A RESULT ON MODULAR FORMS IN CHARACTERISTIC p. Nicholas M. Katz ABSTRACT. d The action of the derivation e = q ~ on the qexpansions of


 Blake Griffin
 1 years ago
 Views:
Transcription
1 Ka I A RESULT ON MODULAR FORMS IN CHARACTERISTIC p Nicholas M. Katz ABSTRACT The action of the erivation e = q ~ on the qexpansions of moular forms in characteristic p is one of the funamental tools in the Serre/SwinnertonDyer theory of mo p moular forms. In this note, we exten the basic results about this action, alreay known for P > 5 an level one, to arbitrary p an arbitrary primetop level.!. Review of moular forms in characteristic p We fix an algebraically close fiel K of characteristic p > 0, an integer N > 3 prime to p, an a primitive N'th root of unity ~ ~ K. The mouli problem "elliptic curves E over Kalgebras with level N structure ~ of eterminant {" is represente by (Euniv i ~univ) M N with M N a smooth affine irreucible curve over K. In terms of the invertible sheaf on M N I = ~, fleuniv/m N ' the grae ring R~ of (not necessarily holomorphic at the cusps) level N moular forms over K is O k) k~ Given a Kalgebra B, a test object (E,~) over B, an a nowherevanishing invariant ifferential ~ on E, any element
2 Ka2 54 f e R N (not necessarily homogeneous) has a value f(e,~,~) e B, an f is etermine by all of its values (cf. [2]). Over B = K((ql/N)), we have the Tate curve Tare(q) with its "canonical" ifferential C0ca n (viewing Tare(q) as ~m/q ~, ~can "is" t/t from ~m ). By evaluating at the level N structures as o of eterminant ~ on Tare(q), all of which are efine over K((ql/n)), we obtain the qexpansions of elements f e R N at the corresponing cusps: fm0(q) fn f(tate(q),~can,~ 0) e K((ql/N)) A homogeneous element f e R k N is uniquely etermine by its weight k an by any one of its qexpansions. A form f c R k N is sai to be holomorphic if all of its qexpansions lie in K[[ql/N]], an to be a cusp form if all of its qexpansions lie in q I/NK[[ql/N]] The holomorphic forms constitute a subring R~,holo of RN, an the cusp forms are a grae ieal in RN,holo. The Hasse invariant A c R is efine moularly as olo follows. Given (E,~,(~) over B, let r i c HI(E,OE ) be the basis ] ual to ~ c H0(E,CE/B). The p'th power enomorphism x ~ x p of O E inuces an enormorphism of HI(E,OE ), which must carry r; to a multiple of itself. So we can write ~P = A(E,~,~),~ in HI(E,OE ), for some A(E,~,~) c B, which is the value of A on (E,~,~). All the qexpansions of the Hasse invariant are ientically ]: A o(q ) = l in K((q~/N)) for each s o. For each level N structure s o on Tare(q), the corresponing qexpansion efines ring homomorphisms whose kernels are precisely the principal ieals (AI)R~ an
3 55 ~3 (Al)R~,holo respectively ([4], [5]). A form f ~ R N k is sai to be of exact filtration k if it is not ivisible by A in R~, or equivalently, if there is no form f' k~ k' R N with ( k which, at some cusp, has the same qexpansion that f oes. I!. Statment of the theorem, an its corollaries The following theorem is ue to Serre an SwinnertonDyer ([4], [5]) in characteristic P > 5, an level N = I. Theorem. (I) There exists a erivation A0:R~ ~ RN+P+~ which increases egrees by p + I, an whose effect upon each qexpansion is q~ : (A0f) (q) = q _ (fso(q)) for each S O So q (2) If f c R k N has exact filtration k, an p oes not ivie k, then A0f has exact filtration k + p + ~, an in particular A0f ~ O. k g c R N. (3) If f c R~ k an A0f = 0, then f = gp for a unique Some Corollaries (]) The operator A0 maps the subring of holomorphic forms to the ieal of cusp forms. (Look at qexpansions.) (2) If f is nonzero an holomorphic, of weight I ~ k ~ p  2, then f has exact filtration k. (For if f = Ag, then g is holomorphic of weight k  (pl) ~ O, hence g = 0.) injective. (3) If ] ~ k ~ p 2, the map 0 k ~ Rk+P+1  A :RN,holo N,holo is (This follows from (2) above an the theorem.) (4) If f is nonzero an holomorphic of weight p  l, an vanishes at some cusp, then f has exact filtration p  ~. (For if f = Ag, then g is holomorphic of weight O, hence constant; as g vanishes at one cusp, it must be zero.)
