The Riemann-Roch Theorem: a Proof, an Extension, and an Application MATH 780, Spring 2019
|
|
- Reginald Tyler
- 5 years ago
- Views:
Transcription
1 The Riemann-Roch Theorem: a Proof, an Extension, and an Application MATH 780, Spring 2019 This handout continues the notational conentions of the preious one on the Riemann-Roch Theorem, with one slight addition We will denote ideles ie, elements of K A = GL 1 with upper-case Latin letters: A, B, The associated diisors will be denoted using the corresponding fraktur font: A, B, In other words, if A = A is an idele, then the corresponding diisor is Recall that for an idele A = A, A = di A = MK ord A ΛA = {b : b A 1 O for all places MK} We first proe the following Theorem 1: Let K be a function field with field of constants F q Then for all ideles A GL 1 with corresponding diisor A, la = dega + 1 g + dim Fq ΛA + K, where g := dim Fq ΛI + K Proof: We claim that for ideles A and B satisfying ΛA ΛB, dim Fq ΛA/ΛB = dega degb To see why, it suffices to consider the case where A 1 O = B 1 O for all but one place But this case is just the definition of the degree of a place and the order: dim Fq A 1 O /B 1 O = [A 1 = ord B 1 O : B 1 O ] deg ord A 1 deg = ord A ord B deg Next, still assuming ΛA ΛB, from the standard isomorphism theorems and Therefore ΛA/ΛB ΛB = + ΛA K /ΛB ΛB + ΛA K ΛB = ΛA ΛA + K = ΛB + ΛA K ΛB + K ΛA K ΛB ΛA K = ΛA K ΛB K = LA LB 1 dega degb = la lb + dim Fq ΛA + K ΛB + K 1
2 By Lemma 2 of the second handout on the adele ring, there is an idele C such that ΛC + K = Let B be an arbitrary idele and choose an idele A with ΛA = ΛB + ΛC Then ΛA ΛB and by 1 dega degb = la lb + dim Fq ΛB + K In particular, the last summand aboe is finite, so that we may rewrite 1 in the form 2 dega degb = la lb + dim Fq ΛB + K dim Fq ΛA + K Now let A and B be arbitrary ideles and choose an idele D satisfying both ΛD ΛA and ΛD ΛB Using 2 twice yields dega dim Fq ΛA + K + la = degd dim Fq This shows that for any idele A, the quantity dega dim Fq ΛD + K + ld = degb dim Fq ΛA + K + la ΛB + K + lb is the same Now considering the case of the identity idele I with corresponding diisor 0, we get dega dim Fq Rearranging yields Theorem 1 ΛA + K + la = deg0 dim Fq = 1 g ΛI + K + l0 Theorem 2 Riemann s Theorem: The genus g may be characterized as the maximum of dega la + 1 oer all diisors A DiK Further, there is an integer z, depending only on K, such that dega la + 1 = g for all diisors A with dega z Proof: By Theorem 1 we hae dega la + 1 = g dim Fq ΛA + K for all diisors A The first part of Theorem 2 is thus tatamount to the existence of a diisor A with = ΛA + K, which was a preious lemma Now choose a diisor A 0 with = ΛA 0 +K, ie, one which satisfies dega 0 la 0 +1 = g, and set z = dega 0 + g Then for any diisor A with dega z we hae la A 0 dega A g z dega g 1 2
3 Specifically, there is a non-zero α LA A 0 Consider B = A + diα; this diisor is linearly equialent to A and satisfies B A 0 We now hae by 2 and the containment ΛB + K ΛA 0 + K dega la = degb lb = dega 0 la 0 + dim Fq dim Fq ΛA 0 + K ΛB + K dega 0 la 0 = g 1 Recall the notion of the algebraic dual space of a ector space: if V is a ector space oer a field F, then the dual space V consists of the linear transformations from V into F For a finite-dimensional ector space, it s well-known that the space and its dual are isomorphic, whence hae the same dimension The Riemann-Roch Theorem is a consequence of Theorem 1 together with an appropriate realization of / ΛA + K as a dual space We ll take the most direct approach gien what we e already proen possible, but do note that there are many approaches, each with their own attributes Definition: A Weil differential is an F q -linear transformation ω : F q whose kernel contains a subset of the form ΛA + K for some idele A For a gien idele A we denote the collection of Weil differentials anishing on ΛA + K by Ω, and denote the set of all Weil differentials by Ω K It s a triial matter to confirm that Ω K is a subspace of the dual space of iewed as a ector space oer F q in the obious manner In fact, it s clear that Ω is the dual space of / ΛA + K, so that 3 dim Fq Ω = dim Fq ΛA + K Directly from the definition we see that Ω Ω K B wheneer the associated diisors satisfy A B, implying ΛA ΛB We may also iew Ω K as a ector space oer K as follows For α K and ω Ω K, set αω a = ω αa for all a In particular, note that if α 0 and ω anishes on ΛA + K, then αω anishes on ΛαA + K Proposition 1: As a ector space oer K, Ω K has dimension 1 Proof: We first note that Ω K {0} Indeed, by Theorem 1 we must hae a non-zero element of Ω wheneer the associated diisor A has degree less than 1, say Fix a non-zero ω Ω K ; we must show that Ω K = ωk Towards that end, suppose ω Ω K If ω = 0, then ω = 0ω and we re done, so we will now assume ω 0 Since ω, ω Ω K, there are ideles A and A with ω Ω and ω Ω These ideles are certainly not unique Indeed, if ω Ω, then ω Ω K B wheneer B A Neertheless, we may certainly fix/choose such ideles For a gien idele B consider the F q -linear and one-toone maps φ : LA + B Ω K B 1 and φ : LA + B Ω K B 1 gien by φα = αω and φ α = αω Exercise 1: Proe that these maps are, indeed, F q -linear and one-to-one 3
