GALOIS REPRESENTATIONS AND THE MODULARITY THEOREM THOMAS MORRILL Abstract. We wish to study the algebraic closure of Q, denoted Q and its associated ring of integers Z. Often classical Galois theory is employed to examine the finite extensions of Q, but this analysis fails to apply to the infinite extension Q. Following a brief review of Galois theory, we introduce the absolute Galois group over Q and examine its group representations in `-adic vector spaces. These Galois representations have fundamental connections to modular forms and elliptic curves. Dedicated to Dr. Mic Jackson. 1. Introduction We are interested in the roots of polynomials. Restricting to polynomials with coefficients in Q yields a rich theory. The roots of polynomials in Z[x] are called algebraic numbers. It can be shown that the set of all algebraic numbers forms a field containing the set of all rational numbers Q. One method of studying algebraic numbers is to examine the finite degree extensions of Q using Galois Theory. Here we assume the reader is familiar with abstract algebra. Foundational material may be found in [DF04]. Evariste´ Galois developed his theory in the early 1800s as a tool to study the solutions of polynomial equations. Rather than study a field extension K=F directly, Galois examined its group of automorphisms, specifically those that fixed some subfield of K. So long as K is Galois over F , there is a one-to-one correspondence between the subgroups of the automor- phism groups of K and fields H that lie between F and K. This fundamental theorem, along with the simplicity of the alternating group A5, was enough to demonstrate that there is no general formula to derive the roots of quintic polynomials using radicals. Algebraic number theory typically begins by applying Galois theory to finite degree extensions of Q. However, because the algebraic numbers are an infinite degree field extension of the rationals, these results do not apply to Q. In Section 2, we begin with a brief overview of algebraic number theory and some intro- ductory results. Although classical Galois theory does not apply to infinite extensions, we may attempt to generalize its methods. We are motivated to study the automorphism group of Q, which we denote GQ. This group becomes our main object of study. In order to get a handle on the structure of GQ, we introduce the inverse limit of topological groups and rings in Section 3. We assume the reader is familiar with the basics of point- set topology, which may be found in [Mun00]. Inverse limits allow us to decompose an automorphism of Q into a sequence of automorphisms of number fields. Moreover, other objects which arise from inverse limits, such as the `-adic numbers Q`, their analog λ-adic number Kλ, and the Tate module of torsion subgroups have a natural affinity with GQ. With the automorphism group in hand, in Section 4 we compare the algebraic structure of Q to the number fields from Section 2. Often in group theory, the problem of studying an 1 abstract group may be simplified by mapping it into matrix groups, or equivalently viewing its group action on vector spaces. These tools allow us to apply linear algebra to the study of GQ. In Section 5 we introduce modular forms, another large topic in number theory, in order to showcase some results using Galois representations. First, we define the modular group Γ , also known as SL2(Z), and establish its group action on the upper half plane H of C. A modular form f(z) is a holomorphic function whose values are influenced by this group action. We assume the reader is familiar with complex analysis. Additional information may be found in [Ash71]. The space of modular forms is a complex vector space, with a basis given by newforms. The Fourier coefficients of newforms give rise to number fields, and subsequently Galois representations ρf . With this in place we prove a theorem by the French mathematician J.P. Serre regarding congruences, prime density, and the irreducibility of ρf . All figures were made in [S+11]. Elliptic curves are introduced in Section 6, and they also give rise to Galois representa- tions. An elliptic curve is the solution set of a cubic polynomial in two variables, as viewed in projective space. As long as the curve is non-singular, it may be equipped with a geomet- rically powered group structure. The torsion subgroups of elliptic curves subsequently allow for the influence of GQ. Finally, in Section 7 we close the paper with some statements of the modularity theorem, a deep connection found between elliptic curves and modular forms. In plain language, the theorem states that all elliptic curves exhibit properties inherited from modular forms. It was the partial proof of this theorem in 1993 by Andrew Wiles that yielded the proof of Fermat's Last Theorem. The proof is too large for the scope of this paper, much less the margin of a textbook, but we give a sketch of how modularity relates to xn + yn = zn. We include it here as Wiles proved his version of modularity as stated in the language of Galois representations. 