4 Ka4 56 k has Aef = 0, (5) (etermination of Ker(Ae)). If f ~ R N then we can uniquely write f = Ar.g p with 0 ~ r < p  I, r + k ~ O(mo p), an g ~ R~ with pf + r(p]) = k. (This is proven by inuction on r, the case r = 0 being part (3) of the theorem. If r ~ O, then O(p), but Aef = 0. Hence by part (2) of k+]p the theorem f = Ah for some h ~ R N Because f an h have the same qexpansions, we have Aeh = O, an h has lower r.) (6) In (5) above, if f is holomorphic (resp. a cusp form, resp. invariant by a subgroup of SL2(Z/NZ)), so is g (by unicity of g).!i!. Beginning of the proof: efining e an A~, an proving part (1) The absolute Frobenius enomorphism F of M N inuces an Flinear enomorphism of H 1 (E /M ~ as follows. The pull DR univ" N/' (F) of is obtaine by iviing Euniv by back Euniv Euniv its finite flat rank p subgroup scheme Ker Fr where Fr:Euniv ~(F) > ~univ is the relative Frobenius morphism. The esire map is Fr * 1 (F) Fr: HDR(Euniv /MN) >4R(Euniv/MN) ~R(Euniv/MN) (F) Lemma ]. The image U of Fr is a locally free submoule of H~(Eunl~n "v/m)n of rank one, with the quotient H~R(Euniv/MN)/U,Hasse locally free of rank one. The open set M N C M N where A is invertible is the largest open set over which U splits the Hoge filtration, i.e., where > R(Euniv/MN). Proof. Because F~ kills HO(Euniv, ~Euniv/MN )(F) it factors through the quotient H](Euniv,O) (F), where it inuces the inclusion map in the "conjugate filtration" short exact sequence
5 57 Ke~ (cf [~], 2.3) 0 > H I (F) Fr nl >4R(Euniv/MN) > HO(Euniv, Euniv/MN) > O. This proves the first part of the lemma. To see where U splits the Hoge filtration, we can work locally on M N. Choose a basis ~, rl of H~R aapte to the Hoge filtration, an satisfying <~,TI>DR = ] Then ~ projects to a basis of HI(E,OE ) ual to ~, an so the matrix of Fr on R is (remembering Fr(~ ( ) = 0) * 4 * ~) (0 A where A is the value of the Hasse invariant. Thus U is spanne by Be + A~, an the conition that e an Be + A~ together span H~R is precisely that A be invertible. Q.E.D. Remark. Accoring to the first part of the lemma, the func tions A an B which occur in the above matrix have no common zero. This will be crucial later. We can now efine a erivation e of RN[I/A] as follows. (Compare [2], A1.4.) Over M~ asse, we have the ecomposition which for each integer k > I inuces a ecomposition Symmk~R ~ _m k(~)(e k1 ( ~ )(~U) _... ~ U k The GaussManin connection ] I inuces, for each k > I a connection Using the KoairaSpencer isomorphism ([2], A.].3.]7) ~ HN/K
6  Ka we can efine a mapping of sheaves k k+ 2 as the composite > Sym:L = (~  Iil DR  "'" f~m R KS Pr 1 } ~ k+ 2 MHasse Passing to global sections over N ' we obtain, for k > I, a map 0,~Hasse 2 ) e " > H0(mN,m :H ~N " " Lemma 2. The effect upon qexpansions is Proof. Consier Tate(q) with its canonical ifferential co 2 can over k((ql/n)) Uner the KoairaSpencer isomorphism, can correspons to q/q, the ual erivation to which is q ~. By the explicit calculations of ([2], A.2.2.7), U is spanne by v(q ~)(~can). Thus given an element k ~O,~Hasse~ _.~, _ k), its local expression as a section of _ f C on (Tate(q), some ~0) is f Thus v(f c~ 0 k q k ) V(q ~~)((q)'~can) q (q)'ecan = f~0 k 2 (q)'~can) ~can : v(q T4)(f% k+ 2 : q q (f~0 (q))" ~can k+ I. V( q ~) (~can) ' + k~f~0(q)" can Because ~ (q ~q) (~can) lies in U, it follows from
