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. 푖=0 Every prime 픭 of 퐹 is maximal in 풪퐹 with finite residue in ℤ[휁푝]. Using this, Kummer proved in an 1850 paper field 풪퐹/픭. The Galois group 퐺 of a finite Galois extension that if 푝 is regular, which is to say that 푝 ∤ ℎ , then 퐸/퐹 acts transitively on the set of primes 픮 of 퐸 containing ℚ(휁푝) Fermat’s last theorem holds in exponent 푝. (Its use lies in 픭. The stabilizer of 픮 in 퐺 is called its decomposition group 푖 the fact that if 푝 is regular, then (푥 + 휁푝푦) cannot be the 퐺픮 and is isomorphic to Gal(퐸픮/퐹픭). The extension 풪퐸/픮 푁픭 푝th power of a nonprincipal ideal.) It’s known that there of 풪퐹/픭 has cyclic Galois group generated by 푥 ↦ 푥 . are infinitely many irregular primes but not that there are This element lifts to an element of 퐺픮 called a Frobenius element at 픮. The lift is unique if 픭 is unramified in 퐸/퐹. infinitely many regular primes, though over sixty percent It is independent of 픮 if 퐺 is also abelian, in which case we of primes up to any given number are expected to be regu- denote it by 휑 . lar. 픭 푘 ≥ 0 푘 퐵 ∈ ℚ The Hilbert class field 퐻 of 퐹 is the largest unramified For , the th Bernoulli number 푘 is the 퐹 푘 0 푥 퐵 = 0 abelian extension of 퐹. Here, “unramified” means that no th derivative at of the function 푒푥−1 . One has 푘 푘 ≥ 3 prime ramifies and no real embedding becomes complex. for odd . Here’s a table of Bernoulli numbers for positive even indices: The Artin map taking the class of a prime 픭 to 휑픭 provides an isomorphism 푘 2 4 6 8 10 12 14 퐵 1 − 1 1 − 1 5 − 691 7 ∼ 푘 6 30 42 30 66 2730 6 Cl퐹 ⟶ Gal(퐻퐹/퐹). Kummer proved the following, which amounts to a spe- For example, the Hilbert class field of ℚ(√−5) is cial case of the class number formula. ℚ(√5, 푖), and the Artin isomorphism tells us whether or Theorem (Kummer). A prime 푝 is irregular if and only if 푝 not a prime 픭 of ℤ[√−5] is principal via the sign in 휑픭(푖) divides the numerator of 퐵푘 for some positive even 푘 < 푝. = ±푖. Class field theory concerns “reciprocity maps” generalizing the Artin isomorphism by relaxing the ramifica- In particular, 691 is irregular as it divides 퐵12. Here’s a tion conditions. These can be used to prove reciprocity table of irregular primes 푝 < 150 and the indices 푛 of the laws generalizing Gauss’ law of quadratic reciprocity. Bernoulli numbers 퐵푛 they divide: 푝 37 59 67 101 103 131 149 Cyclotomic Fields 푛 32 44 58 68 24 22 130 Iwasawa theory has its origins in the study of the arithmetic The prime 157 divides both 퐵62 and 퐵110. The index of of cyclotomic fields, a classical area of number theory that irregularity 푖푝 of 푝 is the number of Bernoulli numbers 퐵푘 dates back to attempts at proving Fermat’s last theorem in with 푘 < 푝 even that 푝 divides. Its values up to an increas- the mid-1800s. This is a fascinating subject in its own right, ingly large bound are expected to fit a Poisson distribution 1 not least for the connections it reveals between Bernoulli with parameter 2 . The latest in a long history of compu- numbers and the structure of cyclotomic class groups. tations is due to Hart, Harvey, and Ong for 푝 < 231 = For a positive integer 푁, the 푁th cyclotomic field ℚ(휁푁) 2147483648. In this range, 푖푝 attains a maximum value is given by adjoining the primitive 푁th root of unity 휁푁 = of 9.