4 We claim that one can choose B such that the images of φ and φ hae non-triial intersection Note that, gien this claim, we then hae α, α K with αω = α ω, so that ω ωω K As for the claim, let z be the quantity in Riemann s Theorem and let B be such that degb max{z dega, z dega, 1, 3g 1 dega dega } Then both degb + A, degb + A z, so by Riemann s Theorem 4 la + B = dega + 1 g + degb, la + B = dega + 1 g + degb On the other hand, by Theorem 1 and 3, Ω K B 1 has dimension 5 dim Fq ΩK B 1 = l B deg B 1 + g = degb 1 + g since deg B = degb 1 Now 4 tells us the dimensions as F q -ector spaces of the respectie images of φ and φ, which when added are larger than the dimension of the codomain Ω K B 1 by 5 and construction Thus these images hae non-triial intersection As remarked in the proof aboe, for any non-zero ω Ω K there are infinitely many ideles A such that ω Ω Howeer, there is a least upper bound of sorts among such ideles/diisors the ordering coming from the inherent partial ordering on the diisor group Proposition 2: Gien any non-zero Weil differential ω, there is a unique diisor, called the diisor of ω and denoted diω, such that ω Ω K diω and for all ideles A with ω Ω we hae A diω For any α K and non-zero ω Ω K, diαω = diα + diω Proof: Let z be the constant in Riemann s Theorem Then a direct consequence of that result and 3 is that Ω = {0} wheneer dega z Thus, there is a diisor diω not necessarily proen to be unique at this point! of maximal degree such that ω Ω K diω We must proe that this diisor is uniquely determined Suppose to the contrary that there is a diisor A diω such that ω Ω Since A diω there is a place 0 MK such that ord 0 A > ord 0 diω Now consider the diisors B := 0 + diω and B = diω Let α ΛB Write α = β + γ where { α if 0, β = 0 if = 0 Then one readily erifies that β ΛB and γ ΛA We now hae ω α = ω β + ω γ = 0, so that ω Ω K B But degb = deg 0 + deg diω > deg diω, contradicting the hypothesis on diω Exercise 2: Complete the proof of Proposition 2 We remark that the definitions imply immediately that Ω consists of those Weil differentials ω satisfying diω A together with 0 Also, Propositions 1 and 2 together show that the following definition makes sense Definition: The canonical class is the equialence class of diisors containing diω for all non-zero ω Ω K 4
5 The last part of our proof of the Riemann-Roch Theorem is the following Theorem 3: Let A DiK and W = diω be a canonical diisor Then the mapping φ: LW A Ω gien by φα = αω is an isomorphism of F q -ector spaces Proof: Let α be a non-zero element of LW A assuming there is one Then by definition diα A W, so that diαω A This clearly implies that αω Ω Now let ω be a non-zero element of Ω By Proposition 1 we must hae ω = αω for some α K This implies that diαω A, so that diα A W, ie, α LW A Exercise 3: Show that φ aboe is one-to-one and F q -linear One can go about things in a similar but somewhat different manner than aboe Consider the topological dual of the locally compact group Gien any locally compact abelian group G, the dual group G consists of continuous homomorphisms called characters of G from G to the unit circle in the complex plane The dual group G inherits a topology from G by using the topology of uniform conergence on compact sets; G is then called the topological dual of G Moreoer, if H is any closed subgroup, the characters of G which induce the triial character on H make up a closed subgroup H of G ; this is called the subgroup associated with H by duality and is isomorphic to the dual of G/H Weil s approach is ia the following Theorem 4: Let V be a finite-dimensional ector space oer a number field or function field K Let χ be a character of, triial on K, and denote the adele ring of V by V A For any V A we hae a unique V A gien by = χ for all V A This determines an isomorphism between VA and V A Moreoer, it maps V to the subgroup of VA associated with the discrete subgroup V We now see how to extend the Riemann-Roch Theorem Suppose K is a function field and consider an element A = A GL n Exercise 4: Proe that A GL n K for all places MK and A O n = O n for almost all places Moreoer, gien such A for all places, A = A GL n Note that the determinant map takes GL n to GL 1, as usual Similar to the onedimensional case, we define and set LA = ΛA K n ΛA = {b KA n : b A 1 O n for all places MK} Exercise 5: Proe that LA is a finite-dimensional ector space oer F q for all A GL n With Exercise 5 in hand, we denote the dimension of LA by la exactly as before The proof of Theorem 1 applies almost word-for-word to get the following Theorem 1+: Let K be a function field with field of constants F q Then for all A GL n la = deg di deta + n1 g + dim Fq K n A ΛA + K n As for the term on the right aboe, ia the isomorphism from K+ΛI 1 to K ΛωI 1, where ω is any idele whose diisor is in the canonical class, one sees that the dual to 5 K n A K n +ΛA is
