Appendix A Appendix B Modular Symbol Algorithms, List of American Autobiographical List of Suggested Writing Prompts Literature How does where you’re from or your family’s heritage influence Computational Number Theory, who you are? Reflect on your geographic, ethnic, cultural From The Best American Essays of the Century. Ed. Joyce Carol or religious background. and the Millennium Problems Oates and Robert What are some unsaid rules in your life (as a woman or man, as Atwan. New York: Houghton Mifflin Company, 2000. a student or teacher, as a young person, as a Hungarian, as Angelou, Maya. “I Know Why the Caged Bird Sings.” 342-357. a member of whichever group with which you identify)? Bourne, Randolph. “The Handicapped.” 57-70. What do you want outsiders to know about your culture? Write Ehrlich, Gretel. “Solace of Open Spaces.” 467-76. about your culture. Stephanie Jakus Herr, Michael. “Illumination Rounds.” 327-341. Write about a time you had to stand up for a belief when most Kingston, Maxine Hong. “No Name Woman.” 383-94. people didn’t agree with you. Momaday, N. Scott. “The Way to Rainy Mountain.” 313-18. Whose voices are part of multicultural Hungary? Who is a Oates, Joyce Carol. “They All Just Went Away.” 553-63. Hungarian? Rodriguez, Richard. “Aria: A Memoir of a Bilingual ...... Childhood.” 447-466. Bibliography Twain, Mark. “Corn-pone Opinions.” 1-5. Hideg, Éva and Erzsébet Nováky. “The Future Orientation of The University of Michigan, Ann Arbor Eötvös Loránd University Wright, Richard. “The Ethics of Living Jim Crow: An Hungarian Youth in the Years 2074 East Hall, 530 Church Street, 1117 Budapest, Pázmány Péter sétány 1/C Autobiographical Sketch.” 159 -70. of the Transformation.”World Futures Studies Federation. Ann Arbor, MI 48109-1043 www.elte.hu From Literature and Society: An Introduction to Fiction, 1997. http://www.wfsf.org/pub http://www.math.lsa.umich.edu/ Adviser: Árpád Tóth Poetry, Drama, Nonfiction. Ed. /publications/Brisbane_97/HIDEGNOV.pdf [email protected] Pamela Annas and Robert Rosen. New Jersey: Prentice-Hall, Raphael, Taffy, Laura Pardo and Kathy Highfield. Book Club: 1994. A Literature-Based ...... Kovic, Ron. From “Born on the Fourth of July.” 1059-69. Curriculum. Massachusetts: Small Planet Communications,
Whitecloud, Thomas. “Blue Winds Dancing.” 1336-41 1997. As one of the Millennium Problems, the solutions to which carry a prize of one million dollars Wright, Richard. “The Man Who Went to Chicago.” 858-82. ---- “Book Club Workshop: Learning About Language and a piece, the Birch Swinnerton-Dyer conjecture, since its introduction in the early 1960’s, has
Kingsolver, Barbara. High Tide in Tucson: Essays from Now or Literacy through Culture.” Ciera. remained both a fundamental unsolved problem in algebraic number theory and one of the Never. USA: Harper April 1999. http://www.ciera.org/library/archive/1999-04/ most challenging problems of the twenty first century. My work on the Fulbright fellowship Perennial, 1996. 1-16. abs-online-99-04.html In J. in computing cohomology groups using the modular symbol method is not in the direction of Michie, Gregory. Holler If You Hear Me. New York: Teachers Many (Ed.), Instructional practices for literacy teacher- proving the Millennium Problem, but in implementing the tools of computational number College Press, 1999. 1- 12. educators. Mahwah, NJ: Erlbaum. 39-49. theory that have developed in the past twenty years of progress on the conjecture, and in the Sandburg, Carl. “Chicago.” http://www.carl-sandburg.com/ direction of expanding these tools for new uses. The mathematics behind the algorithms in this chicago.htm. 7 April 2006. project lies in the intersection of algebraic number theory, homology and cohomology theory, Sedaris, David. Me Talk Pretty One Day. USA: Little Brown, complex analysis, and algebraic geometry, and is well explained in the order of its development 2000. 153-65. using the history of the Birch Swinnerton-Dyer conjecture. Selzer, Richard. From “Confessions of a Knife” Modern American Memoirs. Ed. Annie One of the seven Millennium Problems of the Clay Mathematics Institute in Dillard and Cort Conley. USA: Harper Perennial, 1996. 100-08. Cambridge, Massachusetts is the Birch Swinnerton-Dyer conjecture, named after two British mathematicians, Bryan Birch and Peter Swinnerton-Dyer, who first formulated the conjecture. The conjecture relates the number of infinite order basis elements of
