Elliptic Curves Lecture Notes
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13 Elliptic Function
13 Elliptic Function Recall that a function g : C → Cˆ is an elliptic function if it is meromorphic and there exists a lattice L = {mω1 + nω2 | m, n ∈ Z} such that g(z + ω)= g(z) for all z ∈ C and all ω ∈ L where ω1,ω2 are complex numbers that are R-linearly independent. We have shown that an elliptic function cannot be holomorphic, the number of its poles are finite and the sum of their residues is zero. Lemma 13.1 A non-constant elliptic function f always has the same number of zeros mod- ulo its associated lattice L as it does poles, counting multiplicites of zeros and orders of poles. Proof: Consider the function f ′/f, which is also an elliptic function with associated lattice L. We will evaluate the sum of the residues of this function in two different ways as above. Then this sum is zero, and by the argument principle from complex analysis 31, it is precisely the number of zeros of f counting multiplicities minus the number of poles of f counting orders. Corollary 13.2 A non-constant elliptic function f always takes on every value in Cˆ the same number of times modulo L, counting multiplicities. Proof: Given a complex value b, consider the function f −b. This function is also an elliptic function, and one with the same poles as f. By Theorem above, it therefore has the same number of zeros as f. Thus, f must take on the value b as many times as it does 0. -
Problems About Formal Groups and Cohomology Theories Arizona Winter School 2019
PROBLEMS ABOUT FORMAL GROUPS AND COHOMOLOGY THEORIES ARIZONA WINTER SCHOOL 2019 VESNA STOJANOSKA 1. Formal Group Laws In this document, a formal group law over a commutative ring R is always com- mutative and one-dimensional. Specifically, it is a power series F (x; y) 2 R[[x; y]] satisfying the following axioms: • F (x; 0) = x = F (0; x), • F (x; y) = F (y; x), and • F (x; F (y; z)) = F (F (x; y); z). Sometimes we denote F (x; y) by x +F y. For a non-negative integer n, the n-series of F is defined inductively by [0]F x = 0, and [n + 1]F x = x +F [n]F x. (1) Show that for any formal group law F (x; y) 2 R[[x; y]], x has a formal inverse. In other words, there is a power series i(x) 2 R[[x]] such that F (x; i(x)) = 0. As a consequence, we can define [−n]F x = i([n]F x). (2) Check that the following define formal group laws: (a) F (x; y) = x + y + uxy, where u is any unit in R. Also show that this is isomorphic to the multiplicative formal group law over R. x+y (b) F (x; y) = 1+xy . Hint: Note that if x = tanh(u) and y = tanh(v), then F (x; y) = tanh(x + y). (3) Suppose F; H are formal group laws over a complete local ring R, and we're given a morphism f : F ! H, i.e. f(x) 2 R[[x]] such that f(F (x; y)) = H(f(x); f(y)): Now let C be the category of complete local R-algebras and continuous maps between them; for T 2 C, denote by Spf(T ): C!Set the functor S 7! HomC(T;S). -
Formal Groups, Witt Vectors and Free Probability
Formal Groups, Witt vectors and Free Probability Roland Friedrich and John McKay December 6, 2019 Abstract We establish a link between free probability theory and Witt vectors, via the theory of formal groups. We derive an exponential isomorphism which expresses Voiculescu's free multiplicative convolution as a function of the free additive convolution . Subsequently we continue our previous discussion of the relation between complex cobordism and free probability. We show that the generic nth free cumulant corresponds to the cobordism class of the (n 1)-dimensional complex projective space. This permits us to relate several probability distributions− from random matrix theory to known genera, and to build a dictionary. Finally, we