2. Algebraic Number Theory We are interested in the collection of all roots of polynomials with coefficients over Q, denoted Q, called the algebraic closure of Q. Recall the Fundamental Theorem of Algebra: Theorem 1. If f(x) 2 C[x], then f has a root in C: A proof may be found in Chapter 14 of [DF04]. By inducting on the number of roots of a polynomial f, we see that Q ⊂ C. A number α 2 C is called algebraic if there exists an irreducible polynomial f with coefficients in Z such that f(α) = 0. We call f the minimal polynomial for α if f is irreducible and monic. Complex numbers that are not the root of some polynomial over Z are not algebraic. These are called transcendental numbers. Although transcendence theory is a very rich topic in number theory, we will not explore transcendental numbers further in this paper.Any polynomial g 2 Z[x] that has α as a root must be divisible by f. If α is the root of a irreducible monic (lead coefficient 1) polynomial in Z, then α is called an algebraic integer. The set ofp algebraic integers is denoted by Z. For example, consider the golden ratio φ = (1 + 5)=2. As φ is a root of the irreducible polynomial f(x) = x2 − x − 1; 2 and f is a monic polynomial in Z[x], we see that φ is an algebraic integer. By factoring f(x) in C, p ! p ! 1 + 5 1 − 5 x2 − x − 1 = x − x − ; 2 2 p 0 we find a second algebraic integer, (1 − 5)=2, whichp we denote φ for brevity. Note that the only difference between φ and φ0 is the sign of 5; we will revisit this when dealing with automorphisms of quadratic fields. Also note that −φφ0 = 1. That is, both φ and φ0 are units in Z, as they both have a multiplicative inverse. Units will become important when we try to factor algebraic integers. More generally, if α is an algebraic number with minimal polynomial f and β is another root of f, then we say that α and β are algebraic conjugates. For example, the polynomial x2 + 1 is irreducible in Z[x]. Its complex roots are i and −i, which means that i and −i are algebraic conjugates as well as complex conjugates. In fact, complex conjugation is a special case of algebraic conjugation. To see this, let p(x) be an irreducible polynomial in Z and suppose that α is a root of p. Then p(α) = 0 = 0 = p(α) = p(α): That is, the complex conjugate of α is also one of its algebraic conjugates. However, many algebraic numbers have more than one algebraic conjugate. To clarify terminology, we will refer to the distinct roots of the minimal polynomial of α as the conjugates of α and refer to the operation a + bi = a − bi as complex conjugation. In algebraic number theory, we do not study Q directly; rather, we examine its smaller subfields. A number field F is a field Q ⊆ F ⊆ C which is a finite dimensional vector space over Q. The degree of F over Q, written [F : Q], is the dimension of F as a vector space over Q. If α1; : : : ; αn 2 C, the set Q adjoin α1; : : : ; αn is given by f(α1; : : : ; αn) Q(α1; : : : ; αn) := f; g 2 Q[x1; : : : ; xn]; g(α1; : : : ; αn) 6= 0 : g(α1; : : : ; αn) We see that Q(α1; : : : ; αn) is a subfield of C. Likewise, Z adjoin α1; : : : ; αn is given by Z[α1; : : : ; αn] := ff(α1; : : : ; αn) j f 2 Z[x1; : : : ; xn]g ; which forms a subring of C. Note that Q(α) contains multiplicative inverses for all its elements, whiles Z[α] does not. We state the following theorem without proof. Theorem 2. A number α 2 C is algebraic if and only if Q(α) is a number field. Moreover, if F is a number field, then there exists α 2 Q such that F = Q(α). A proof may be found in Chapters 1 and 2 of [ST02]. We immediately see that if β is an element of the number field F = Q(α), then Q(β) must be a finite dimensional Q vector space, as it is a subspace of Q(α). Therefore β is algebraic. We now consider our first examples of number fields.