18 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 1 Regularity of 푝 is equivalent to the triviality of the Sylow By a reflection principle of Leopoldt, an even (푘) (푝−푘) (푘) 푝-subgroup 퐴 of Clℚ(휁푝). We call 퐴 the 푝-part of the class eigenspace 퐴 vanishes if 퐴 = 0, while 퐴 = 0 group. So, what more does the fact that an odd 푝 divides implies that 퐴(푝−푘) is cyclic. The proof of this uses class a particular Bernoulli number tell us about 퐴? The answer field theory and a general duality between Galois andfield is found in the action of elements known as Kummer theory. × Δ = Gal(ℚ(휁푝)/ℚ) ≅ (ℤ/푝ℤ) Classical Iwasawa Theory on 퐴 induced by the Δ-action on field elements. For each 훿 ∈ (ℤ/푝ℤ)×, there is a unique element 휔(훿) Iwasawa theory concerns the growth of arithmetic objects × in towers of number fields. More precisely, it concerns the ∈ ℤ푝 of order dividing 푝−1 that reduces to 훿. The result- 푝 ing homomorphism growth of -parts of class groups, and more general ob- × × jects called Selmer groups, in towers of number fields of 휔∶ (ℤ/푝ℤ) → ℤ푝 푝-power degree. This growth exhibits a certain regularity is a splitting of the reduction modulo 푝 map. The group that in good circumstances can be partially described by a 퐴 breaks up as a direct sum 푝-adic variant of a complex-valued 퐿-function. The simplest sort of tower consists of a sequence 퐹 of 퐴 = ⨁ 퐴(푖) 푛 푖∈ℤ/(푝−1)ℤ fields ∞ of subgroups 퐹 = 퐹0 ⊂ 퐹1 ⊂ 퐹2 ⊂ ⋯ ⊂ 퐹∞ = ⋃ 퐹푛 퐴(푖) = {푎 ∈ 퐴 ∣ 훿(푎) = 휔(훿)푖푎 for all 훿 ∈ Δ} 푛=0 푛 for integers 푖 modulo 푝 − 1. For obvious reasons, these with 퐹푛/퐹 cyclic of degree 푝 . The Galois group Γ = Δ 퐴 Gal(퐹∞/퐹) of the tower is the inverse limit of the groups are often called -eigenspaces of . We shall seek to study 푛 such eigenspaces, beginning with the following important Γ푛 = Gal(퐹푛/퐹) ≅ ℤ/푝 ℤ and as such is isomorphic to theorem. the additive group of ℤ푝. It is thus a compact group under the 푝-adic topology. Every number field 퐹 has a cyclo- Theorem (Herbrand–Ribet). If 푘 is a positive even integer tomic ℤ푝-extension defined as the unique ℤ푝-extension of 푘 < 푝 푝 퐵 퐴(푝−푘) ≠ 0 with , then divides 푘 if and only if . 퐹 inside the union of the fields 퐹(휁푝푛 ). Herbrand proved in 1932 that 퐴(푝−푘) = 0 unless 푝 di- If 퐹 has more than one ℤ푝-extension, it has infinitely vides 퐵푘. Ribet proved the converse in a 1976 paper [Ri]. many. Yet, the Galois group of the compositum of all ℤ푝- The proof of Herbrand’s theorem relies on a direct con- extensions still has finite ℤ푝-rank 푡 ≥ 푟2(퐹)+1. Leopoldt struction of an annihilator of the class group in ℤ[Δ] due conjectured this to be an equality in 1962. This notori- to Stickelberger. Ribet’s proof is more delicate, involving a ously difficult conjecture can be phrased as the equality of × congruence between modular forms