6 isomorphic to K n ΛωA, where A is the inerse transpose of A GL n We thus arrie at the following extension of the Riemann-Roch Theorem Theorem 5: Let K be a function field and let ω K A canonical class Then for all A GL n we hae la = deg di deta + n1 g + lωa be an idele such that diω is in the Of course, the case n = 1 here is just the usual Riemann-Roch Theorem Recall that we may iew K n A as our analog for Euclidean n-space, and then Kn is the analog of the integer lattice Z n We get a generic lattice ia an element of the general linear group: AK n Here the adelic absolute alue called the adelic module of an idele a GL 1 is a A = q deg dia Thus the determinant of lattice AK n would be q deg di deta Our analog of the unit ball is B A = On Thus LA is precisely the set of lattice points in the unit ball, so that q la is the number of such lattice points This is where the analogy would break down entirely if the field of constants were infinite Theorem 5 tells us exactly how many such lattice points there are Sticking with this identification, the analog of the length of a lattice point would be as follows Gien any A GL n and x K n \ {0}, the adelic length of Ax is gien by Ax A = min { a A : Ax ab A } a GL 1 = min a GL 1 { a A : a 1 Ax B A }, = min a GL 1 { a A : x Λa 1 A} Of course, 0 A is defined to be 0 It s sometimes simpler to deal with the logarithm rather than the length itself, so we will gie it a name: the logarithmic height of the lattice point Ax is h Ax = log q Ax A We can now try our hand at some classic geometry of numbers but in the function field setting The successie minima λ 1 A λ n A are gien by λ i A = min{m : there are i linearly independent x 1,, x i K n with h Ax i m} Minkowski s First Theorem: For any A GL n we hae nλ 1 A ng deg di deta Proof: We ll cheat just a little bit here and assume the image of the degree function on DiK is all of Z which it is Let a be an idele with deg dia = λ 1 A 1 Then by definition laa = 0 Since the far righthand term in Theorem 5 is necessarily non-negatie it s a dimension, after all, we see that 0 = laa deg di detaa + n1 g = deg di deta + n deg dia + 1 g = deg di deta + nλ 1 A ng 6
7 Compare this with the classical result: for any lattice Λ R n, the first minima satisfies λ 1 Λ n VolB 2 n detλ, where VolB is the olume of the unit ball in R n and the successie minima are the actual lengths rather than logarithms of lengths - whence the multiplicatie nature of the statement aboe rather than the additie statement of the function field ersion The 2 n term arises from conexity: B + B = 2B In our case B A + B A = B A, so we don t expect to see this term So what is the olume of B A? Note that we set the determinant of AK n to be q deg di deta In other words, we set K n to hae determinant 1 This means we hae some sort of measure technically a haar measure on KA n where the discete subgroup Kn has coolume 1 This turns out to imply that B A has olume q n1 g Indeed, if we take the haar measure on KA n obtained by setting the measure of each O n to be 1 a reasonable enough choice, then we need to normalize multiply by q n1 g in order for K n to hae coolume 1 Therefore, our function field Minkowski s First Theorem is just like the classical case, but with the 2 n term replaced by q n That term is actually a result of the discrete nature of our lengths as compared to the Euclidean case Indeed, we hae the following Minkowski s First Theorem Alternate Version: We hae AK n B A {0} for all A GL n with deta A < VolB A This is exactly the classical ersion sans the 2 n term, which we don t expect in the function field case as explained aboe Note that AK n B A {0} is equialent to λ 1 A 0 Exercise 6: Assuming the image of deg : DiK Z is all of Z, show that the function field Minkowski s First Theorem is equialent to the Alternate Version 7
OVERVIEW 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 informationBc. Dominik Lachman. Bruhat-Tits buildings
MASTER THESIS Bc. Dominik Lachman Bruhat-Tits buildings Department of Algebra Superisor of the master thesis: Study programme: Study branch: Mgr. Vítězsla Kala, Ph.D. Mathematics Mathematical structure
More informationThe Riemann-Roch Theorem
The Riemann-Roch Theorem In this lecture F/K is an algebraic function field of genus g. Definition For A D F, is called the index of specialty of A. i(a) = dim A deg A + g 1 Definition An adele of F/K
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationRIEMANN SURFACES. max(0, deg x f)x.
