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Iwasawa Theory: a Climb up the Tower

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Iwasawa Theory: a Climb up the Tower Iwasawa Theory: A Climb up the Tower Romyar Sharifi Iwasawa theory is an area of number theory that emerged Through such descriptions, Iwasawa theory unveils in- from the foundational work of Kenkichi Iwasawa in the tricate links between algebraic, geometric, and analytic ob- late 1950s and onward. It studies the growth of arithmetic jects of an arithmetic nature. The existence of such links objects, such as class groups, in towers of number fields. is a common theme in many central areas within arith- Its key observation is that a part of this growth exhibits metic geometry. So it is that Iwasawa theory has found a remarkable regularity, which it aims to describe in terms itself a subject of continued great interest. This year’s Ari- of values of meromorphic functions known as 퐿-functions, zona Winter School attracted nearly 300 students hoping such as the Riemann zeta function. to learn about it! The literature on Iwasawa theory is vast and often tech- Romyar Sharifi is a professor of mathematics at UCLA. His email addressis nical, but the underlying ideas possess undeniable beauty. [email protected]. I hope to convey some of this while explaining the origi- Communicated by Notices Associate Editor Daniel Krashen. nal questions of Iwasawa theory and giving a sense of the This material is based upon work supported by the National Science Foundation directions in which the area is heading. under Grant Nos. 1661658 and 1801963. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti/1759 16 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 1 Algebraic Number Theory in which case it has dense image in ℂ. The numbers of real and complex-conjugate pairs of complex embeddings are To understand Iwasawa theory requires some knowledge respectively denoted 푟1(퐹) and 푟2(퐹). × of the background out of which it arose. We attempt to The unit group 풪퐹 of invertible elements in 풪퐹 under chart a course, beginning with a whirlwind tour of the el- multiplication is deeply intertwined with the class group ements of algebraic number theory. We make particular Cl퐹. In fact, these groups are the kernel and cokernel of × note of two algebraic objects, the class group and the unit the map 퐹 → 퐼퐹 taking an element to its principal frac- × group of a number field, that play central roles for us. tional ideal. Dirichlet’s unit theorem says that 풪퐹 is a di- Algebraic numbers are the roots inside the complex num- rect product of the group of roots of unity in 퐹 and a free bers of nonzero polynomials in a single variable with ra- abelian group of rank 푟 = 푟1(퐹) + 푟2(퐹) − 1. This is tional coefficients. They lie in finite field extensions of ℚ proven using logarithms of absolute values of units with called number fields. The set of algebraic numbers forms respect to archimedean embeddings. The regulator 푅퐹 of a subfield ℚ of ℂ known as an algebraic closure of ℚ. In- 퐹 is a nonzero real number defined as a determinant ofa side ℚ sits a subring ℤ of algebraic integers, consisting of matrix formed out of such logarithms. the roots of monic polynomials with integer coefficients. The prototypical example is 퐹 = ℚ(√−5), for which The field automorphisms of ℚ form a huge group called 풪퐹 = ℤ[√−5]. One has 푟1(퐹) = 0, 푟2(퐹) = 1, and × the absolute Galois group 퐺ℚ = Gal(ℚ/ℚ). These au- 풪퐹 = {±1}. We have two factorizations of 6 into irre- tomorphisms permute the roots of each rational polyno- ducible elements of ℤ[√−5] that don’t differ up to units: mial, and consequently this action preserves the algebraic 6 = 2 ⋅ 3 = (1 + √−5)(1 − √−5). integers. We’ll use 퐹 to denote a number field. The integer ring The unique factorization into primes that resolves this for 풪퐹 of 퐹 is the subring of algebraic integers in 퐹. It is a our purposes is PID if and only if it’s a UFD, but unlike 풪ℚ = ℤ, it need (6) = (2, 1 + √−5)2(3, 1 + √−5)(3, 1 − √−5). not in general be either. Rather, 풪퐹 is what is known as a Dedekind domain. As such, it has the property that every The class number of ℚ(√−5) is 2, so the class group is nonzero ideal factors uniquely up to ordering into a prod- generated by the class of any nonprincipal ideal, such as uct of prime ideals. This property provides a