204 205 AY 2005-2006 Stephanie Jakus: Symbol Algorithms, Computational Number Theory the group of rational points on an elliptic is, “all preface from beginning to end,” In the early part of the nineteenth sequence formed by counting the number curve to what is called the L-function and is aimed at a general audience century, Louis Joel Mordell (1888-1972), of solutions to the equation of an elliptic of the curve. As one of the Millennium without omitting technical terminology, an American born mathematician who curve modulus a prime for each prime Problems, the solutions to which carry a but also without developing it to any worked for most of his life in England, number. Taniyama worked with Goro prize of one million dollars a piece, the extent. The reader is referred to [3],[8], made contributions to the theory of Shimura through 1957 on refining the Birch Swinnerton-Dyer conjecture, since or [12] for details about the mathematics. modular forms by using what is now conjecture, but in 1958, for reasons that its introduction in the early 1960’s, has The history of the relationship between know as Hecke operators to prove one were allusive even to Taniyama himself, remained both a fundamental unsolved modular forms and elliptic curves that of Ramanujan’s conjectures [9]. This he committed suicide. Shortly after, his problem in algebraic number theory and lead to modular symbol algorithms was followed by the introduction of the fiancee also committed suicide because one of the most challenging problems of and to the partial solution of the Birch L-function by Erich Hecke (1887-1947), she had vowed to Taniyama never to the twenty first century. The history of the Swinnerton-Dyer conjecture, begins with and his research on the properties of part from him [14]. For a long time the problem is noteworthy, as the predecessor such mathematical luminaries as Srinivasa the algebra of Hecke operators, two conjecture was not even recognized as to the Birch Swinnerton-Dyer conjecture, Ramanujan, Felix Klein, and Henri crucial steps in the direction of the being correct, let alone of significance. the Taniyama Shimura theorem led to Poincare. Srinivasa Aiyangar Ramanujan Birch Swinnerton-Dyer conjecture [5]. It was not until 1980 that the German one of the most well-publicized results in (1887-1920), the self-taught Indian Hecke operators play a role as averaging mathematician Gerhard Frey suggested number theory, the solution of Fermat’s number theorist who is credited with operators on the space of modular forms, that the Taniyama Shimura conjecture Last Theorem, while the development over three thousand theorems and had and L-functions are functions with two implied Fermat’s Last Theorem. Fermat’s of the mathematics instigated by incredible insight into the relevance of inputs, an elliptic curve and a number Last Theorem is the statement that it is work on the conjecture has led to new modular forms in number theory, studied from the complex plane. An elliptic curve impossible to find integer solutions to an error-correcting codes from the algebraic a particular modular form, the cusp form is a polynomial of the form y2=f(x), where equation of the form of the Pythagorean geometry of elliptic curves and an during the beginning of the twentieth f(x) is a polynomial with degree three, or theorem, but with exponents, or degree, improved method for factoring integers century [10]. At the same time Felix the highest power of x being x3. It wasn’t greater than two. In the seventeenth based on elliptic curves over finite fields. Klein (1849-1925) and Henri Poincare until the end of the 1950’s, however, that century, Pierre de Fermat had scribbled a My work on the Fulbright fellowship (1854-1912) studied automorphic the extent of the relationship between note in the margin of a math book he was in computing cohomology groups using functions and Klein summarized his work modular forms and the theory of elliptic studying, saying that he had discovered the modular symbol method is not in on automorphic and elliptic modular curves was hypothesized. a marvelous proof