7 59 Ka7 (q)). the efinition of e that we have (ef)~o(q) = q q (f~0 Q.E.D. k ~k+p+1 Lemma 3. For k ~ ], there is a unique map Ae:R N > m N such that the iagram below commutes HO ( 4asse, )_ ~ k U 9_~_> re'o". Hassek~N, k+2~ A > n"o'kmn~asse,~ k+ p+ ] ) RNk = A0 > ~k+p+l ~N = HO(MN,fk+P+?) Proof. Again we work locally on M N. be a local I basis of, ~ the local basis of ~MN/K corresponing to by the KoairaSpencer isomorphism, D the local basis of DerNN/K ual to ~, an ~' = ~ ~R" Then = I, (this characterizes D), so an m' form a basis of ] HDR, aapte to the Hoge filtration, in terms of which the matrix of Fr is 0B) (0 A ~asse with A = A(E,e). Let u e U be the basis of U over M N which is ual to e. Then u is proportional to Be + Ae', an satisfies <e'~'>dr = I, so that B u = K ~ + e l In terms of all this, we will compute 9f for f e RN, k an show that it has at worst a single power of A in its enominator. k k Locally, f is the section of ~, with fl holomorphic. v(f~ ~k) = v(d)(f1~ k).~ = V(D)(flco 2 k+ 2 k+ 1, = D(f] ).co + B =D(f I) k+2 + kflco k+1(u _ K ~)
8 Ka8 60 B k+2!<+i : (D(fl) kf 1 K) + kfl "u Thus from the efinition of e it follows that the local expression of ~(f) is 0(f) = (D(f~)  kf I ~) k+2 Q.E.D. We can now conclue the proof of Part (1) of the theorem. Up to now, we have only efine A on elements of R~ of positive egree. But as R~ has units which are homogeneous of positive egree (e.g., A), the erivation Ae extens uniquely to all of RN by the explicit formula Aef A~(f'aPr) for r >> 0 AP r The local expression for Ae(f) k Ae(f) = (AD(fl)kfiB)~ k+p+1 for f e R N remains vali. IV. Conclusion of the proof: Parts (2) an (3) Suppose f e R k N has exact filtration k. This means that f is not ivisible by A in R~, i.e., that at some zero of A, f has a lower orer zero (as section of k) than A oes (as section of pl). (In fact, we know by Igusa [3] that A has simple zeros, so in fact f must be invertible at some zero of A. Rather surprisingly, we will not make use of this fact.) Locally on MN, we pick a basis ~ of ~. Then f becomes Ok f].~, an A~(f) is given by A~(f~ ~k) = (kd(fl)_kbfl) Suppose now that k is not ivisible by p. Recall that B is invertible at all zeros of A (of the remark following Lemma ]). Thus if x s M N is a zero of A where orx(f]) < Orx(A),
9 61 Ka9 we easily compute orx(ad(f l)kbf I) = orx(f I) < orx(a). This proves Part (2) of the theorem. To prove Part (3), let expression for Ae(f) gives f c R pk N have = 0= The local An(f I ) k+p+1 = 0 an hence D(fi) = 0. Because M N is a smooth curve over a perfect fiel of characteristic p, this implies that fl is a pth power say fl = (gl)p" Thus fl ~ kp = (g1~ k) p, so that f, as section of _~kp, is, locally on M N, the pth power of a k (necessarily unique) section g of ci~ By unicity, these local g's patch together. Q.E.D. REFERE NCE S 171] [2] [3] [4] N. Katz, Algebraic solutions of ifferential equations, Inventiones Math. 18 (1972), Paic properties of moular schemes an moular forms, Proceeings of the 1972 Antwerp International Summer School on Moular Functions, Springer Lecture Notes in Mathematics 350 (1973) J. Igusa, Class number of a efinite quaternion with prime iscriminant, Proe. Nat'l. Aca. Sci. 44 (1958) J. P. Serre Congruences et formes moulaires ('apres H. P. F. SwinnertonDyer) 416, S4minaire N. Bourbaki, 1971/72, Springer Lecture Notes in Mathematics 317 (1973), H. P. F. SwinnertonDyer, On ~aic representations an congruences for coefficients of moular forms, Proceeings of the 1972 Antwerp International Su~mmer School on Moular Forms Springer Lecture Notes in Mathematics 350 (1973), N. Katz Departement of Mathematics Fine Hall Princeton University Princeton, N.J
