This appendix first gives some background material on counting methods. Many probability problems require counting techniques. In particular, these techniques are extremely useful for computing probabilities in a chance experiment in which all possible outcomes are equally likely. In such experiments, one needs effective methods to count the number of outcomes in any specific event. In counting problems, it is important to know whether the order in which the elements are counted is relevant or not. After the discussion on counting methods, the Appendix summarizes a number of properties of the famous number e and the exponential function ex both playing an important role in probability.

Permutations

How many different ways can you arrange a number of different objects such as letters or numbers? For example, what is the number of different ways that the three letters A, B, and C can be arranged? By writing out all the possibilities ABC, ACB, BAC, BCA, CAB, and CBA, you can see that the total number is six. This brute-force method of writing down all the possibilities and counting them is naturally not practical when the number of possibilities gets large, for example the number of different ways to arrange the 26 letters of the alphabet. You can also determine that the three letters A, B, and C can be written down in six different ways by reasoning as follows. For the first position, there are three available letters to choose from, for the second position there are two letters over to choose from, and only one letter for the third position.

In the first part of this book, we worked many times with models of random variables. In performing a chance experiment, one is often not interested in the particular outcome that occurs but in a specific numerical value associated with that outcome. Any function that assigns a real number to each outcome in the sample space of the experiment is called a random variable. The purpose of this chapter is to familiarize the reader with a number of basic rules for calculating characteristics of random variables such as the expected value and the variance. These rules are easiest explained and understood in the context of discrete random variables. Therefore, the discussion in this chapter is restricted to the case of discrete random variables. However, the rules for discrete random variables apply with obvious modifications to other types of random variables as well. In Chapter 10, we discuss so-called continuous random variables. Such random variables have a continuous interval as the range of possible values.

Random variables

Intuitively, a random variable is a variable that takes on its values by chance. The convention is to use capital letters such as X, Y, Z to denote random variables. Formally, a random variable is defined as a real-valued function on the sample space of a chance experiment. A random variable X assigns a numerical value X(ω) to each element ω of the sample space. For example, if X is the sum of the dots when rolling twice one fair die, the random variable X assigns the numerical value i + j to the outcome (i, j) of the chance experiment.

Constructing the mathematical foundations of probability theory has proven to be a long-lasting process of trial and error. The approach consisting of defining probabilities as relative frequencies in cases of repeatable experiments leads to an unsatisfactory theory. The frequency view of probability has a long history that goes back to Aristotle. It was not until 1933 that the great Russian mathematician Andrej Nikolajewitsch Kolmogorov (1903–1987) laid a satisfactory mathematical foundation of probability theory. He did this by taking a number of axioms as his starting point, as had been done in other fields of mathematics. Axioms state a number of minimal requirements that the mathematical objects in question (such as points and lines in geometry) must satisfy. In the axiomatic approach of Kolmogorov, probability figures as a function on subsets of a sample space. The axioms are the basis for the mathematical theory of probability. As a milestone, the law of large numbers can be deduced from the axioms by logical reasoning. The law of large numbers confirms our intuition that the probability of an event in a repeatable experiment can be estimated by the relative frequency of its occurrence in many repetitions of the experiment. This law, which has already been discussed in Chapter 2, is the fundamental link between theory and the real world. The purpose of this chapter is to discuss the axioms of probability theory in more detail and to derive from the axioms the most basic rules for the calculation of probabilities.

When I was a student, a class in topology made a great impression on me. The teacher asked me and my classmates not to take notes during the first hour of his lectures. In that hour, he explained ideas and concepts from topology in a nonrigorous, intuitive way. All we had to do was listen in order to grasp the concepts being introduced. In the second hour of the lecture, the material from the first hour was treated in a mathematically rigorous way and the students were allowed to take notes. I learned a lot from this approach of interweaving intuition and formal mathematics.

This book, about probability as it applies to our daily lives, is written very much in the same spirit. It introduces the reader to the world of probability in an informal way. It is not written in a theorem-proof style. Instead, it aims to teach the novice the concepts of probability through the use of motivating and insightful examples. In the book, no mathematics are introduced without specific examples and applications to motivate the theory. Instruction is driven by the need to answer questions about probability problems that are drawn from real-world contexts. Most of the book can easily be read by anyone who is not put off by a few numbers and some high school algebra. The informal yet precise style of the book makes it suited for classroom use, particularly when more self-activation is required from students.

How fast can we compute the value of an L-function at the center of the critical strip?

We will divide this question into two separate questions while also making it more precise. Fix an elliptic curve E defined over ℚ and let L(E, s) be its L-series. For each fundamental discriminant D let L(E, D, s) be the Lseries of the twist ED of E by the corresponding quadratic character; note that L(E, 1, s) = L(E, s).