RIEMANN SURFACES 10. Weeks 11 12: Riemann-Roch theorem and applications 10.1. Divisors. The notion of a divisor looks very simple. Let X be a compact Riemann surface. A divisor is an expression a x x x
More informationDISCRETE SUBGROUPS, LATTICES, AND UNITS.
DISCRETE SUBGROUPS, LATTICES, AND UNITS. IAN KIMING 1. Discrete subgroups of real vector spaces and lattices. Definitions: A lattice in a real vector space V of dimension d is a subgroup of form: Zv 1
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 informationMAT 445/ INTRODUCTION TO REPRESENTATION THEORY
MAT 445/1196 - INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 Representation Theory of Groups - Algebraic Foundations 1.1 Basic definitions, Schur s Lemma 1.2 Tensor products 1.3 Unitary representations
More informationBalanced Partitions of Vector Sequences
Balanced Partitions of Vector Sequences Imre Bárány Benjamin Doerr December 20, 2004 Abstract Let d,r N and be any norm on R d. Let B denote the unit ball with respect to this norm. We show that any sequence
More information11 Annihilators. Suppose that R, S, and T are rings, that R P S, S Q T, and R U T are bimodules, and finally, that
11 Annihilators. In this Section we take a brief look at the important notion of annihilators. Although we shall use these in only very limited contexts, we will give a fairly general initial treatment,
More information15 Dirichlet s unit theorem
18.785 Number theory I Fall 2017 Lecture #15 10/30/2017 15 Dirichlet s unit theorem Let K be a number field. The two main theorems of classical algebraic number theory are: The class group cl O K is finite.
More informationALGEBRAIC TOPOLOGY IV. Definition 1.1. Let A, B be abelian groups. The set of homomorphisms ϕ: A B is denoted by
ALGEBRAIC TOPOLOGY IV DIRK SCHÜTZ 1. Cochain complexes and singular cohomology Definition 1.1. Let A, B be abelian groups. The set of homomorphisms ϕ: A B is denoted by Hom(A, B) = {ϕ: A B ϕ homomorphism}
More informationAN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES
AN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES DOMINIC L. WYNTER Abstract. We introduce the concepts of divisors on nonsingular irreducible projective algebraic curves, the genus of such a curve,
More informationdifferent formulas, depending on whether or not the vector is in two dimensions or three dimensions.
