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

Size: px
Start display at page:

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

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 q-expansions of moular forms in characteristic p is one of the funamental tools in the Serre/Swinnerton-Dyer 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 prime-to-p 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 K-algebras 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 K-algebra B, a test object (E,~) over B, an a nowhere-vanishing invariant ifferential ~ on E, any element

2 Ka-2 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 q-expansions 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 q-expansions. A form f c R k N is sai to be holomorphic if all of its q-expansions lie in K[[ql/N]], an to be a cusp form if all of its q-expansions 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 q-expansions 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 q-expansion efines ring homomorphisms whose kernels are precisely the principal ieals (A-I)R~ an

3 55 ~-3 (A-l)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 q-expansion that f oes. I!. Statment of the theorem, an its corollaries The following theorem is ue to Serre an Swinnerton-Dyer ([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 q-expansion 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 q-expansions.) (2) If f is non-zero 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 - (p-l) ~ 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 non-zero 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 Ka-4 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 q-expansions, 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 F-linear 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 k-1 ( ~ )(~U) _... ~ U k The Gauss-Manin connection ] I inuces, for each k > I a connection Using the Koaira-Spencer 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 q-expansions is Proof. Consier Tate(q) with its canonical ifferential co 2 can over k((ql/n)) Uner the Koaira-Spencer 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 Ka-7 (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 Koaira-Spencer 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 Ka-8 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 p-l). (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 Ka-9 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), P-aic 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. Swinnerton-Dyer) 416, S4minaire N. Bourbaki, 1971/72, Springer Lecture Notes in Mathematics 317 (1973), H. P. F. Swinnerton-Dyer, 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

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 information

Factoring Dickson polynomials over finite fields

Factoring 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 information

Pythagorean Triples Over Gaussian Integers

Pythagorean Triples Over Gaussian Integers International Journal of Algebra, Vol. 6, 01, no., 55-64 Pythagorean Triples Over Gaussian Integers Cheranoot Somboonkulavui 1 Department of Mathematics, Faculty of Science Chulalongkorn University Bangkok

More information

Chapter 13: Basic ring theory

Chapter 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 information

The Dirichlet Unit Theorem

The 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 information

1. Recap: What can we do with Euler systems?

1. Recap: What can we do with Euler systems? Euler Systems and L-functions 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 information

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I. 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 information

Proof. The map. G n i. where d is the degree of D.

Proof. 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 information

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE 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 information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 16

INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 16 INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 16 RAVI VAKIL Contents 1. Valuation rings (and non-singular points of curves) 1 1.1. Completions 2 1.2. A big result from commutative algebra 3 Problem sets back.

More information

U.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 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 information

EXERCISES FOR THE COURSE MATH 570, FALL 2010

EXERCISES 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 information

it is easy to see that α = a

it 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 information

Research Article On the Rank of Elliptic Curves in Elementary Cubic Extensions

Research 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 information

CONSEQUENCES OF THE SYLOW THEOREMS

CONSEQUENCES 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 information

Continued Fractions and the Euclidean Algorithm

Continued 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 information

Introduction to finite fields

Introduction 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 information

26 Ideals and Quotient Rings

26 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 information

Galois representations with open image

Galois 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 information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional 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 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

The 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

The 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 information

Alex, I will take congruent numbers for one million dollars please

Alex, 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.ohio-state.edu One of the most alluring aspectives of number theory

More information

ELLIPTIC CURVES AND LENSTRA S FACTORIZATION ALGORITHM

ELLIPTIC 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 information

Factoring of Prime Ideals in Extensions

Factoring 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 information

11 Ideals. 11.1 Revisiting Z

11 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 information

Introduction to Algebraic Geometry. Bézout s Theorem and Inflection Points

Introduction 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 information

NOTES ON LINEAR TRANSFORMATIONS

NOTES 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 information

MSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUM-LIKELIHOOD ESTIMATION

MSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUM-LIKELIHOOD ESTIMATION MAXIMUM-LIKELIHOOD ESTIMATION The General Theory of M-L Estimation In orer to erive an M-L estimator, we are boun to make an assumption about the functional form of the istribution which generates the

More information

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

COMMUTATIVITY 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 information

7. Some irreducible polynomials

7. 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 information

Similarity and Diagonalization. Similar Matrices

Similarity 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 information

GROUP ALGEBRAS. ANDREI YAFAEV

GROUP 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 information

The 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 (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 information

Zeros of Polynomial Functions

Zeros of Polynomial Functions Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

More information

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 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 information

Group Theory. Contents

Group 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 information

FIBER PRODUCTS AND ZARISKI SHEAVES

FIBER 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 information

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

COMMUTATIVITY 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 information

Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

Linear 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 information

Classification of Cartan matrices

Classification 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 information

University 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 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 information

Lecture 14: Section 3.3

Lecture 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 information

FACTORING IN QUADRATIC FIELDS. 1. Introduction. This is called a quadratic field and it has degree 2 over Q. Similarly, set

FACTORING 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 information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM 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 information

MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu

MOP 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 information

SOME PROPERTIES OF SYMBOL ALGEBRAS OF DEGREE THREE

SOME 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 information

ADDITIVE GROUPS OF RINGS WITH IDENTITY

ADDITIVE 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 torsion-free

More information

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22

FOUNDATIONS 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 information

NOTES ON CATEGORIES AND FUNCTORS

NOTES 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 information

Math 231b Lecture 35. G. Quick

Math 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 J-homomorphism could be defined by

More information

MASSACHUSETTS 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 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 information

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS

POLYNOMIAL 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 information

FACTORING AFTER DEDEKIND

FACTORING 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 information

G = G 0 > G 1 > > G k = {e}

G = 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 information

Applying this to X = A n K it follows that Cl(U) = 0. So we get an exact sequence

Applying 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 information

Topological Algebraic Geometry Workshop Oslo, September 4th-8th, 2006. Spectra associated with Artin-Schreier curves. Doug Ravenel

Topological Algebraic Geometry Workshop Oslo, September 4th-8th, 2006. Spectra associated with Artin-Schreier curves. Doug Ravenel Topological Algebraic Geometry Workshop Oslo, September 4th-8th, 2006 Spectra associated with Artin-Schreier curves Doug Ravenel University of Rochester September 6, 2006 1 2 1. My favorite part of the

More information

Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2

Sign 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 information

Lecture 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 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 information

3 1. Note that all cubes solve it; therefore, there are no more

3 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 information

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING 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 information

QUADRATIC RECIPROCITY IN CHARACTERISTIC 2

QUADRATIC 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 information

DIVISORS AND LINE BUNDLES

DIVISORS 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 information

COMMUTATIVE RINGS. Definition: A domain is a commutative ring R that satisfies the cancellation law for multiplication:

COMMUTATIVE 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 information

BANACH AND HILBERT SPACE REVIEW

BANACH 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 information

On Bhargava s representations and Vinberg s invariant theory

On 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 information

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE 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 information

Mannheim curves in the three-dimensional sphere

Mannheim curves in the three-dimensional sphere Mannheim curves in the three-imensional sphere anju Kahraman, Mehmet Öner Manisa Celal Bayar University, Faculty of Arts an Sciences, Mathematics Department, Muraiye Campus, 5, Muraiye, Manisa, urkey.

More information

Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces

Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Communications in Mathematical Physics Manuscript-Nr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan Bechtluft-Sachs, Marco Hien Naturwissenschaftliche

More information

EMBEDDING DEGREE OF HYPERELLIPTIC CURVES WITH COMPLEX MULTIPLICATION

EMBEDDING 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 information

FACTORING IN THE HYPERELLIPTIC TORELLI GROUP

FACTORING 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 information

FINITE FIELDS KEITH CONRAD

FINITE 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 information

MATH1231 Algebra, 2015 Chapter 7: Linear maps

MATH1231 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 information

Introduction to Modern Algebra

Introduction 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 information

3. Equivalence Relations. Discussion

3. 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 information

ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES 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 information

Galois Theory III. 3.1. Splitting fields.

Galois 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 information

A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field.

A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field. A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field Jean-Pierre Serre Let k be a field. Let Cr(k) be the Cremona group of rank 2 over k,

More information

Math 230.01, Fall 2012: HW 1 Solutions

Math 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 information

The Open University s repository of research publications and other research outputs

The 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 degree-diameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:

More information

Online publication date: 11 March 2010 PLEASE SCROLL DOWN FOR ARTICLE

Online 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 information

Monomial Ideals, Binomial Ideals, Polynomial Ideals

Monomial 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 information

Cyclotomic Extensions

Cyclotomic 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

( ) 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 information

APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS 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 information

Euler Class Construction

Euler 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 information

minimal polyonomial Example

minimal 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 information

Here the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and

Here 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 information

5.1 Commutative rings; Integral Domains

5.1 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 information

SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS

SOME 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 r-th

More information

A Generalization of Sauer s Lemma to Classes of Large-Margin Functions

A Generalization of Sauer s Lemma to Classes of Large-Margin Functions A Generalization of Sauer s Lemma to Classes of Large-Margin 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 information

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?

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? 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 information

COHOMOLOGY OF GROUPS

COHOMOLOGY 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 information

Elements of Abstract Group Theory

Elements 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 information

ZORN S LEMMA AND SOME APPLICATIONS

ZORN 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 non-constructive existence theorems throughout mathematics. We will

More information

MCS 563 Spring 2014 Analytic Symbolic Computation Wednesday 9 April. Hilbert Polynomials

MCS 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 information

Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2)

Modern 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 information

A number field is a field of finite degree over Q. By the Primitive Element Theorem, any number

A 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 information

CONNECTIONS ON PRINCIPAL G-BUNDLES

CONNECTIONS ON PRINCIPAL G-BUNDLES CONNECTIONS ON PRINCIPAL G-BUNDLES RAHUL SHAH Abstract. We will describe connections on principal G-bundles via two perspectives: that of distributions and that of connection 1-forms. We will show that

More information

A result of Gabber. by A.J. de Jong

A 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 quasi-compact and separated. 1.1 Theorem.

More information