1. A. How fast can we compute the central value L(E, 1)?

2. B. How fast can we compute L(E, D, 1) for D in some interval say a ≤ D ≤ b?

These questions are obviously related but, as we will argue below, are not identical.

We should perhaps clarify what to compute means. First of all, we know, thanks to the work of Wiles and others, that L(E, s) = L(f, s) for some modular form f of weight 2; hence, L(E, s), first defined on the half-plane ℜ(s)> 3/2, extends to an analytic function on the whole s-plane which satisfies a functional equation as s goes to 2 – s. In particular, it makes sense to talk about the value L(E, 1) of our L-function at the center of symmetry s = 1. The same reasoning applies to L(E, D, s).

As a first approximation to our question we may simply want to know the real number L(E, D, 1) to some precision given in advance; but we can expect something better.

Abstract

We briefly describe how to get the power of logarithm in the asymptotic for the number of vanishings in the family of even quadratic twists of a given elliptic curve. There are four different possibilities, largely dependent on the rational 2-torsion structure of the curve we twist.

Introduction

Let E be a rational elliptic curve of conductor N and Δ its discriminant, with Ed its dth quadratic twist. The seminal paper modelled the value-distribution of L(Ed, 1) via random matrix theory and applied a discretisation process to the coefficients of an associated modular form of weight 3/2. This led to the conjecture that asymptotically there are cEX3/4 (log X)3/8–1 twists by prime p < X with even functional equation and L(Ep; 1) = 0, where the 3/8 comes from random matrix theory, and the –1 comes from the prime number theorem.

We wish to determine a similar heuristic for the asymptotic for the number of twists by all fundamental discriminants |d| such that L(Ed, s) has even functional equation and L(Ed, 1) = 0. We find that the power of logarithm that we obtain depends on the growth rate of various local Tamagawa numbers of twists of E. Because of this, it is somewhat unfortunate that isogenous curves need not have the same local Tamagawa numbers. This is most particularly a problem when we have a curve with full rational 2-torsion and it is isogenous to one that only has one rational 2-torsion point; in this case, we should work with the curve with full 2-torsion.

This survey paper contains two parts. The first one is a written version of a lecture given at the “Random Matrix Theory and L-functions” workshop organized at the Newton Institute in July 2004. This was meant as a very concrete and down to earth introduction to elliptic curves with some description of how random matrices become a tool for the (conjectural) understanding of the rank of Mordell-Weil groups by means of the Birch and Swinnerton-Dyer Conjecture; the reader already acquainted with the basics of the theory of elliptic curves can certainly skip it. The second part was originally the write-up of a lecture given for a workshop on the Birch and Swinnerton-Dyer Conjecture itself, in November 2003 at Princeton University, dealing with what is known and expected about the variation of the rank in families of elliptic curves. Thus it is also a natural continuation of the first part. In comparison with the original text and in accordance with the focus of the first part, more details about the input and confirmations of Random Matrix Theory have been added.

Acknowledgments. I would like to thank the organizers of both workshops for inviting me to gives these lectures, and H. Helfgott, C. Hall, C. Delaunay, S. Miller, M. Young and M. Rubinstein for helpful remarks, in particular for informing me of work in process of publication or in progress that I was unaware at the time of the talks.

The goal of this survey is to give some insight into how well-distributed sets of matrices in classical groups arise from families of L-functions in the context of the middle column of Weil's trilingual inscription, namely function fields of curves over finite fields. The exposition is informal and no proofs are given; rather, our aim is to illustrate what is true by considering key examples.

In the first section, we give the basic definitions and examples of function fields over finite fields and the connection with algebraic curves over function fields. The language is a throwback to Weil's Foundations, which is quite out of fashion but which gives good insight with a minimum of baggage.

Abstract

We discuss the idea of a “family of L-functions” and describe various methods which have been used to make predictions about L-function families. The methods involve a mixture of random matrix theory and heuristics from number theory. Particular attention is paid to families of elliptic curve L-functions. We describe two random matrix models for elliptic curve families: the Independent Model and the Interaction Model.

Introduction

Using ensembles of random matrices to model the statistical properties of a family of L-functions has led to a wealth of interesting conjectures and results in number theory. In this paper we survey recent results in the hopes of conveying our best current answers to these questions:

1. What is a family of L-functions?

2. How do we model a family of L-functions?

3. What properties of the family can the model predict?

In the remainder of this section we briefly review some commonly studied families and describe some of the properties which have been modeled using ideas from random matrix theory. In Section 2 we provide a definition of “family of L-functions” which has been successful in permitting precise conjectures, and we briefly describe how to model such a family. In Section 3 we discuss families of elliptic curve L-functions and show that there is an additional subtlety which requires us to slightly broaden the class of random matrix models we use.