ectors The word ector comes from the Latin word ectus which means carried. It is best to think of a ector as the displacement from an initial point P to a terminal point Q. Such a ector is expressed as
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 43 RAVI VAKIL CONTENTS 1. Facts we ll soon know about curves 1 1. FACTS WE LL SOON KNOW ABOUT CURVES We almost know enough to say a lot of interesting things about
More informationMath 306 Topics in Algebra, Spring 2013 Homework 7 Solutions
Math 306 Topics in Algebra, Spring 203 Homework 7 Solutions () (5 pts) Let G be a finite group. Show that the function defines an inner product on C[G]. We have Also Lastly, we have C[G] C[G] C c f + c
More informationNOTES ON DIVISORS AND RIEMANN-ROCH
NOTES ON DIVISORS AND RIEMANN-ROCH NILAY KUMAR Recall that due to the maximum principle, there are no nonconstant holomorphic functions on a compact complex manifold. The next best objects to study, as
More informationTATE S THESIS BAPTISTE DEJEAN
TATE S THESIS BAPTISTE DEJEAN Abstract. L-functions are of great interest to number theorists. Key to their study are their meromorphic extensions and functional equations. Hecke defined a class of L-functions
More information1 :: Mathematical notation
1 :: Mathematical notation x A means x is a member of the set A. A B means the set A is contained in the set B. {a 1,..., a n } means the set hose elements are a 1,..., a n. {x A : P } means the set of
More informationTopics in linear algebra
Chapter 6 Topics in linear algebra 6.1 Change of basis I want to remind you of one of the basic ideas in linear algebra: change of basis. Let F be a field, V and W be finite dimensional vector spaces over
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 informationReview of Matrices and Vectors 1/45
Reiew of Matrices and Vectors /45 /45 Definition of Vector: A collection of comple or real numbers, generally put in a column [ ] T "! Transpose + + + b a b a b b a a " " " b a b a Definition of Vector
More informationMath 121 Homework 5: Notes on Selected Problems
Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements
More informationa (b + c) = a b + a c
Chapter 1 Vector spaces In the Linear Algebra I module, we encountered two kinds of vector space, namely real and complex. The real numbers and the complex numbers are both examples of an algebraic structure
More informationPatterns of Non-Simple Continued Fractions
Patterns of Non-Simple Continued Fractions Jesse Schmieg A final report written for the Uniersity of Minnesota Undergraduate Research Opportunities Program Adisor: Professor John Greene March 01 Contents
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationgφ(m) = ω l (g)φ(g 1 m)) where ω l : Γ F l
Global Riemann-Roch formulas Let K be a number field, Γ = Gal( K/K), M a finite Γ-module of exponent m; ie mm = (0) If S is a finite set of places of K we let Γ S = Gal(K S /K), where K S is the union
More information(a + b)c = ac + bc and a(b + c) = ab + ac.
2. R I N G S A N D P O LY N O M I A L S The study of vector spaces and linear maps between them naturally leads us to the study of rings, in particular the ring of polynomials F[x] and the ring of (n n)-matrices
More information1.8 Dual Spaces (non-examinable)
2 Theorem 1715 is just a restatement in terms of linear morphisms of a fact that you might have come across before: every m n matrix can be row-reduced to reduced echelon form using row operations Moreover,
More informationMATH 101: ALGEBRA I WORKSHEET, DAY #3. Fill in the blanks as we finish our first pass on prerequisites of group theory.
MATH 101: ALGEBRA I WORKSHEET, DAY #3 Fill in the blanks as we finish our first pass on prerequisites of group theory 1 Subgroups, cosets Let G be a group Recall that a subgroup H G is a subset that is
More informationTHE BRAUER-MANIN OBSTRUCTION FOR CURVES. December 16, Introduction
THE BRAUER-MANIN OBSTRUCTION FOR CURVES VICTOR SCHARASCHKIN December 16, 1998 1. Introduction Let X be a smooth projectie ariety defined oer a number field K. A fundamental problem in arithmetic geometry
More informationMath 210C. The representation ring
Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let
More informationHolomorphy of the 9th Symmetric Power L-Functions for Re(s) >1. Henry H. Kim and Freydoon Shahidi
IMRN International Mathematics Research Notices Volume 2006, Article ID 59326, Pages 1 7 Holomorphy of the 9th Symmetric Power L-Functions for Res >1 Henry H. Kim and Freydoon Shahidi We proe the holomorphy
More information4.4 Noetherian Rings
4.4 Noetherian Rings Recall that a ring A is Noetherian if it satisfies the following three equivalent conditions: (1) Every nonempty set of ideals of A has a maximal element (the maximal condition); (2)
More informationLemma 1.3. The element [X, X] is nonzero.
Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group
More informationSome consequences of the Riemann-Roch theorem
Some consequences of the Riemann-Roch theorem Proposition Let g 0 Z and W 0 D F be such that for all A D F, dim A = deg A + 1 g 0 + dim(w 0 A). Then g 0 = g and W 0 is a canonical divisor. Proof We have
More informationNORMS EXTREMAL WITH RESPECT TO THE MAHLER MEASURE
NORMS EXTREMAL WITH RESPECT TO THE MAHLER MEASURE PAUL FILI AND ZACHARY MINER Abstract. In this paper, we introduce and study seeral norms which are constructed in order to satisfy an extremal property
More informationFirst we introduce the sets that are going to serve as the generalizations of the scalars.
Contents 1 Fields...................................... 2 2 Vector spaces.................................. 4 3 Matrices..................................... 7 4 Linear systems and matrices..........................