discuss aspects of free probability and the asymptotic representation theory of the symmetric group from a conformal field theoretic perspective and show that every distribution with mean zero is embeddable into the Universal Grassmannian of Sato-Segal-Wilson. MSC 2010: 46L54, 55N22, 57R77, Keywords: Free probability and free operator algebras, formal group laws, Witt vectors, complex cobordism (U- and SU-cobordism), non-crossing partitions, conformal field theory. Contents 1 Introduction 2 2 Free probability and formal group laws4 2.1 Generalised free probability . .4 arXiv:1204.6522v3 [math.OA] 5 Dec 2019 2.2 The Lie group Aut( )....................................5 O 2.3 Formal group laws . .6 2.4 The R- and S-transform . .7 2.5 Free cumulants . 11 3 Witt vectors and Hopf algebras 12 3.1 The affine group schemes . 12 3.2 Witt vectors, λ-rings and necklace algebras . 15 3.3 The induced ring structure . -
Formal Group Rings of Toric Varieties 3
FORMAL GROUP RINGS OF TORIC VARIETIES WANSHUN WONG Abstract. In this paper we use formal group rings to construct an alge- braic model of the T -equivariant oriented cohomology of smooth toric vari- eties. Then we compare our algebraic model with known results of equivariant cohomology of toric varieties to justify our construction. Finally we construct the algebraic counterpart of the pull-back and push-forward homomorphisms of blow-ups. 1. Introduction Let h be an algebraic oriented cohomology theory in the sense of Levine-Morel [14], where examples include the Chow group of algebraic cycles modulo rational equivalence, algebraic K-theory, connective K-theory, elliptic cohomology, and a universal such theory called algebraic cobordism. It is known that to any h one can associate a one-dimensional commutative formal group law F over the coefficient ring R = h(pt), given by c1(L1 ⊗ L2)= F (c1(L1),c1(L2)) for any line bundles L1, L2 on a smooth variety X, where c1 is the first Chern class. Let T be a split algebraic torus which acts on a smooth variety X. An object of interest is the T -equivariant cohomology ring hT (X) of X, and we would like to build an algebraic model for it. More precisely, instead of using geometric methods to compute hT (X), we want to use algebraic methods to construct another ring A (our algebraic model), such that A should be easy to compute and it gives infor- mation about hT (X). Of course the best case is that A and hT (X) are isomorphic. -
ELLIPTIC FUNCTIONS (Approach of Abel and Jacobi)
Math 213a (Fall 2021) Yum-Tong Siu 1 ELLIPTIC FUNCTIONS (Approach of Abel and Jacobi) Significance of Elliptic Functions. Elliptic functions and their associated theta functions are a new class of special functions which play an impor- tant role in explicit solutions of real world problems. Elliptic functions as meromorphic functions on compact Riemann surfaces of genus 1 and their associated theta functions as holomorphic sections of holomorphic line bun- dles on compact Riemann surfaces pave the way for the development of the theory of Riemann surfaces and higher-dimensional abelian varieties. Two Approaches to Elliptic Function Theory. One approach (which we call the approach of Abel and Jacobi) follows the historic development with motivation from real-world problems and techniques developed for solving the difficulties encountered. One starts with the inverse of an elliptic func- tion defined by an indefinite integral whose integrand is the reciprocal of the square root of a quartic polynomial. An obstacle is to show that the inverse function of the indefinite integral is a global meromorphic function on C with two R-linearly independent primitive periods. The resulting dou- bly periodic meromorphic functions are known as Jacobian elliptic functions, though Abel was actually the first mathematician who succeeded in inverting such an indefinite integral. Nowadays, with the use of the notion of a Rie- mann surface, the inversion can be handled by using the fundamental group of the Riemann surface constructed to make the square root of the quartic polynomial single-valued. The great advantage of this approach is that there is vast literature for the properties of the Jacobain elliptic functions and of their associated Jacobian theta functions. -