Ideal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationFactoring Dickson polynomials over finite fields
Factoring Dickson polynomials over finite fiels Manjul Bhargava Department of Mathematics, Princeton University. Princeton NJ 08544 manjul@math.princeton.eu Michael Zieve Department of Mathematics, University
More informationPythagorean Triples Over Gaussian Integers
International Journal of Algebra, Vol. 6, 01, no., 5564 Pythagorean Triples Over Gaussian Integers Cheranoot Somboonkulavui 1 Department of Mathematics, Faculty of Science Chulalongkorn University Bangkok
More informationChapter 13: Basic ring theory
Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More information1. Recap: What can we do with Euler systems?
Euler Systems and Lfunctions These are notes for a talk I gave at the Warwick Number Theory study group on Euler Systems, June 2015. The results stated make more sense when taken in step with the rest
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationProof. The map. G n i. where d is the degree of D.
7. Divisors Definition 7.1. We say that a scheme X is regular in codimension one if every local ring of dimension one is regular, that is, the quotient m/m 2 is one dimensional, where m is the unique maximal
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationINTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 16
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 16 RAVI VAKIL Contents 1. Valuation rings (and nonsingular points of curves) 1 1.1. Completions 2 1.2. A big result from commutative algebra 3 Problem sets back.
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationEXERCISES FOR THE COURSE MATH 570, FALL 2010
EXERCISES FOR THE COURSE MATH 570, FALL 2010 EYAL Z. GOREN (1) Let G be a group and H Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p 2 (p a prime
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More informationResearch Article On the Rank of Elliptic Curves in Elementary Cubic Extensions
Numbers Volume 2015, Article ID 501629, 4 pages http://dx.doi.org/10.1155/2015/501629 Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions Rintaro Kozuma College of International
More informationCONSEQUENCES OF THE SYLOW THEOREMS
CONSEQUENCES OF THE SYLOW THEOREMS KEITH CONRAD For a group theorist, Sylow s Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing. Geoff Robinson 1.
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationIntroduction to finite fields
Introduction to finite fields Topics in Finite Fields (Fall 2013) Rutgers University Swastik Kopparty Last modified: Monday 16 th September, 2013 Welcome to the course on finite fields! This is aimed at
More information26 Ideals and Quotient Rings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 26 Ideals and Quotient Rings In this section we develop some theory of rings that parallels the theory of groups discussed
More informationGalois representations with open image
Galois representations with open image Ralph Greenberg University of Washington Seattle, Washington, USA May 7th, 2011 Introduction This talk will be about representations of the absolute Galois group
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationThe wave equation is an important tool to study the relation between spectral theory and geometry on manifolds. Let U R n be an open set and let
1. The wave equation The wave equation is an important tool to stuy the relation between spectral theory an geometry on manifols. Let U R n be an open set an let = n j=1 be the Eucliean Laplace operator.