More informationarxiv: v1 [math.co] 25 Apr 2016
On the zone complexity of a ertex Shira Zerbib arxi:604.07268 [math.co] 25 Apr 206 April 26, 206 Abstract Let L be a set of n lines in the real projectie plane in general position. We show that there exists
More informationQuadratic reciprocity (after Weil) 1. Standard set-up and Poisson summation
(December 19, 010 Quadratic reciprocity (after Weil Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ I show that over global fields k (characteristic not the quadratic norm residue symbol
More information8 The Socle and Radical.
8 The Socle and Radical. Simple and semisimple modules are clearly the main building blocks in much of ring theory. Of coure, not every module can be built from semisimple modules, but for many modules
More informationAbstract Vector Spaces and Concrete Examples
LECTURE 18 Abstract Vector Spaces and Concrete Examples Our discussion of linear algebra so far has been devoted to discussing the relations between systems of linear equations, matrices, and vectors.
More informationMath 762 Spring h Y (Z 1 ) (1) h X (Z 2 ) h X (Z 1 ) Φ Z 1. h Y (Z 2 )
Math 762 Spring 2016 Homework 3 Drew Armstrong Problem 1. Yoneda s Lemma. We have seen that the bifunctor Hom C (, ) : C C Set is analogous to a bilinear form on a K-vector space, : V V K. Recall that
More informationThe Inverse Function Theorem
Inerse Function Theorem April, 3 The Inerse Function Theorem Suppose and Y are Banach spaces and f : Y is C (continuousl differentiable) Under what circumstances does a (local) inerse f exist? In the simplest
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 informationMATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous:
MATH 51H Section 4 October 16, 2015 1 Continuity Recall what it means for a function between metric spaces to be continuous: Definition. Let (X, d X ), (Y, d Y ) be metric spaces. A function f : X Y is
More informationLESSON 4: INTEGRATION BY PARTS (I) MATH FALL 2018
LESSON 4: INTEGRATION BY PARTS (I) MATH 6 FALL 8 ELLEN WELD. Integration by Parts We introduce another method for ealuating integrals called integration by parts. The key is the following : () u d = u
More informationPlaces of Number Fields and Function Fields MATH 681, Spring 2018
Places of Number Fields and Function Fields MATH 681, Spring 2018 From now on we will denote the field Z/pZ for a prime p more compactly by F p. More generally, for q a power of a prime p, F q will denote
More informationUnit 11: Vectors in the Plane
135 Unit 11: Vectors in the Plane Vectors in the Plane The term ector is used to indicate a quantity (such as force or elocity) that has both length and direction. For instance, suppose a particle moes
More informationFrom now on we assume that K = K.
Divisors From now on we assume that K = K. Definition The (additively written) free abelian group generated by P F is denoted by D F and is called the divisor group of F/K. The elements of D F are called
More informationTHE TATE MODULE. Seminar: Elliptic curves and the Weil conjecture. Yassin Mousa. Z p
THE TATE MODULE Seminar: Elliptic curves and the Weil conjecture Yassin Mousa Abstract This paper refers to the 10th talk in the seminar Elliptic curves and the Weil conjecture supervised by Prof. Dr.
More informationQuadratic 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 informationMath 144 Activity #9 Introduction to Vectors
144 p 1 Math 144 ctiity #9 Introduction to Vectors Often times you hear people use the words speed and elocity. Is there a difference between the two? If so, what is the difference? Discuss this with your
More informationNORMS EXTREMAL WITH RESPECT TO THE MAHLER MEASURE
NORMS EXTREMAL WITH RESPECT TO THE MAHLER MEASURE PAUL FILI AND ZACHARY MINER Abstract. In a preious paper, the authors introduced seeral ector space norms on the space of algebraic numbers modulo torsion
More informationSimpler form of the trace formula for GL 2 (A)
Simpler form of the trace formula for GL 2 (A Last updated: May 8, 204. Introduction Retain the notations from earlier talks on the trace formula and Jacquet Langlands (especially Iurie s talk and Zhiwei
More informationCover Page. The handle holds various files of this Leiden University dissertation
Cover Page The handle http://hdl.handle.net/1887/32076 holds various files of this Leiden University dissertation Author: Junjiang Liu Title: On p-adic decomposable form inequalities Issue Date: 2015-03-05
More informationV (v i + W i ) (v i + W i ) is path-connected and hence is connected.
Math 396. Connectedness of hyperplane complements Note that the complement of a point in R is disconnected and the complement of a (translated) line in R 2 is disconnected. Quite generally, we claim that
More informationIsogeny invariance of the BSD formula
Isogeny invariance of the BSD formula Bryden Cais August 1, 24 1 Introduction In these notes we prove that if f : A B is an isogeny of abelian varieties whose degree is relatively prime to the characteristic
More informationINVERSE LIMITS AND PROFINITE GROUPS
INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological
More informationGRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.
GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,
More informationSmith theory. Andrew Putman. Abstract
Smith theory Andrew Putman Abstract We discuss theorems of P. Smith and Floyd connecting the cohomology of a simplicial complex equipped with an action of a finite p-group to the cohomology of its fixed
More informationMath 797W Homework 4
Math 797W Homework 4 Paul Hacking December 5, 2016 We work over an algebraically closed field k. (1) Let F be a sheaf of abelian groups on a topological space X, and p X a point. Recall the definition
More informationStat 451: Solutions to Assignment #1
Stat 451: Solutions to Assignment #1 2.1) By definition, 2 Ω is the set of all subsets of Ω. Therefore, to show that 2 Ω is a σ-algebra we must show that the conditions of the definition σ-algebra are
More informationExercise Solutions to Functional Analysis
Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n
More information10. Noether Normalization and Hilbert s Nullstellensatz
10. Noether Normalization and Hilbert s Nullstellensatz 91 10. Noether Normalization and Hilbert s Nullstellensatz In the last chapter we have gained much understanding for integral and finite ring extensions.
More informationMath 396. Quotient spaces
Math 396. Quotient spaces. Definition Let F be a field, V a vector space over F and W V a subspace of V. For v, v V, we say that v v mod W if and only if v v W. One can readily verify that with this definition
More informationA Version of the Grothendieck Conjecture for p-adic Local Fields
A Version of the Grothendieck Conjecture for p-adic Local Fields by Shinichi MOCHIZUKI* Section 0: Introduction The purpose of this paper is to prove an absolute version of the Grothendieck Conjecture
More informationDIVISOR THEORY ON TROPICAL AND LOG SMOOTH CURVES
DIVISOR THEORY ON TROPICAL AND LOG SMOOTH CURVES MATTIA TALPO Abstract. Tropical geometry is a relatively new branch of algebraic geometry, that aims to prove facts about algebraic varieties by studying
More informationTopics in Representation Theory: Roots and Weights
Topics in Representation Theory: Roots and Weights 1 The Representation Ring Last time we defined the maximal torus T and Weyl group W (G, T ) for a compact, connected Lie group G and explained that our
More informationand this makes M into an R-module by (1.2). 2
1. Modules Definition 1.1. Let R be a commutative ring. A module over R is set M together with a binary operation, denoted +, which makes M into an abelian group, with 0 as the identity element, together
More informationRoberto s Notes on Linear Algebra Chapter 1: Geometric vectors Section 8. The dot product
Roberto s Notes on Linear Algebra Chapter 1: Geometric ectors Section 8 The dot product What you need to know already: What a linear combination of ectors is. What you can learn here: How to use two ectors
More informationSection 1.7. Linear Independence
Section 1.7 Linear Independence Motiation Sometimes the span of a set of ectors is smaller than you expect from the number of ectors. Span{, w} w Span{u,, w} w u This means you don t need so many ectors
More informationThe Riemann Roch Theorem
The Riemann Roch Theorem Well, a Riemann surface is a certain kind of Hausdorf space You know what a Hausdorf space is, don t you? Its also compact, ok I guess it is also a manifold Surely you know what
More informationSection II.1. Free Abelian Groups
II.1. Free Abelian Groups 1 Section II.1. Free Abelian Groups Note. This section and the next, are independent of the rest of this chapter. The primary use of the results of this chapter is in the proof
More information4-vectors. Chapter Definition of 4-vectors
Chapter 12 4-ectors Copyright 2004 by Daid Morin, morin@physics.harard.edu We now come to a ery powerful concept in relatiity, namely that of 4-ectors. Although it is possible to derie eerything in special
More informationREPRESENTATION THEORY OF S n
REPRESENTATION THEORY OF S n EVAN JENKINS Abstract. These are notes from three lectures given in MATH 26700, Introduction to Representation Theory of Finite Groups, at the University of Chicago in November
More informationEIGENVALUES AND EIGENVECTORS
CHAPTER 6 EIGENVALUES AND EIGENVECTORS SECTION 6. INTRODUCTION TO EIGENVALUES In each of Problems we first list the characteristic polynomial p( λ) A λi of the gien matrix A and then the roots of p( λ
More informationMath 231b Lecture 16. G. Quick
Math 231b Lecture 16 G. Quick 16. Lecture 16: Chern classes for complex vector bundles 16.1. Orientations. From now on we will shift our focus to complex vector bundles. Much of the theory for real vector
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 informationwhere m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism
8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the
More informationMath 213br HW 12 solutions
Math 213br HW 12 solutions May 5 2014 Throughout X is a compact Riemann surface. Problem 1 Consider the Fermat quartic defined by X 4 + Y 4 + Z 4 = 0. It can be built from 12 regular Euclidean octagons
More informationLinear and Bilinear Algebra (2WF04) Jan Draisma
Linear and Bilinear Algebra (2WF04) Jan Draisma CHAPTER 3 The minimal polynomial and nilpotent maps 3.1. Minimal polynomial Throughout this chapter, V is a finite-dimensional vector space of dimension
More informationMath 210C. A non-closed commutator subgroup
Math 210C. A non-closed commutator subgroup 1. Introduction In Exercise 3(i) of HW7 we saw that every element of SU(2) is a commutator (i.e., has the form xyx 1 y 1 for x, y SU(2)), so the same holds for
More informationLecture 29: Free modules, finite generation, and bases for vector spaces
Lecture 29: Free modules, finite generation, and bases for vector spaces Recall: 1. Universal property of free modules Definition 29.1. Let R be a ring. Then the direct sum module is called the free R-module
More information1 Fields and vector spaces
1 Fields and vector spaces In this section we revise some algebraic preliminaries and establish notation. 1.1 Division rings and fields A division ring, or skew field, is a structure F with two binary
More informationAlgebra I Fall 2007
MIT OpenCourseWare http://ocw.mit.edu 18.701 Algebra I Fall 007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.701 007 Geometry of the Special Unitary
More informationExercises on chapter 0
Exercises on chapter 0 1. A partially ordered set (poset) is a set X together with a relation such that (a) x x for all x X; (b) x y and y x implies that x = y for all x, y X; (c) x y and y z implies that
More informationMath 425 Lecture 1: Vectors in R 3, R n
Math 425 Lecture 1: Vectors in R 3, R n Motiating Questions, Problems 1. Find the coordinates of a regular tetrahedron with center at the origin and sides of length 1. 2. What is the angle between the
More informationThe most important result in this section is undoubtedly the following theorem.
28 COMMUTATIVE ALGEBRA 6.4. Examples of Noetherian rings. So far the only rings we can easily prove are Noetherian are principal ideal domains, like Z and k[x], or finite. Our goal now is to develop theorems
More informationTHE DESCRIPTIVE COMPLEXITY OF SERIES REARRANGEMENTS
THE DESCRIPTIVE COMPLEXITY OF SERIES REARRANGEMENTS MICHAEL P. COHEN Abstract. We consider the descriptie complexity of some subsets of the infinite permutation group S which arise naturally from the classical
More informationTopics in Representation Theory: Fourier Analysis and the Peter Weyl Theorem
Topics in Representation Theory: Fourier Analysis and the Peter Weyl Theorem 1 Fourier Analysis, a review We ll begin with a short review of simple facts about Fourier analysis, before going on to interpret
More informationDualities in Mathematics: Locally compact abelian groups
Dualities in Mathematics: Locally compact abelian groups Part III: Pontryagin Duality Prakash Panangaden 1 1 School of Computer Science McGill University Spring School, Oxford 20-22 May 2014 Recall Gelfand
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then
More informationPresentation 1
18.704 Presentation 1 Jesse Selover March 5, 2015 We re going to try to cover a pretty strange result. It might seem unmotivated if I do a bad job, so I m going to try to do my best. The overarching theme
More informationi=1 β i,i.e. = β 1 x β x β 1 1 xβ d
66 2. Every family of seminorms on a vector space containing a norm induces ahausdorff locally convex topology. 3. Given an open subset Ω of R d with the euclidean topology, the space C(Ω) of real valued
More informationA RIEMANN-ROCH THEOREM FOR EDGE-WEIGHTED GRAPHS
A RIEMANN-ROCH THEOREM FOR EDGE-WEIGHTED GRAPHS RODNEY JAMES AND RICK MIRANDA Contents 1. Introduction 1 2. Change of Rings 3 3. Reduction to Q-graphs 5 4. Scaling 6 5. Reduction to Z-graphs 8 References
More information9. Finite fields. 1. Uniqueness
9. Finite fields 9.1 Uniqueness 9.2 Frobenius automorphisms 9.3 Counting irreducibles 1. Uniqueness Among other things, the following result justifies speaking of the field with p n elements (for prime
More informationMath 121 Homework 4: Notes on Selected Problems
Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W
More informationarxiv: v2 [math.co] 12 Jul 2009
A Proof of the Molecular Conjecture Naoki Katoh and Shin-ichi Tanigawa arxi:0902.02362 [math.co] 12 Jul 2009 Department of Architecture and Architectural Engineering, Kyoto Uniersity, Kyoto Daigaku Katsura,
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 information