of this theorem, but the direction of proving the Millennium functions in a four volume treatise [6],[7]. The connection between modular the margin did not contain enough room Problem, but in implementing the tools A modular function is a meromorphic forms and elliptic curves appeared in for the proof. Many speculate that he of computational number theory that function on the complex upper-half 1955 as Yutaka Taniyama’s conjecture could not have found such a proof, and, have developed in the past twenty years plane that is invariant under the action that all elliptic curves over the rationals in any case, he never wrote it down. of progress on the conjecture, and in of certain groups of matrices on the are modular. By modular elliptic curve, The conjecture remained unproven for the direction of expanding these tools upper-half plane, while a modular form Taniyama meant that the elliptic curve three hundred fifty seven years. In the for new uses. The mathematics behind is holomorphic on the upper-half plane could be associated with a modular early nineteen nineties, Ken Ribet of the algorithms in this project lies in the union a point at infinity with the same form in the following manner: being the University of California, Berkeley intersection of algebraic number theory, invariance. The terms holomorphic and holomorphic, every modular form has a proved that the Taniyama Shimura homology and cohomology theory, meromorphic from complex analysis Fourier transform (a special kind of power conjecture would indeed prove Fermat’s complex analysis, and algebraic geometry, refer to the existence of a power series series representation in terms of sine Last Theorem, and in 1999 Andrew and is well explained in the order of its representation for the function at all and cosine) and Taniyama conjectured Wiles of Princeton University proved the development using the history of the points on which the function is defined that the sequence of coefficients in the Taniyama Shimura conjecture [13]. Birch Swinnerton-Dyer conjecture. This or on all but a discrete set of points Fourier transform corresponds to a In 1993, Wiles gave a lecture on his essay, in the words of Benoit Mandelbrot respectively.
206 207 AY 2005-2006 Stephanie Jakus: Symbol Algorithms, Computational Number Theory progress on the Taniyama Shimura are modular, thus advancing the scope of IV.,” published by Springer-Verlag, Just as modular forms can be defined conjecture to the Issac Newton Institute the earlier proofs [11]. which contains the proceedings of on the upper-half plane, so can they be at Cambridge University, and, in 1995, Since Wiles’ proof numerous books the International Summer School on, defined on Riemann surfaces that arise Wiles submitted a manuscript of what have been written on the subject of “Modular functions of one variable and from the upper-half plane by considering he thought was a proof of the conjecture Fermat’s last theorem, the most popular arithmetical applications.” The Antwerp a subsection, a so called fundamental to Inventiones Mathematicae, one of of which is Simon Singh’s book, “Fermat’s Tables published Birch and Swinnerton- domain, that is formed by taking Springer-Verlag’s journals. A team of Enigma: The Epic Quest to Solve Dyer’s results as well as the those of other equivalence classes of points in the half six was organized by Barry Mazur of the World’s Greatest Mathematical mathematicians who had been computing plane under a the action of a matrix group Harvard University, editor of Inventiones Problem.” While Fermat’s Last invariants of elliptic curves such as A. O. L. on the plane where two points in the plane Mathematicae, to review Wiles’ proof Theorem may not be the world’s greatest Atkin [12]. The Antwerp Tables were the are equivalent if there is a group element and included Ken Ribet, Nick Katz of mathematical problem, efforts to prove first site for publishing and disseminating that takes one to the other. Modular Princeton University, and Richard Taylor the theorem have certainly led to the the previously uncollected results of these forms on these hyperbolic surfaces are of Harvard University. The proof of the development of algebraic number theory. mathematicians. In the 1990’s Henri elements of the first