Fernando Q. Gouvêa

p-adic Numbers

Having built our foundation, we can now apply the general theory to the specific case of the field ℚ of rational numbers. Extending our scope to include all fields of algebraic numbers (i.e., finite extensions of ℚ), or even to include what the experts call “global fields” in general, would not be very hard. Nevertheless, we have preferred to stick, at first, to the most concrete example available. In a later chapter, we will consider some aspects of the problem of extending valuations from ℚ to larger fields. More details about the theory of valuations on global fields can be found in several of the references.

The square-free sieve and the rank of elliptic curves

Let E be an elliptic curve over Q. A celebrated theorem of Mordell asserts that E(Q), the (abelian) group of rational points of E, is finitely generated. By the rank of E we mean the rank of E(Q). Thus the rank of E is positive if and only if E possesses an infinity of rational points. Relatively few general qualitative assertions can be made about the rank as E varies. How large can the rank get? Although we expect that there are elliptic curves over Q with arbitrarily high ranks, this is presently unknown. What is the average size of the rank? Recently Brumer and McGuinness have reported on their study of 310,716 elliptic curves of prime conductor less than 108 where they have found that 20.06% of those curves have even rank > 2. (Cf. [BM]; also see forthcoming work of Brumer where, subject to a number of standard conjectures, he shows that 2.3 is an upper bound for the average rank for all elliptic curves over Q ordered in terms of their Faltings height.) What is the behavior of the rank over the family of twists of a given elliptic curve? Here, one has three natural kinds of families of twists: (1) Quadratic twists. One can take any elliptic curve and systematically twist it by all quadratic characters (this is the type of family of elliptic curves we are concerned with in this paper). Specifically, if E is an elliptic curve over Q given by the Weierstrass equation Y2 = X + A * X + B, and D is any squarefree integer, the (quadratic) twist of E by D, ED, is given by the equation

Families of modular eigenforms

This article is an expansion of the notes to a one-hour lecture for an MSRI workshop on computational number theory. The editors of Mathematics of Computation kindly asked us to submit these notes for publication, and we are enormously pleased to do so. Our original audience did not consist of experts in the field of modular forms, and we have tried to keep this article accessible to nonexperts. We have made an experimental investigation of certain arithmetic conjec- tures using MACSYMA. This investigation requires a search for certain mod- ular eigenforms of high weight. These computations pose problems which we feel may be interesting on their own. We are novices here, and we seek advice from people more experienced in making computations of an analogous sort. We are, in fact, deeply grateful to J. F. Mestre, who came to our aid and who vastly extended our computations using PARI.1 Mestre has graciously allowed us to present his computations in this article. The kind of families we have in mind, as in the title of this lecture, has as its prototypical example the standard family of classical Eisenstein series of level 1 and weight k for k = 4, 6, 8, ... , whose Fourier expansions are given by oo

Non-Ordinary Primes: A Story

A normalized modular eigenform f is said to be ordinary at a prime p if P does not divide the p-th Fourier coefficient of f. We take f to be a modular form of level 1 and weight k ∈ {12, 16, 18, 20, 22, 26} and search for primes where f is not ordinary. To do this, we need an efficient way to compute the reduction modulo p of the p-th Fourier coefficient. A convenient formula was known for k = 12; trying to understand it leads to generalized Rankin–Cohen brackets and thence to formulas that we can useto look for non-ordinary primes. We do this for p ≤ 1 000 000.

Arithmetic of Diagonal Hypersurfaces over Finite Fields

1. Twisted Jacobi sums 2. Cohomology groups of n=nnm(c) 3. Twisted Fermat motives 4. The inductive structure and the Hodge and Newton polygons 5. Twisting and the Picard numbers n=nmn(c) 6. Brauer numbers associated to twisted Jacobi sums 7. Evaluating the polynomials Q(n,T) at T=q-r 8. The Lichtenbaum-Milne conjecture for n=nnm(c) 9. Observations and open problems.

Deforming galois representations: Controlling the conductor

where k is a finite field of characteristic p, was introduced recently by Mazur in [ 171. In the special case where IZ = 2, Mazur went on to examine several examples in some depth, and to pose the question whether the representations that arise are associated, in some sense, to modular forms. This question relates to recent work of Hida [ 10-131, Mazur and Tilouine [19], Wiles [25], and the author [S, 93. A summary of Mazur’s basic set-up is presented in the first section of this paper. Suppose that we begin with a representation

A guide to groups, rings, and fields

Preface A guide to this guide 1. Algebra: classical, modern, and ultramodern 2. Categories 3. Algebraic structures 4. Groups and their representations 5. Rings and modules 6. Fields and skew fields Bibliography Index of notations Index.

Quadratic Twists of Rigid Calabi–Yau Threefolds Over ℚ

We consider rigid Calabi–Yau threefolds defined over \(\mathbb{Q}\) and the question of whether they admit quadratic twists. We give a precise geometric definition of the notion of a quadratic twists in this setting. Every rigid Calabi–Yau threefold over \(\mathbb{Q}\) is modular so there is attached to it a certain newform of weight 4 on some Γ 0(N). We show that quadratic twisting of a threefold corresponds to twisting the attached newform by quadratic characters and illustrate with a number of obvious and not so obvious examples. The question is motivated by the deeper question of which newforms of weight 4 on some Γ 0(N) and integral Fourier coefficients arise from rigid Calabi–Yau threefolds defined over \(\mathbb{Q}\) (a geometric realization problem).

A Guide to Groups, Rings, and Fields by Fernando Gouvêa

Preface A guide to this guide 1. Algebra: classical, modern, and ultramodern 2. Categories 3. Algebraic structures 4. Groups and their representations 5. Rings and modules 6. Fields and skew fields Bibliography Index of notations Index.

Math through the Ages: A Gentle History for Teachers and Others

History in the mathematics classroom The history of mathematics in a large nutshell Sketches: 1. Keeping count - writing whole numbers 2. Reading and writing arithmetic - where the symbols came from 3. Nothing becomes a number - the story of zero 4. Broken numbers - writing fractions 5. Something less than nothing? - negative numbers 6. By tens and tenths - metric measurement 7. Measuring the circle - the story of p 8. The Cossic art - writing algebra with symbols 9. Linear thinking - solving first degree equations 10. A square and things - quadratic equations 11. Intrigue in renaissance Italy - solving cubic equations 12. A cheerful fact - the Pythagorean theorem 13. A marvelous proof - Fermats last theorem 14. On beauty bare - Euclids plane geometry 15. In perfect shape - the Platonic solids 16. Shapes by the numbers - coordinate geometry 17. Impossible, imaginary useful - complex numbers 18. Half is better - sine and cosine 19. Strange new worlds - the non-Euclidean geometries 20. In the eye of the beholder - projective geometry 21. Whats in a game - the start of probability theory 22. Making sense of data - statistics becomes a science 23. Machines that think - electronic computers 24. Beyond counting - infinity and the theory of sets What to read next Bibliography Index.