that occurs when 푝 di- the ℤ-rank of the unit group 풪퐹 and the ℤ푝-rank of the vides 퐵푘 and its consequences for a Galois representation. closure of its image inside the direct sum of its local com- We’ll explain his method later. pletions at primes over 푝. Leopoldt’s conjecture is known There is also a coarser decomposition of 퐴 as 퐴+ ⊕ 퐴−, for abelian extensions of ℚ and imaginary quadratic fields where 퐴± is the subgroup of elements on which complex by 1967 work of Brumer. conjugation acts as ±1. Then 퐴± is the direct sum of the Δ- Let’s fix a ℤ푝-extension 퐹∞ of 퐹 in our discussion. In eigenspaces of 퐴 for even/odd 푖. The Herbrand–Ribet the- 1959, Iwasawa proved a result on the growth of the orders orem concerns 퐴−, but Kummer had already shown that of the 푝-parts 퐴푛 of the class groups of the fields 퐹푛 [Iw1]. 퐴− = 0 퐴 = 0 implies . Theorem (Iwasawa). There exist nonnegative integers 휆 and The order of 퐴+ is the highest power of 푝 dividing the + + 휇 and an integer 휈 such that class number ℎ of the fixed field ℚ(휁푝) of complex con- 푝푛휇+푛휆+휈 jugation. In 1920, Vandiver rediscovered and later popu- |퐴푛| = 푝 + larized a conjecture of Kummer’s that 퐴 = 0. Hart, Har- for all sufficiently large 푛. vey, and Ong have verified this conjecture for 푝 < 231. Vandiver’s conjecture can be rephrased as a question As a 푝-group, 퐴푛 is a module over ℤ푝. Since it also has about units. That is, the group of cyclotomic units in ℤ[휁푝] a commuting Γ푛-action, 퐴푛 is a module for the group ring is generated by ℤ푝[Γ푛], which consists of finite formal sums of elements 푗−1 of Γ푛 with ℤ푝-coefficients. 1 + 휁푝 + ⋯ + 휁푝 for 1 < 푗 < 푝. We can compare the 퐴푛 via norm maps 퐴푛+1 → 퐴푛 for The class number formula implies that the index of this every 푛 ≥ 0, as well as via maps 퐴푛 → 퐴푛+1 induced by × + subgroup of the unit group ℤ[휁푝] is ℎ . Vandiver’s con- the inclusion of 퐹푛 in 퐹푛+1. These maps are compatible jecture asserts that 푝 does not divide this index. with the action of ℤ푝[Γ푛+1] on both sides, with the action

JANUARY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 19 on 퐴푛 arising through the restriction map Γ푛+1 → Γ푛. The There is then a unique continuous isomorphism of com- Iwasawa algebra pact ℤ푝-algebras that takes 훾 − 1 to 푇. We shall use such an isomorphism to identify Λ and ℤ푝J푇K. Λ = lim ℤ푝[Γ푛] ⟵ The structure theory of finitely generated modules over 푛 Λ mimics the theory of finitely generated modules over then acts on both the inverse limit lim 퐴푛 and the direct ⟵푛 a PID if one treats Λ-modules as being defined up to fi- limit 퐴∞ = lim 퐴푛퐴푛. The Iwasawa algebra is a com- nite submodules and quotient modules. The idea is that ⟶푛 Λ becomes a PID upon localization at any principal prime pletion of the usual group ring ℤ푝[Γ], and as such it is a