Lecture 9 Riemann Surfaces, Elliptic Functions Laplace Equation
Lecture 9 Riemann surfaces, elliptic functions Laplace equation Consider a Riemannian metric gµν in two dimensions. In two dimensions it is always possible to choose coordinates (x,y) to diagonalize it as ds2 = Ω(x,y)(dx2 + dy2). We can then combine them into a complex combination z = x+iy to write this as ds2 =Ωdzdz¯. It is actually a K¨ahler metric since the condition ∂[i,gj]k¯ = 0 is trivial if i, j = 1. Thus, an orientable Riemannian manifold in two dimensions is always K¨ahler. In the diagonalized form of the metric, the Laplace operator is of the form, −1 ∆=4Ω ∂z∂¯z¯. Thus, any solution to the Laplace equation ∆φ = 0 can be expressed as a sum of a holomorphic and an anti-holomorphid function. ∆φ = 0 φ = f(z)+ f¯(¯z). → In the following, we assume Ω = 1 so that the metric is ds2 = dzdz¯. It is not difficult to generalize our results for non-constant Ω. Now, we would like to prove the following formula, 1 ∂¯ = πδ(z), z − where δ(z) = δ(x)δ(y). Since 1/z is holomorphic except at z = 0, the left-hand side should vanish except at z = 0. On the other hand, by the Stokes theorem, the integral of the left-hand side on a disk of radius r gives, 1 i dz dxdy ∂¯ = = π. Zx2+y2≤r2 z 2 I|z|=r z − This proves the formula. Thus, the Green function G(z, w) obeying ∆zG(z, w) = 4πδ(z w), − should behave as G(z, w)= log z w 2 = log(z w) log(¯z w¯), − | − | − − − − near z = w. -
Curvature Theorems on Polar Curves
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES Volume 33 March 15, 1947 Number 3 CURVATURE THEOREMS ON POLAR CUR VES* By EDWARD KASNER AND JOHN DE CICCO DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY, NEW YoRK Communicated February 4, 1947 1. The principle of duality in projective geometry is based on the theory of poles and polars with respect to a conic. For a conic, a point has only one kind of polar, the first polar, or polar straight line. However, for a general algebraic curve of higher degree n, a point has not only the first polar of degree n - 1, but also the second polar of degree n - 2, the rth polar of degree n - r, :. ., the (n - 1) polar of degree 1. This last polar is a straight line. The general polar theory' goes back to Newton, and was developed by Bobillier, Cayley, Salmon, Clebsch, Aronhold, Clifford and Mayer. For a given curve Cn of degree n, the first polar of any point 0 of the plane is a curve C.-1 of degree n - 1. If 0 is a point of C, it is well known that the polar curve C,,- passes through 0 and touches the given curve Cn at 0. However, the two curves do not have the same curvature. We find that the ratio pi of the curvature of C,- i to that of Cn is Pi = (n -. 2)/(n - 1). For example, if the given curve is a cubic curve C3, the first polar is a conic C2, and at an ordinary point 0 of C3, the ratio of the curvatures is 1/2. -
Congruences Between the Coefficients of the Tate Curve Via Formal Groups