cohomology group theorem depended on something called Much of the theoretical mathematics that Cohen of the Université Bordeaux I, also of the surface, a topological invariant an Euler System, but the review team was developed to prove Fermat’s Last the inventor of the Pari computer algebra of the surface. The i-dimensional found an error with the use of the Euler Theorem is contained in such books as system designed for fast computations in cohomology groups and their dual System, and so Wiles was left to correct Yves Hellegouarch’s, “Invitation to the number theory, with Nils-Peter Skoruppa homology groups measure the same his proof. He invited Richard Taylor, who Mathematics of Fermat-Wiles,” and of the Universität Siegen and Don Bernard property of a space, which is intuitively had formerly been his Ph.D. student, to perhaps the definitive text on the subject, Zagier, Skoruppa’s doctoral advisor, made the number of i-dimensional rooms return to Princeton where they worked though requiring a lot of mathematical extensive tables of elliptic curves but never in a space. The equivalent to modular together, and Taylor eventually found a background, is Gary Cornell, Joseph published them in a book [12]. forms in cohomology, in homology are way around using the Euler System, thus Silverman, and Glenn Stevens,’ “Modular In 1992, and again in 1997, J.E. called modular symbols, and these play completing the proof. The final version of Forms and Fermat’s Last Theorem.” [12] Cremona of the University of Nottingham an important role in computing elliptic Wiles’ proof was one hundred eight pages In addition to the theory behind published a book, “Algorithms for curves over the rationals. and a complementary paper published by Fermat’s Last Theorem, the field of Modular Elliptic Curves,” with the In the chapter, “Modular Symbol Taylor on the techniques used to finish computational algebraic number theory Cambridge University Press. The second Algorithms,” Cremona focuses on the proof added another nineteen pages developed to test conjectures related to chapter of the book, “Modular Symbol computing modular symbols of the [13]. the theorem. One of the first algorithms Algorithms,” was the basis for my hyperbolic surface that arises from Wiles’s proof of the Taniyama Shimura written in computational algebraic number Fulbright project. Cambridge University considering the upper half plane under conjecture was a major breakthrough for theory was that of Bryan Birch and Peter Press decided not to republish Cremona’s the action of gamma zero, one of the the Birch Swinnerton-Dyer conjecture Swinnerton-Dyer when they tested the book after the 1997 edition, and so modular groups. Though he wrote code because in 1990 and 1991 Victor Kolyvagin Birch Swinnerton-Dyer conjecture on the Cremona put the book on the web, and it in the programming language Algol68 of The Graduate Center at CUNY and large EDSAC computer at Cambridge in became a well known tool and reference to implement his algorithms, in his book Karl Rubin of Stanford University had the early 1960’s. Since then, a number of for algebraic number theorists wishing to he gives an overview of the mathematics made significant advances on proving mathematicians have written programs write algorithms to compute the modular behind computing modular symbols, the Birch Swinnerton-Dyer conjecture, to compute invariants of elliptic curves. symbols of elliptic curves [3]. The second and does not give an explicit algorithm but these advances applied only to those One of the most well known references chapter of his book, “Modular Symbol for computing them [13]. While rational elliptic curves that were modular. for data on elliptic curves is the 1975 Algorithms,” gives a detailed outline of programming in standard computer The proof of the Taniyama Shimura Antwerp Tables, formally known as the steps needed to write an algorithm to languages is perhaps optimal in terms of established that all rational elliptic curves “Modular Functions of One-Variable. compute what is called a modular symbol. efficiency, it is also time consuming to