compact topological ring. Modules over Λ are also known ideal, and there are no nonzero finite modules over the lo- as Iwasawa modules. calization. A homomorphism 푓∶ 푀 → 푁 of finitely generated Λ- The Artin isomorphism identifies 퐴푛 with the Galois modules is called a pseudo-isomorphism if it has finite group Gal(퐿푛/퐹푛) of the maximal unramified abelian 푝- kernel and cokernel. The notion of pseudo-isomorphism extension (i.e., of 푝-power degree) 퐿푛 of 퐹푛. These maps are compatible with norms on class groups and restriction gives an equivalence relation on any set of finitely gener- Λ on Galois groups. ated, torsion -modules. The inverse limit of Artin isomorphisms identifies Theorem (Iwasawa, Serre). For any finitely generated, torsion lim 퐴푛 with the Galois group ⟵푛 Λ-module 푀, there is a pseudo-isomorphism 푋 = Gal(퐿 /퐹 ), 푟 푠 ∞ ∞ ∞ 푘푖 푚푗 푀 → ⨁ Λ/(푓푖 ) ⊕ ⨁ Λ/(푝 ), where 퐿∞ = ⋃푛 퐿푛. As an inverse limit of finite 푝-groups, 푖=1 푗=1 푋∞ is said to be pro-푝, and 퐿∞ is the maximal unramified where 푟, 푠 ≥ 0, each 푓푖 is a monic irreducible polynomial in abelian pro-푝-extension of 퐹∞. deg 푓푖 ℤ푝[푇] satisfying 푓푖 ≡ 푇 mod 푝, and each 푘푖 and 푚푗 is The Γ-action on the inverse limit of the 퐴 is identified 푛 a positive integer. via the Artin isomorphisms with the conjugation action of Γ on 푋∞. For this, one lifts 훾 ∈ Γ to ̃훾∈ Gal(퐿∞/퐹) and In the notation of the theorem, we set allows 훾 to act on 휎 ∈ 푋∞ by 푟 푠 휆(푀) = 푘 deg 푓 and 휇(푀) = 푚 . 훾∶ 휎 ↦ ̃훾휎̃훾−1. ∑ 푖 푖 ∑ 푗 푖=1 푗=1 푋 Λ This gives ∞ the structure of a -module, and we refer to We can also associate to 푀 its characteristic ideal char(푀) 푋 퐴 ∞ as the unramified Iwasawa module. In that each 푛 is in Λ. That is, given the above pseudo-isomorphism, we 푋 Λ finite, ∞ is finitely generated and torsion over . define The Λ-module 푋∞ is compact, while 퐴∞ is discrete. To 푟 휇(푀) 푘푖 obtain a compact Λ-module from 퐴∞, we take its Pontrya- char(푀) = (푝 ∏ 푓푖 ) . ∨ 푖=1 gin dual 퐴∞ = Hom(퐴∞,ℚ푝/ℤ푝), which is again finitely 푟 푘푖 generated and torsion. Its structure is very closely related The polynomial ∏푖=1 푓푖 is the usual characteristic poly- to that of 푋∞. nomial of 푇 acting on the finite-dimensional ℚ푝-vector 퐹 In good circumstances, such as when ∞ is the cyclo- space 푀 ⊗ℤ푝 ℚ푝. tomic ℤ푝-extension of 퐹 = ℚ(휁푝), we can recover 퐴푛 In the case of the unramified Iwasawa module 푋∞, the 푝푛 from 푋∞ as the largest quotient of 푋∞ upon which Γ quantities 휆 = 휆(푋∞) and 휇 = 휇(푋∞) are those in Iwa- ∨ acts trivially. Crucial to this is the fact that Γ is a pro-푝- sawa’s growth formula. In fact, we also have 휆 = 휆(퐴∞) ∨ group. Because of this, Λ is a local ring, and one can em- and 휇 = 휇(퐴∞), with the characteristic ideals of 푋∞ and ∨ −1 ploy Nakayama’s lemma. This stands in stark contrast to 퐴∞ differing by the change of