) D) 5. i If - i = a + ib, then a =, b = B) a =, b = a =, b = D) a =, b= 3. Find the integral values for
More informationA GRAPHICAL REPRESENTATION OF RINGS VIA AUTOMORPHISM GROUPS
A GRAPHICAL REPRESENTATION OF RINGS VIA AUTOMORPHISM GROUPS N. MOHAN KUMAR AND PRAMOD K. SHARMA Abstract. Let R be a commutative ring with identity. We define a graph Γ Aut R (R) on R, with vertices elements
More informationUltraproducts of Finite Groups
Ultraproducts of Finite Groups Ben Reid May 11, 010 1 Background 1.1 Ultrafilters Let S be any set, and let P (S) denote the power set of S. We then call ψ P (S) a filter over S if the following conditions
More informationThe L-series Attached to a CM Elliptic Curve
The L-series Attached to a CM Elliptic Curve Corina E. Pǎtraşcu patrascu@fas.harvard.edu May 19, 2005 Abstract In this paper we present the L-series attached to an elliptic curve with complex multiplication.
More informationLecture 7: Etale Fundamental Group - Examples
Lecture 7: Etale Fundamental Group - Examples October 15, 2014 In this lecture our only goal is to give lots of examples of etale fundamental groups so that the reader gets some feel for them. Some of
More informationPart II. Number Theory. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section I 1G 70 Explain what is meant by an Euler pseudoprime and a strong pseudoprime. Show that 65 is an Euler
More informationA SIMPLE PROOF OF KRONECKER-WEBER THEOREM. 1. Introduction. The main theorem that we are going to prove in this paper is the following: Q ab = Q(ζ n )
A SIMPLE PROOF OF KRONECKER-WEBER THEOREM NIZAMEDDIN H. ORDULU 1. Introduction The main theorem that we are going to prove in this paper is the following: Theorem 1.1. Kronecker-Weber Theorem Let K/Q be
More informationOVERVIEW OF TATE S THESIS
OVERVIEW O TATE S THESIS ALEX MINE Abstract. This paper gies an oeriew of the main idea of John Tate s 1950 PhD thesis. I will explain the methods he used without going into too much technical detail.
More informationAlgebraic Number Theory
TIFR VSRP Programme Project Report Algebraic Number Theory Milind Hegde Under the guidance of Prof. Sandeep Varma July 4, 2015 A C K N O W L E D G M E N T S I would like to express my thanks to TIFR for
More information(January 14, 2009) q n 1 q d 1. D = q n = q + d
(January 14, 2009) [10.1] Prove that a finite division ring D (a not-necessarily commutative ring with 1 in which any non-zero element has a multiplicative inverse) is commutative. (This is due to Wedderburn.)
More informationThe Kronecker-Weber Theorem
The Kronecker-Weber Theorem November 30, 2007 Let us begin with the local statement. Theorem 1 Let K/Q p be an abelian extension. Then K is contained in a cyclotomic extension of Q p. Proof: We give the
More informationABSOLUTE VALUES AND VALUATIONS
ABSOLUTE VALUES AND VALUATIONS YIFAN WU, wuyifan@umich.edu Abstract. We introduce the basis notions, properties and results of absolute values, valuations, discrete valuation rings and higher unit groups.
More informationQuasi-reducible Polynomials
Quasi-reducible Polynomials Jacques Willekens 06-Dec-2008 Abstract In this article, we investigate polynomials that are irreducible over Q, but are reducible modulo any prime number. 1 Introduction Let
More informationTORSION AND TAMAGAWA NUMBERS
TORSION AND TAMAGAWA NUMBERS DINO LORENZINI Abstract. Let K be a number field, and let A/K be an abelian variety. Let c denote the product of the Tamagawa numbers of A/K, and let A(K) tors denote the finite
More informationAlgebraic Cryptography Exam 2 Review
Algebraic Cryptography Exam 2 Review You should be able to do the problems assigned as homework, as well as problems from Chapter 3 2 and 3. You should also be able to complete the following exercises:
More informationNorm Groups with Tame Ramification
ECTURE 3 orm Groups with Tame Ramification et K be a field with char(k) 6=. Then K /(K ) ' {continuous homomorphisms Gal(K) Z/Z} ' {degree étale algebras over K} which is dual to our original statement
More informationPoints of Finite Order
Points of Finite Order Alex Tao 23 June 2008 1 Points of Order Two and Three If G is a group with respect to multiplication and g is an element of G then the order of g is the minimum positive integer
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 information