More informationAlex, I will take congruent numbers for one million dollars please
Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohiostate.edu One of the most alluring aspectives of number theory
More informationELLIPTIC CURVES AND LENSTRA S FACTORIZATION ALGORITHM
ELLIPTIC CURVES AND LENSTRA S FACTORIZATION ALGORITHM DANIEL PARKER Abstract. This paper provides a foundation for understanding Lenstra s Elliptic Curve Algorithm for factoring large numbers. We give
More informationFactoring of Prime Ideals in Extensions
Chapter 4 Factoring of Prime Ideals in Extensions 4. Lifting of Prime Ideals Recall the basic AKLB setup: A is a Dedekind domain with fraction field K, L is a finite, separable extension of K of degree
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationMSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUMLIKELIHOOD ESTIMATION
MAXIMUMLIKELIHOOD ESTIMATION The General Theory of ML Estimation In orer to erive an ML estimator, we are boun to make an assumption about the functional form of the istribution which generates the
More informationCOMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS
COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V. EROVENKO AND B. SURY ABSTRACT. We compute commutativity degrees of wreath products A B of finite abelian groups A and B. When B
More information7. Some irreducible polynomials
7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationGROUP ALGEBRAS. ANDREI YAFAEV
GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite
More informationThe fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit
The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationGroup Theory. Contents
Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation
More informationFIBER PRODUCTS AND ZARISKI SHEAVES
FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also
More informationCOMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS
Bull Austral Math Soc 77 (2008), 31 36 doi: 101017/S0004972708000038 COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V EROVENKO and B SURY (Received 12 April 2007) Abstract We compute
More informationLinear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)
MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of
More informationClassification of Cartan matrices
Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationFACTORING IN QUADRATIC FIELDS. 1. Introduction. This is called a quadratic field and it has degree 2 over Q. Similarly, set
FACTORING IN QUADRATIC FIELDS KEITH CONRAD For a squarefree integer d other than 1, let 1. Introduction K = Q[ d] = {x + y d : x, y Q}. This is called a quadratic field and it has degree 2 over Q. Similarly,
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationMOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu
Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing
More informationSOME PROPERTIES OF SYMBOL ALGEBRAS OF DEGREE THREE
SOME PROPERTIES OF SYMBOL ALGEBRAS OF DEGREE THREE CRISTINA FLAUT and DIANA SAVIN Communicated by the former editorial board In this paper, we study some properties of the matrix representations of the
More informationADDITIVE GROUPS OF RINGS WITH IDENTITY
ADDITIVE GROUPS OF RINGS WITH IDENTITY SIMION BREAZ AND GRIGORE CĂLUGĂREANU Abstract. A ring with identity exists on a torsion Abelian group exactly when the group is bounded. The additive groups of torsionfree
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it
More informationNOTES ON CATEGORIES AND FUNCTORS
NOTES ON CATEGORIES AND FUNCTORS These notes collect basic definitions and facts about categories and functors that have been mentioned in the Homological Algebra course. For further reading about category
More informationMath 231b Lecture 35. G. Quick
Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the Jhomomorphism could be defined by
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS Contents 1. Moment generating functions 2. Sum of a ranom number of ranom variables 3. Transforms
More informationPOLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS
POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).
More informationFACTORING AFTER DEDEKIND
FACTORING AFTER DEDEKIND KEITH CONRAD Let K be a number field and p be a prime number. When we factor (p) = po K into prime ideals, say (p) = p e 1 1 peg g, we refer to the data of the e i s, the exponents
More informationG = G 0 > G 1 > > G k = {e}
Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same
More informationApplying this to X = A n K it follows that Cl(U) = 0. So we get an exact sequence