variables 푇 ↦ (1+푇) −1. the case of finite Galois extensions of prime-to-푝 degree, Iwasawa conjectured that 휇 = 0 if 퐹∞ is the cyclotomic for which one has far less control over the growth of 푝- ℤ푝-extension of 퐹. Ferrero and Washington proved that parts of class groups. Nevertheless, Washington showed this holds when 퐹/ℚ is abelian [FW]. In this case, Iwasawa − in 1979 that the 푝-parts eventually stop growing in the cy- also showed that the (−1)-eigenspace 푋∞ for the action clotomic ℤℓ-extension for ℓ ≠ 푝 of an abelian extension of complex conjugation has no finite Λ-submodule. of ℚ. For 퐹∞ the cyclotomic ℤ푝-extension of 퐹 = ℚ(휇푝), the As observed by Serre, the Iwasawa algebra Λ is isomor- Iwasawa module 푋∞ has an action of Δ = Gal(퐹/ℚ) phic to a power series ring ℤ푝J푇K in a single variable 푇. that commutes with its Λ-action, so we can again break For this, we fix a topological generator 훾 of Γ, which is up 푋∞ as a direct sum of Δ-eigenspaces. In the computa- to say an element generating a dense subgroup or, equiv- tions of Hart–Harvey–Ong, not a single Δ-eigenspace of alently, an element that restricts to a generator of each Γ푛. 푋∞ has 휆-invariant greater than 1. Such an eigenspace

20 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 1 (푖) (푖) 푋∞ is nonzero if and only if 퐴 is nonzero, so these com- Theorem (Iwasawa main conjecture, Mazur–Wiles). For putations imply that 휆 ≤ 9 for 푝 < 231. any even integer 푘 ≢ 0 mod 푝 − 1, we have char(푋(푝−푘)) = (푓 ). 푝-adic 퐿-Functions ∞ 푘 For any positive integer 푛, the Riemann zeta function sat- The main conjecture implies that the highest power of 푝 푘 isfies dividing 퐿푝(휔 , 푠) is also the order of the largest quotient 퐵푛 (푝−푘) 푠 휁(1 − 푛) = − . of 푋∞ upon which 푇 acts as (1 + 푝) − 1. Taking 푛 푠 = 0, the main conjecture gives the order of 퐴(푝−푘); it The prime 푝 divides the denominator of 퐵푛 in lowest form does not, however, tell us the isomorphism class (though if and only if 푝−1 divides 푛, by an 1840 result of Clausen refinements do exist). and von Staudt. Let’s assume 푝−1 does not divide 푛, since The main conjecture can also be formulated in terms of 퐴(1) otherwise it’s known that the relevant eigenspace the 푝-ramified Iwasawa module 픛∞ = Gal(푀∞/퐹∞) for (푝−푛) = 퐴 is 0. 푀∞ the union of the maximal abelian 푝-extensions of the For 푚 ≡ 푛 mod 푝 − 1, Kummer showed in 1851 that 퐹푛 ramified only at the unique primes (1 − 휁푝푛+1 ) over 푝. 