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 3, Pages 677{681 S 0002-9939(99)05133-3 Article electronically published on July 6, 1999 CONGRUENCES BETWEEN THE COEFFICIENTS OF THE TATE CURVE VIA FORMAL GROUPS ANTONIOS BROUMAS (Communicated by David E. Rohrlich) 2 3 Abstract. Let Eq : Y + XY = X + h4X + h6 be the Tate curve with canonical differential, ! = dX=(2Y + X). If the characteristic is p>0, then the Hasse invariant, H, of the pair (Eq;!) should equal one. If p>3, then calculation of H leads to a nontrivial separable relation between the coefficients h4 and h6.Ifp=2orp= 3, Thakur related h4 and h6 via elementary methods and an identity of Ramanujan. Here, we treat uniformly all characteristics via explicit calculation of the formal group law of Eq. Our analysis was motivated by the study of the invariant A which is an infinite Witt vector generalizing the Hasse invariant. 1. Introduction The Tate curve in characteristic either zero or positive is an elliptic curve with canonical Weierstrass model: 2 3 (1) Eq : Y + XY = X + h4X + h6 5S +7S nk where h = 5S and h = 3 5 for S = : 4 − 3 6 − 12 k 1 qn n 1 X≥ − Note that the denominator appearing in the expression of h6 is simply a nota- tional convenience and in fact Eq is defined over Z[[q]]. Hence, since all coefficients are integral, we can specialize to positive characteristic p for any prime p of our choice and obtain Eq defined over Fp[[q]]. In all positive characteristics the Hasse invariant of the pair (Eq;!)for!the canonical invariant differential, ! = dX=(2Y + X), is equal to 1. -
NOTES on ELLIPTIC CURVES Contents 1
NOTES ON ELLIPTIC CURVES DINO FESTI Contents 1. Introduction: Solving equations 1 1.1. Equation of degree one in one variable 2 1.2. Equations of higher degree in one variable 2 1.3. Equations of degree one in more variables 3 1.4. Equations of degree two in two variables: plane conics 4 1.5. Exercises 5 2. Cubic curves and Weierstrass form 6 2.1. Weierstrass form 6 2.2. First definition of elliptic curves 8 2.3. The j-invariant 10 2.4. Exercises 11 3. Rational points of an elliptic curve 11 3.1. The group law 11 3.2. Points of finite order 13 3.3. The Mordell{Weil theorem 14 3.4. Isogenies 14 3.5. Exercises 15 4. Elliptic curves over C 16 4.1. Ellipses and elliptic curves 16 4.2. Lattices and elliptic functions 17 4.3. The Weierstrass } function 18 4.4. Exercises 22 5. Isogenies and j-invariant: revisited 22 5.1. Isogenies 22 5.2. The group SL2(Z) 24 5.3. The j-function 25 5.4. Exercises 27 References 27 1. Introduction: Solving equations Solving equations or, more precisely, finding the zeros of a given equation has been one of the first reasons to study mathematics, since the ancient times. The branch of mathematics devoted to solving equations is called Algebra. We are going Date: August 13, 2017. 1 2 DINO FESTI to see how elliptic curves represent a very natural and important step in the study of solutions of equations. Since 19th century it has been proved that Geometry is a very powerful tool in order to study Algebra. -
Lectures on the Combinatorial Structure of the Moduli Spaces of Riemann Surfaces
LECTURES ON THE COMBINATORIAL STRUCTURE OF THE MODULI SPACES OF RIEMANN SURFACES MOTOHICO MULASE Contents 1. Riemann Surfaces and Elliptic Functions 1 1.1. Basic Definitions 1 1.2. Elementary Examples 3 1.3. Weierstrass Elliptic Functions 10 1.4. Elliptic Functions and Elliptic Curves 13 1.5. Degeneration of the Weierstrass Elliptic Function 16 1.6. The Elliptic Modular Function 19 1.7. Compactification of the Moduli of Elliptic Curves 26 References 31 1. Riemann Surfaces and Elliptic Functions 1.1. Basic Definitions. Let us begin with defining Riemann surfaces and their moduli spaces. Definition 1.1 (Riemann surfaces). A Riemann surface is a paracompact Haus- S dorff topological space C with an open covering C = λ Uλ such that for each open set Uλ there is an open domain Vλ of the complex plane C and a homeomorphism (1.1) φλ : Vλ −→ Uλ −1 that satisfies that if Uλ ∩ Uµ 6= ∅, then the gluing map φµ ◦ φλ φ−1 (1.2) −1 φλ µ −1 Vλ ⊃ φλ (Uλ ∩ Uµ) −−−−→ Uλ ∩ Uµ −−−−→ φµ (Uλ ∩ Uµ) ⊂ Vµ is a biholomorphic function. Remark. (1) A topological space X is paracompact if for every open covering S S X = λ Uλ, there is a locally finite open cover X = i Vi such that Vi ⊂ Uλ for some λ. Locally finite means that for every x ∈ X, there are only finitely many Vi’s that contain x. X is said to be Hausdorff if for every pair of distinct points x, y of X, there are open neighborhoods Wx 3 x and Wy 3 y such that Wx ∩ Wy = ∅. -