208 209 AY 2005-2006 Stephanie Jakus: Symbol Algorithms, Computational Number Theory write code and find algorithms to compute available to students and researchers in ELSE K:=(-1)^(I-1)*Q[I]*Q[I-1]^(-1) MOD N; END_IF; standard mathematical operations. it’s most user friendly form for shareware A[K+1]:=A[K+1]+1; In the years since Cremona first wrote prices or with a more primitive interface END_FOR; his book, a wealth of so called computer for free download off of the internet. END_IF; /* WE COMPUTE THE PARTIAL FRACTION CONVERGENTS FROM THE SECOND MODULAR algebra systems, software programs with While other computer algebra systems SYMBOL */ many built in mathematical operations such as Henri Cohen’s PARI, mentioned IF D=0 THEN A[1]:=A[1]-1; ELSE Y:=OP(CONTFRAC(B/D),1); and settings, were developed to aid above, are also available as freeware, and QBAR:=ARRAY(-2..NOPS(Y)-1); QBAR[-2]:=1; QBAR[-1]:=0; QBAR[0]:=1; mathematical research. Two of the most much faster for computations in number FOR I FROM 1 TO NOPS(Y)-1 DO QBAR[I]:= Y[I+1]*QBAR[I-1]+QBAR[I-2];END_FOR; famous of these computer algebra systems, theory, computing and writing algorithms FOR I FROM -1 TO NOPS(Y)-1 DO Mathematica and Maple, are used and in the general computer algebra systems IF QBAR[I-1] MOD N=0 THEN K:=N: sold widely in the United States. The is easier and more versatile. ELSE K:=(-1)^(I-1)*QBAR[I]*QBAR[I-1]^(-1) MOD N; newer competitor, MuPad, developed Therefore, I implemented the END_IF; by the MuPad Research Group at the mathematics in Cremona’s, “Modular A[K+1]:=A[K+1]-1; END_FOR; University of Paderborn in Germany Symbol Algorithms,” in MuPad by writing END_IF; using the earlier commercial software as a loop that computes modular symbols A; prototypes, is virtually unheard of in the for gamma zero over a prime modulus END_PROC PROC COEFFICIENT(A, B, C, D) ... END United States, is very similar to the well for a given set of Hecke operators. The known systems, a bit easier to use, and annotated loop is as follows: /* HERE WE BEGIN THE PROCEDURE HECKE, WHICH, COMPUTES THE HECKE EIGENVALUES */ HECKE:=PROC(P) BEGIN /* N IS THE LEVEL FOR GAMMA ZERO */ /* HERE WE LIST THE HECKE OPERATORS */ N:=11: MATRIX:=(): /* CONSTRUCTOR IS THE FUNCTION THAT WILL GENERATE THE ARRAY OF COEFFICIENTS OF FOR S FROM 0 TO P-1 DO COSET REPRESENTATIVES*/ T(S):=MATRIX([[1,S],[0,P]]) CONSTRUCTOR:=DOM::MATRIX(DOM::RATIONAL); END_FOR: /* HERE WE WRITE THE MATRICIES THAT CORRESPOND TO THE COEFFICIENTS IN THE ABOVE T(P):=MATRIX([[P,0],[0,1]]): ARRAY */ /* HERE WE COMPUTE THE HECKE MATRIX */ /* WE WON’T USE THEM UNTIL THE HECKE PROCEDURE */ FOR T FROM 1 TO N+1 DO FOR K FROM 1 TO N DO ADD:=CONSTRUCTOR(1,N+1); M(K):=MATRIX([[1,0],[K-1,1]]); FOR S FROM 0 TO P DO END_FOR: A:=(T(S)*M(T))[1,1];B:=(T(S)*M(T))[1,2];C:=(T(S)*M(T))[2,1];D:=(T(S)*M(T))[2,2]; M(N+1):=MATRIX([[0,-1],[1,0]]): ADD:=ADD + COEFFICIENT(A,B,C,D); /* HERE WE BEGIN THE PROCEDURE COEFFICIENT, THE INPUT OF WHICH ARE THE ENTRIES OF END_FOR; A MATRIX */ MATRIX:=MATRIX.CONSTRUCTOR::TRANSPOSE(ADD); /* AND THE OUTPUT OF WHICH IS THE COEFFICIENT ARRAY */ END_FOR; COEFFICIENT:=PROC(A,B,C,D) H:=CONSTRUCTOR::TRANSPOSE(MATRIX); BEGIN LINALG::EIGENVALUES(H); A:=CONSTRUCTOR(1,N+1); END_PROC /* HERE WE COMPUTE THE PARTIAL FRACTION CONVERGENTS FROM THE FIRST MODULAR PROC HECKE(P) ... END SYMBOL */ /* THE TECHNIQUE IS OUTLINED IN CREMONA CHAPTER 2 */ HECKE(2) IF C=0 THEN A[1]:=A[1]+1; ELSE X:=OP(CONTFRAC(A/C),1); {-2, 0, 3} Q:=ARRAY(-2..NOPS(X)-1); HECKE(3) Q[-2]:=1; Q[-1]:=0; Q[0]:=1; {-1, 0, 1, 4} FOR I FROM 1 TO NOPS(X)-1 DO Q[I]:= X[I+1]*Q[I-1]+Q[I-2];END_FOR; HECKE(5) FOR I FROM -1 TO NOPS(X)-1 DO {0, 1, 6} IF Q[I-1] MOD N=0 HECKE(7) THEN K:=N: {-2, 0, 8} HECKE(11) {0, 2}
210 211 AY 2005-2006 Stephanie Jakus: Symbol Algorithms, Computational Number Theory