휁(1 − 푚) ≡ 휁(1 − 푛) mod 푝. This version of the main conjecture asserts that (푘) This explains why our criterion for regularity requires only char(픛∞ ) = (푔푘), indices 푛 < 푝. Even better, if 푚 ≡ 푛 mod 푝푗−1(푝 − 1) where 푔푘 is given by the change of variables 푔푘(푇) = for some 푗 ≥ 1, then we have −1 푓푘((1 + 푝)(1 + 푇) − 1). The equivalence follows from 푚−1 푛−1 푗 (1 − 푝 )휁(1 − 푚) ≡ (1 − 푝 )휁(1 − 푛) mod 푝 . an Iwasawa-theoretic version of Kummer duality. In [Iw2], Iwasawa proved his main conjecture assuming Fixing an even integer 푘, these congruences imply the (푘) 푘 that 퐴 = 0. In this case, the equality of characteristic existence of a continuous ℤ푝-valued function 퐿푝(휔 , 푠) ideals becomes an isomorphism of a 푝-adic variable 푠 ∈ ℤ푝 satisfying (푝−푘) 푘 푛−1 퐿푝(휔 , 1 − 푛) = (1 − 푝 )휁(1 − 푛) 푋∞ ≅ Λ/(푓푘). for all 푛 ≡ 푘 mod 푝 − 1. Here, as before, 휔 denotes the It’s worth understanding how Iwasawa’s argument goes, × × 푝-adic character 휔∶ (ℤ/푝ℤ) → ℤ푝 . as through it one obtains a form of the main conjecture 푘 The 푝-adic 퐿-functions 퐿푝(휔 , 푠) were constructed by free from 퐿-functions asserting an equality of characteris- Kubota and Leopoldt. Their values at other nonnegative tic ideals of Iwasawa modules coming from unit and class integers are similarly given by special values of Dirichlet groups. 퐿-functions of complex-valued characters of Δ. The Rie- Iwasawa studied the image of the cyclotomic units in- mann zeta function is the Dirichlet 퐿-function for the triv- side the local units of the completion at the prime over ial character. 푝, working up the tower of completions at 푝 of the fields 퐹푛 by considering sequences of elements compatible un- The Iwasawa Main Conjecture der norm maps. The Λ-module 풰∞ of norm-compatible ℰ 풞 The Iwasawa main conjecture describes the characteristic sequences of local units contains submodules ∞ and ∞ (푝−푘) generated by the sequences of global units and cyclotomic ideal of 푋∞ for an odd prime 푝 and an even integer 푘 in 푘 (1) units, respectively. terms of the 푝-adic 퐿-function 퐿푝(휔 , 푠). As 푋∞ is trivial Class field theory provides an exact sequence since 퐴(1) = 0, let us suppose that 푘 ≢ 0 mod 푝 − 1. 푘 0 → ℰ /풞 → 풰 /풞 → 픛 → 푋 → 0. Iwasawa showed that 퐿푝(휔 , 푠) is determined on 푠 ∈ ∞ ∞ ∞ ∞ ∞ ∞ ℤ푝 by a unique power series 푓푘 ∈ Λ satisfying This in turn yields an exact sequence with each module re- 푠 푘 푘 (푘) 푓푘((1 + 푝) − 1) = 퐿푝(휔 , 푠). placed by its 휔 -eigenspace under Δ. If 푋∞ = 0, then the (푘) (푘) class number formula can be used to see that ℰ∞ = 풞∞ Here, we’ve taken the variable 푇 to correspond to 훾 − 1 as well. The four-term exact sequence therefore reduces to for the topological generator 훾 of Γ that raises all roots of 푠 an isomorphism unity of 푝-power order to the power 1+푝, and (1+푝) is 푠푛 the limit of the sequence of (1+푝) for 푠푛 ∈ ℤ satisfying 풰(푘)/풞(푘) ≅ 픛(푘). 푛 ∞ ∞ ∞ 푠푛 ≡ 푠 mod 푝 ℤ푝. (푘) The following conjecture of Iwasawa’s, formulated in Iwasawa then obtains that 픛∞ ≅ Λ/(푔푘) from the follow- the late 1960s, was given a proof by Mazur and Wiles in ing unconditional theorem, which amounts to a 푝-adic a 1984 paper [MW]. regulator computation on cyclotomic units.