3. Divisors on toric varieties We start with computing the class group of a toric variety. Recall that the class group is the group of Weil divisors modulo linear equivalence. We denote the class group
More informationTopological Algebraic Geometry Workshop Oslo, September 4th8th, 2006. Spectra associated with ArtinSchreier curves. Doug Ravenel
Topological Algebraic Geometry Workshop Oslo, September 4th8th, 2006 Spectra associated with ArtinSchreier curves Doug Ravenel University of Rochester September 6, 2006 1 2 1. My favorite part of the
More informationSign changes of Hecke eigenvalues of Siegel cusp forms of degree 2
Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Ameya Pitale, Ralf Schmidt 2 Abstract Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree
More informationLecture Notes for the Summer School on Algebraic Geometry and Number Theory Galatasaray University, Istanbul, Turkey June 2 10, 2014
Lecture Notes for the Summer School on Algebraic Geometry and Number Theory Galatasaray University, Istanbul, Turkey June 2 10, 2014 INTRODUCTION TO SHEAVES LORING W. TU Introduced in the 1940s by Leray
More information3 1. Note that all cubes solve it; therefore, there are no more
Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationQUADRATIC RECIPROCITY IN CHARACTERISTIC 2
QUADRATIC RECIPROCITY IN CHARACTERISTIC 2 KEITH CONRAD 1. Introduction Let F be a finite field. When F has odd characteristic, the quadratic reciprocity law in F[T ] (see [4, Section 3.2.2] or [5]) lets
More informationDIVISORS AND LINE BUNDLES
DIVISORS AND LINE BUNDLES TONY PERKINS 1. Cartier divisors An analytic hypersurface of M is a subset V M such that for each point x V there exists an open set U x M containing x and a holomorphic function
More informationCOMMUTATIVE RINGS. Definition: A domain is a commutative ring R that satisfies the cancellation law for multiplication:
COMMUTATIVE RINGS Definition: A commutative ring R is a set with two operations, addition and multiplication, such that: (i) R is an abelian group under addition; (ii) ab = ba for all a, b R (commutative
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationOn Bhargava s representations and Vinberg s invariant theory
On Bhargava s representations and Vinberg s invariant theory Benedict H. Gross Department of Mathematics, Harvard University Cambridge, MA 02138 gross@math.harvard.edu January, 2011 1 Introduction Manjul
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationMannheim curves in the threedimensional sphere
Mannheim curves in the threeimensional sphere anju Kahraman, Mehmet Öner Manisa Celal Bayar University, Faculty of Arts an Sciences, Mathematics Department, Muraiye Campus, 5, Muraiye, Manisa, urkey.
More informationSets of Fibre Homotopy Classes and Twisted Order Parameter Spaces
Communications in Mathematical Physics ManuscriptNr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan BechtluftSachs, Marco Hien Naturwissenschaftliche
More informationEMBEDDING DEGREE OF HYPERELLIPTIC CURVES WITH COMPLEX MULTIPLICATION
EMBEDDING DEGREE OF HYPERELLIPTIC CURVES WITH COMPLEX MULTIPLICATION CHRISTIAN ROBENHAGEN RAVNSHØJ Abstract. Consider the Jacobian of a genus two curve defined over a finite field and with complex multiplication.
More informationFACTORING IN THE HYPERELLIPTIC TORELLI GROUP
FACTORING IN THE HYPERELLIPTIC TORELLI GROUP TARA E. BRENDLE AND DAN MARGALIT Abstract. The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially
More informationFINITE FIELDS KEITH CONRAD
FINITE FIELDS KEITH CONRAD This handout discusses finite fields: how to construct them, properties of elements in a finite field, and relations between different finite fields. We write Z/(p) and F p interchangeably
More informationMATH1231 Algebra, 2015 Chapter 7: Linear maps
MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter
More informationIntroduction to Modern Algebra
Introduction to Modern Algebra David Joyce Clark University Version 0.0.6, 3 Oct 2008 1 1 Copyright (C) 2008. ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write
More information3. Equivalence Relations. Discussion
3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationGalois Theory III. 3.1. Splitting fields.
Galois Theory III. 3.1. Splitting fields. We know how to construct a field extension L of a given field K where a given irreducible polynomial P (X) K[X] has a root. We need a field extension of K where
More informationA Minkowskistyle bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field.