While elliptic curves over the rationals Gaussian Integers. The first algorithm In the process of writing these two The current era has seen a rapid expansion are well understood because of the work that computes modular symbols has a algorithms, I learned mathematics from in the techniques of computational of Wiles and his predecessors, elliptic special step that requires MuPad’s built basic homology and cohomology theory algebraic number theory, due, in part to curves over other number fields are not in function that computes continued to the introductory presentation of a great interest in the subject generated nearly as well understood. As part of my fractions with real numbers as input. algebraic number theory dealing with by the fame of Andrew Wiles’ proof of Fulbright project, I wrote an algorithm Unfortunately, this continued fraction modular forms given in Koblitz’s book, the Taniyama Shimura conjecture and that decomposes two by two matrices with function cannot be used with input from “Introduction to Elliptic Curves and Fermat’s Last Theorem, and by the high Gaussian Integer entries into a product of such special number fields as the Gaussian Modular Forms.” Based on these studies stature of the Birch Swinnerton-Dyer simpler matrices using a combination of Integers. and continuing an interest in the subject, conjecture as one of the seven Millennium the Euclidean Algorithm and Gaussian The annotated loop for computing I plan to attend the MSRI (Mathematical Problems of the Clay Mathematics Elimination via matrix multiplication, continued fractions for Gaussian Integers Sciences Research Institute) Summer Institute. My contribution towards the and from this simplified a few steps to is as follows: 2006 Graduate Workshop in development of these computational create a continued fraction algorithm for Computational Number Theory, techniques on the Fulbright program “Computing with Modular Forms.” The has been to implement J. E. Cremona’s /* HERE WE BEGIN THE PROCEDURE, ENTRIES, WHICH COMPUTES THE CONTINUED FRACTION workshop is organized by William A. suggested modular symbol algorithm OF A/B */ Stein of the University of Washington for gamma zero, and to begin to extend /* NOTE THAT C AND D ARE PLACE HOLDERS IN THIS ALGORITHM, THEIR PURPOSE WILL at Seattle, who worked with many of the algorithm for use over other number BECOME CLEAR */ the mathematicians at the University of fields. I hope to continue with this project /* IN LATER ALGORITHMS */ California at Berkeley involved in the and learn more computational techniques ENTRIES:= PROC(A,B,C,D) proof of Fermat’s Last Theorem, and at the MSRI workshop on computational BEGIN who maintains a very useful website algebraic number theory this summer. J:=1: titled, “The Modular Forms Database.” A:=MATRIX([[A,B],[C,D]]): William Stein has designed a computer [1] “Andrew Wiles.” Wikipedia, The Free Encyclopedia. 27 Apr A0:=MATRIX([[A,B],[C,D]]): algebra system called SAGE, Software for 2006, 21:51 UTC. 29 Apr 2006, 18:08
212 213 [5] “Erich Hecke.” The MacTutor History of Mathematics [10] “Srinivasa Ramanujan.” Wikipedia, The Free Archive. 10 Apr 2006, 02:18 UTC. 29 Apr 2006, 10:11 [7] “Henri Poincaré.” Wikipedia, The Free Encyclopedia. 25 Apr %93Shimura_theorem&oldid=48939074>. 2006, 15:52 UTC. 29 Apr 2006, 18:21 [8] Koblitz, Neal, Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Math. No. 97, Springer-Verlag, New [13] “Andrew Wiles.” Wikipedia, The Free Encyclopedia. 3 May York, 1984. Second edition, 1993. 2006, 19:57 UTC. 3 May 2006, 20:29 [9] “Louis Mordell.” Wikipedia, The Free Encyclopedia. 11 Apr 2006, 10:19 UTC. 29 Apr 2006, 18:12 ...... Hungary has produced the greatest number of successful concert pianists per capita in the world in the last century. Those trained inside of Hungary can be immediately identified by their distinctive approach to the piano, the sound created from the piano, and their technical prowess. Several “piano schools” exist in the world today; their individual styles are often affiliated with their national histories and the character of their different peoples. To classify the unique approach to pianism in Hungary, this paper will trace how the country’s history and connection between Western and Eastern Europe has developed its ideals of piano performance, demanding both stylistic accuracy in performance practice, musical sensibility, and technical perfection. Then, it will examine the educational process of training a young pianist and why it is so effective. The last part of this paper will briefly analyze a set of pieces written by Franz Liszt, Deux Legendes. Liszt is credited with founding the music school in Budapest, and his ideas influence the way music is taught today. By examining these compositions, this paper will also show how his musical ideas reflect those still emphasized today in the Hungarian school of pianism. 214 215