JANUARY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 21 Theorem (Iwasawa). There is an isomorphism of Λ-modules for all 푛 ≥ 1. The Fourier coefficients 푎푛(푓) of a normalized eigenform 푓 are algebraic numbers that are integral for 풰(푘)/풞(푘) ≅ Λ/(푔 ). ∞ ∞ 푘 푛 ≥ 1, and the coefficient field they generate is a number As characteristic ideals are multiplicative in exact field. sequences of finitely generated, torsion Λ-modules, this Eisenstein series form a class of modular forms that are also tells us unconditionally that the main conjecture is not cusp forms. For instance, for a positive even integer equivalent to the equality 푘 ≥ 4, we have an Eisenstein series ∞ (푘) (푘) (푘) 퐵푘 푘−1 푛 char(ℰ∞ /풞∞ ) = char(푋∞ ) 퐸푘 = − + ∑ ∑ 푑 푞 2푘 푛=1 푑∣푛 in which no 퐿-functions appear. which is an eigenform of weight 푘 for Γ = SL2(ℤ) itself. Modular Forms When an odd prime 푝 divides 휁(1 − 푘) = −퐵푘/푘, the constant term of 퐸푘 is zero modulo 푝. In this case, the re- One might say that the theme of the work of Ribet and duction of 퐸푘 modulo 푝 may be lifted to a cuspidal eigen- Mazur–Wiles is that the study of the geometry of varieties form with coefficients in the ring of integers of anumber over ℚ can be used to solve arithmetic questions. Specifi- field. For example, 퐸12 is congruent modulo 691ℤJ푞K to cally, their work makes use of modular curves and congru- the unique normalized cusp form ences between modular forms. The Galois representations ∞ attached to modular forms are two-dimensional, present- 푛 24 푞 ∏(1 − 푞 ) ing a natural next class of objects to study beyond the one- 푛=1 dimensional abelian characters of class field theory. We of weight 12 for SL2(ℤ). embark upon another brief tour. To a normalized cuspidal eigenform 푓 of weight 푘 ≥ 2, The group of matrices in GL (ℝ) with positive deter- 2 work of Shimura and Deligne attached a 푝-adic Galois rep- minant acts by Möbius transformations on the upper half- resentation 휌푓 ∶ 퐺ℚ → GL2(퐾푓), with 퐾푓 the field ob- plane ℍ of complex numbers with positive imaginary part. tained by adjoining to ℚ푝 the Fourier coefficients of 푓. A modular curve is a quotient of ℍ by the action of a sub- Equivalently, it is a two-dimensional 퐾푓-vector space 푉푓 group of SL2(ℤ) that is determined by congruences among with a commuting 퐺ℚ-action. This representation 휌푓 is its entries. This quotient can be compactified by adding in irreducible and odd (i.e., it has determinant −1 on com- the equivalence classes of the cusps, which are the rational plex conjugation), and it has the property that the trace numbers and infinity. of 휌푓(휑 ) for a Frobenius element 휑 of an unramified A modular form 푓 is a holomorphic function of ℍ that ℓ ℓ prime over a rational prime ℓ is equal to 푎 (푓). transforms under a congruence subgroup Γ (in the stan- ℓ Inside 푉 , there is a rank-two module 퐿 for the val- dard notation, but not to be confused with the Galois 푓 푓 uation ring 풪 of 퐾 that is preserved by the 퐺 -action. group appearing in Iwasawa theory) in a manner prescribed 푓 푓 ℚ Roughly, this says that 휌 can be viewed as taking values by its “weight” 푘, and which is bounded and holomorphic 푓 in GL2(풪푓). So, it makes sense to reduce 휌푓 modulo the at the cusps. Specifically, if ( 푎 푏 ) ∈ Γ, then 푐 푑 maximal ideal of 풪푓 and talk about the resulting repre- 푎푧 + 푏 sentation 휌푓̄ ∶ 퐺ℚ → GL2(픽푞) over the residue field 픽푞. 푓 ( ) = (푐푧 + 푑)푘푓(푧). This residual representation is unique up to isomorphism 푐푧 + 푑 if and only if it is irreducible. Otherwise, its isomorphism A modular form is a cusp form if it is zero at all cusps. class depends upon the choice of 퐿푓. 1 1 If ( 0 1 ) ∈ Γ, then 푓(푧 + 1) = 푓(푧), so 푓 has a Fourier expansion about the cusp at ∞ of the form The Method of Ribet–Mazur–Wiles ∞ Ribet and Mazur–Wiles employed congruences between 푛 푓 = ∑ 푎푛(푓)푞 cusp forms and Eisenstein series to construct unramified 푛=0 abelian extensions of cyclotomic fields. As we’ve noted, if 2휋푖푧 with 푞 = 푒 for 푧 ∈ ℍ. If 푓 is a cusp form, then 푝 ∣ 퐵푘 for an even 푘 < 푝, then there exists a cuspidal eigen- 푎0(푓) = 0. form 푓 congruent to 퐸푘 modulo the maximal ideal of 풪푓. There are Hecke operators 푇푛 for each 푛 ≥ 1 that act The fixed field of the kernel of the Galois representation on modular forms by summing over the action of repre- 휌푓 is ramified only at the prime 푝. Ribet used this to con- 1 0 푝 퐹 = ℚ(휁 ) sentatives of the double coset Γ ( 0 푛 ) Γ as a union of right struct an unramified abelian -extension of 푝 cosets. A modular form is an eigenform if it is a simulta- on which Δ acts through 휔푝−푘. By class field theory, if neous eigenform for all Hecke operators. If an eigenform the extension is nontrivial, then 퐴(푝−푘) is nonzero, which is normalized so that 푎1(푓) = 1, then 푇푛(푓) = 푎푛(푓)푓 is Ribet’s converse to Herbrand’s theorem.