A Minkowskistyle bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field JeanPierre Serre Let k be a field. Let Cr(k) be the Cremona group of rank 2 over k,
More informationMath 230.01, Fall 2012: HW 1 Solutions
Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The
More informationThe Open University s repository of research publications and other research outputs
Open Research Online The Open University s repository of research publications and other research outputs The degreediameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:
More informationOnline publication date: 11 March 2010 PLEASE SCROLL DOWN FOR ARTICLE
This article was downloaded by: [Modoi, George Ciprian] On: 12 March 2010 Access details: Access Details: [subscription number 919828804] Publisher Taylor & Francis Informa Ltd Registered in England and
More informationMonomial Ideals, Binomial Ideals, Polynomial Ideals
Trens in Commutative Algebra MSRI Publications Volume 51, 2004 Monomial Ieals, Binomial Ieals, Polynomial Ieals BERNARD TEISSIER Abstract. These lectures provie a glimpse of the applications of toric geometry
More informationCyclotomic Extensions
Chapter 7 Cyclotomic Extensions A cyclotomic extension Q(ζ n ) of the rationals is formed by adjoining a primitive n th root of unity ζ n. In this chapter, we will find an integral basis and calculate
More information( ) which must be a vector
MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are
More informationAPPLICATIONS OF THE ORDER FUNCTION
APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and
More informationEuler Class Construction
Euler Class Construction Mrinal Kanti Das Satya Mandal Department of Mathematics University of Kansas, Lawrence, Kansas 66045 email:{mdas,mandal}@math.ku.edu August 24, 2004 1 Introduction The proof of
More informationminimal polyonomial Example
Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We
More informationHere the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and
Lecture 9 : Derivatives of Trigonometric Functions (Please review Trigonometry uner Algebra/Precalculus Review on the class webpage.) In this section we will look at the erivatives of the trigonometric
More information5.1 Commutative rings; Integral Domains
5.1 J.A.Beachy 1 5.1 Commutative rings; Integral Domains from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 23. Let R be a commutative ring. Prove the following
More informationSOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS
SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS Ivan Kolář Abstract. Let F be a fiber product preserving bundle functor on the category FM m of the proper base order r. We deduce that the rth
More informationA Generalization of Sauer s Lemma to Classes of LargeMargin Functions
A Generalization of Sauer s Lemma to Classes of LargeMargin Functions Joel Ratsaby University College Lonon Gower Street, Lonon WC1E 6BT, Unite Kingom J.Ratsaby@cs.ucl.ac.uk, WWW home page: http://www.cs.ucl.ac.uk/staff/j.ratsaby/
More information9. Quotient Groups Given a group G and a subgroup H, under what circumstances can we find a homomorphism φ: G G ', such that H is the kernel of φ?
9. Quotient Groups Given a group G and a subgroup H, under what circumstances can we find a homomorphism φ: G G ', such that H is the kernel of φ? Clearly a necessary condition is that H is normal in G.
More informationCOHOMOLOGY OF GROUPS
Actes, Congrès intern. Math., 1970. Tome 2, p. 47 à 51. COHOMOLOGY OF GROUPS by Daniel QUILLEN * This is a report of research done at the Institute for Advanced Study the past year. It includes some general
More informationElements of Abstract Group Theory
Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for
More informationZORN S LEMMA AND SOME APPLICATIONS
ZORN S LEMMA AND SOME APPLICATIONS KEITH CONRAD 1. Introduction Zorn s lemma is a result in set theory that appears in proofs of some nonconstructive existence theorems throughout mathematics. We will
More informationMCS 563 Spring 2014 Analytic Symbolic Computation Wednesday 9 April. Hilbert Polynomials
Hilbert Polynomials For a monomial ideal, we derive the dimension counting the monomials in the complement, arriving at the notion of the Hilbert polynomial. The first half of the note is derived from
More informationModern Algebra Lecture Notes: Rings and fields set 4 (Revision 2)
Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Kevin Broughan University of Waikato, Hamilton, New Zealand May 13, 2010 Remainder and Factor Theorem 15 Definition of factor If f (x)
More informationA number field is a field of finite degree over Q. By the Primitive Element Theorem, any number
Number Fields Introduction A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number field K = Q(α) for some α K. The minimal polynomial Let K be a number field and
More informationCONNECTIONS ON PRINCIPAL GBUNDLES
CONNECTIONS ON PRINCIPAL GBUNDLES RAHUL SHAH Abstract. We will describe connections on principal Gbundles via two perspectives: that of distributions and that of connection 1forms. We will show that
More informationA result of Gabber. by A.J. de Jong
A result of Gabber by A.J. de Jong 1 The result Let X be a scheme endowed with an ample invertible sheaf L. See EGA II, Definition 4.5.3. In particular, X is supposed quasicompact and separated. 1.1 Theorem.
More information