replacement (2, 1 + √−5) or (3, 1 ± √−5). for unique factorization of elements. A “prime” of 퐹 is a Ramification in an extension of number fields isakin nonzero prime ideal of 풪퐹. to the phenomenon of branching in branched covers in A fractional ideal of 퐹 is a nonzero, finitely generated topology. As we’ve implicitly noted, the factorization of 풪 퐹 풪 2 퐹-submodule of . Nonzero ideals of 퐹 are fractional (2) in ℤ[√−5] is (2, 1 + √−5) . The square is telling us 풪 1 ideals, but so are, for instance, all 퐹-multiples of 2 . The that the same prime is occurring (at least) twice in the fac- 퐼 set 퐹 of fractional ideals is an abelian group under mul- torization: this is the branching. Whenever this happens 풪 Cl 퐹 tiplication with identity 퐹. The class group 퐹 of is in an extension of number fields, we say that the prime of 퐼 the quotient of 퐹 by its subgroup of principal fractional the base field ramifies in the extension. Only finitely many 퐹 Cl ideals generated by nonzero elements of . The group 퐹 primes ramify in an extension of number fields. 풪 is trivial if and only if 퐹 is a PID. The Dedekind zeta function of 퐹 is the unique mero- Cl ℎ Remarkably, 퐹 is finite. Its order 퐹 is known as the morphic continuation to ℂ of the series class number. Like the class group itself, it is the subject −푠 of many open questions. For instance, work of Heegner, 휁퐹(푠) = ∑ (푁픞) , Baker, and Stark in the 1950s and ’60s solved a problem 픞⊂풪퐹 of Gauss by showing that there are exactly nine imaginary where 픞 runs over the nonzero ideals of 풪퐹 and 푁픞 is the quadratic fields with class number one: ℚ(푖), ℚ(√−2), index of 픞 in 풪퐹. It has a simple pole at 푠 = 1 and sat- ℚ(√−3), ℚ(√−7),…. On the other hand, Gauss’ con- isfies a functional equation relating 휁퐹(푠) and 휁퐹(1 − 푠) jecture that there are infinitely many such real quadratic that involves factors coming from the archimedean embed- fields is still open. dings. The Dedekind zeta function of ℚ is the ubiquitous A number field 퐹 can be viewed as a subfield of ℂ in Riemann zeta function 휁(푠). multiple ways. That is, any 휎 ∈ 퐺ℚ gives an isomorphism The functional equation tells us that 휁퐹(푠) vanishes to × 휎∶ 퐹 → 휎(퐹), and 휎(퐹) is a subfield of ℂ as well, so order the rank 푟 of 풪퐹 at 푠 = 0. The leading term in precomposition with 휎 yields a different “archimedean” its Taylor expansion is given by the analytic class number embedding of 퐹 in ℂ. We may then place a metric on 퐹 by formula restricting the usual distance function. An embedding of 퐹 휁(푟)(0) ℎ 푅 퐹 = − 퐹 퐹 , in ℂ is real if it has image in ℝ and complex if it is not real, 푟! 푤퐹 JANUARY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 17 2휋푖/푁 where 푤퐹 is the number of roots of unity in 퐹. This pro- 푒 to ℚ. The Kronecker–Weber theorem states that ev- vides an important instance of a meromorphic function ery finite abelian extension of the rationals is contained in intertwining the unit and class groups. More broadly, it’s a cyclotomic field. If 푁 > 1, the prime ideals that ram- a first fundamental example of the links between analytic ify in ℚ(휁2푁) are exactly those dividing 푁. The integer and algebraic objects of arithmetic. ring of ℚ(휁푁) is ℤ[휁푁], so every cyclotomic integer is a To a prime 픭 of 퐹, we can attach a 픭-adic metric under ℤ-linear combination of roots of unity. which elements are closer together if their difference lies Each element 휎 of Gal(ℚ(휁푁)/ℚ) carries 휁푁 to an- 픭 퐹 푖 in a higher power of . We may complete with respect other primitive 푁th root of unity, which has the form 휁푁 to this metric to obtain a complete “local” field 퐹픭 that for some 푖 prime to 푁. The map taking 휎 to the unit 푖 has a markedly non-Euclidean topology. It has a compact modulo 푁 provides an isomorphism 풪 valuation ring 픭 equal to the open and closed unit ball ∼ × about 0. Gal(ℚ(휁푁)/ℚ) ⟶(ℤ/푁ℤ) For example, the 푝-adic metric 푑푝 on ℚ is defined by known as the 푁th cyclotomic character. −푛 푑푝(푥, 푦) = 푝 for the largest 푛 ∈ ℤ such that 푥 − 푦 ∈ For 푥, 푦, and 푧 satisfying the Fermat equation in odd 푛 푝 ℤ. The completion of ℚ with respect to 푑푝 is called the prime exponent 푝, we have a factorization 푝-adic numbers ℚ푝, which has valuation ring the 푝-adic 푝−1 integers ℤ푝. Alternatively, ℤ푝 is the inverse limit of the 푝 푝 푖 푝 푛 푥 + 푦 = ∏ (푥 + 휁푝푦) = 푧 rings ℤ/푝 ℤ under the reduction maps between them.
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