Applications of Elliptic Functions in Classical and Algebraic Geometry
APPLICATIONS OF ELLIPTIC FUNCTIONS IN CLASSICAL AND ALGEBRAIC GEOMETRY Jamie Snape Collingwood College, University of Durham Dissertation submitted for the degree of Master of Mathematics at the University of Durham It strikes me that mathematical writing is similar to using a language. To be understood you have to follow some grammatical rules. However, in our case, nobody has taken the trouble of writing down the grammar; we get it as a baby does from parents, by imitation of others. Some mathe- maticians have a good ear; some not... That’s life. JEAN-PIERRE SERRE Jean-Pierre Serre (1926–). Quote taken from Serre (1991). i Contents I Background 1 1 Elliptic Functions 2 1.1 Motivation ............................... 2 1.2 Definition of an elliptic function ................... 2 1.3 Properties of an elliptic function ................... 3 2 Jacobi Elliptic Functions 6 2.1 Motivation ............................... 6 2.2 Definitions of the Jacobi elliptic functions .............. 7 2.3 Properties of the Jacobi elliptic functions ............... 8 2.4 The addition formulæ for the Jacobi elliptic functions . 10 2.5 The constants K and K 0 ........................ 11 2.6 Periodicity of the Jacobi elliptic functions . 11 2.7 Poles and zeroes of the Jacobi elliptic functions . 13 2.8 The theta functions .......................... 13 3 Weierstrass Elliptic Functions 16 3.1 Motivation ............................... 16 3.2 Definition of the Weierstrass elliptic function . 17 3.3 Periodicity and other properties of the Weierstrass elliptic function . 18 3.4 A differential equation satisfied by the Weierstrass elliptic function . 20 3.5 The addition formula for the Weierstrass elliptic function . 21 3.6 The constants e1, e2 and e3 ..................... -
Geometry of Algebraic Curves
Geometry of Algebraic Curves Fall 2011 Course taught by Joe Harris Notes by Atanas Atanasov One Oxford Street, Cambridge, MA 02138 E-mail address: [email protected] Contents Lecture 1. September 2, 2011 6 Lecture 2. September 7, 2011 10 2.1. Riemann surfaces associated to a polynomial 10 2.2. The degree of KX and Riemann-Hurwitz 13 2.3. Maps into projective space 15 2.4. An amusing fact 16 Lecture 3. September 9, 2011 17 3.1. Embedding Riemann surfaces in projective space 17 3.2. Geometric Riemann-Roch 17 3.3. Adjunction 18 Lecture 4. September 12, 2011 21 4.1. A change of viewpoint 21 4.2. The Brill-Noether problem 21 Lecture 5. September 16, 2011 25 5.1. Remark on a homework problem 25 5.2. Abel's Theorem 25 5.3. Examples and applications 27 Lecture 6. September 21, 2011 30 6.1. The canonical divisor on a smooth plane curve 30 6.2. More general divisors on smooth plane curves 31 6.3. The canonical divisor on a nodal plane curve 32 6.4. More general divisors on nodal plane curves 33 Lecture 7. September 23, 2011 35 7.1. More on divisors 35 7.2. Riemann-Roch, finally 36 7.3. Fun applications 37 7.4. Sheaf cohomology 37 Lecture 8. September 28, 2011 40 8.1. Examples of low genus 40 8.2. Hyperelliptic curves 40 8.3. Low genus examples 42 Lecture 9. September 30, 2011 44 9.1. Automorphisms of genus 0 an 1 curves 44 9.2.