22 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 1 The method works by playing two facts forced by the conjecture, which Rubin completed to a full proof. The congruence off of each other. The first is that 푓 is ordi- method uses what Kolyvagin termed an Euler system, a first nary in the sense that 푎푝(푓) is a 푝-adic unit. For ordinary example of which consists of cyclotomic units in abelian forms, there exists a basis of 푉푓 such that 휌푓 restricted to extensions of ℚ. This system of elements is used to bound a decomposition group at 푝 is upper triangular as a map the order of an even eigenspace of the 푝-part of the class to GL2(풪푓). If needed, one can rescale so that the map group by the order of an eigenspace of the quotient of the 휙∶ 퐺ℚ → 픽푞 to the residue field 픽푞 given by the reduction global units by the cyclotomic units. of the lower left-hand corner of 휌푓 modulo the maximal The key property is a norm compatibility from ℚ(휁푁ℓ) ideal of 풪푓 is nonzero on the larger group 퐺ℚ. to ℚ(휁푁) for 푝 dividing 푁 and a prime ℓ. Explicitly, if The second fact is that the residual Galois representa- ℓ ∤ 푁, one has tion 휌푓̄ is reducible. The above basis can actually be cho- 1 − 휁푁 sen so that 휌푓̄ has the form 푁 (1 − 휁 ) = . ℚ(휁푁ℓ)/ℚ(휁푁) 푁ℓ ℓ−1 1 − 휁푁 휔푘−1(휎) 0 휌̄ (휎) = ( ) 푓 휙(휎) 1 With this relation, one applies a Galois-theoretic derivative construction to elements 1 − 휁푁ℓ for good choices on 휎 ∈ 퐺ℚ, viewing 휔 as a character of 퐺ℚ. The re- of primes ℓ congruent to ±1 modulo a sufficiently high striction of 휙 to Gal(ℚ/퐹) is then a homomorphism that power of 푝 to obtain field elements that are powers of cho- is unramified at the prime over 푝 by construction, so it sen nonprincipal ideals. factors through the Galois group of a nontrivial unram- Up the cyclotomic tower, the method of Euler systems ified abelian 푝-extension 퐻 of 퐹 that is Galois over ℚ. shows that The group Δ acts on Gal(퐻/퐹) compatibly with conju- 푝−푘 (푘) (푘) (푘) gation of matrices, which is to say that it acts by 휔 = char(푋∞ ) ∣ char(ℰ∞ /풞∞ ) (휔푘−1)−1. 푘 ∈ ℤ Mazur and Wiles generalized and refined Ribet’s method for even . This is equivalent to the opposite divisi- in the context of Iwasawa theory in order to prove the main bility to that of Mazur–Wiles, and again the analytic class conjecture and its generalization to arbitrary abelian exten- number formula yields equality. sions of ℚ. By studying the Galois actions on Jacobians of modular curves, they construct an unramified abelian ex- Totally Real and CM Fields tension of the field 퐹∞ of all 푝-power roots of unity with The Iwasawa main conjecture generalizes directly from ℚ (푝−푘) Galois group a Λ-module quotient of 푋∞ that has char- to totally real fields, those number fields with only real acteristic ideal (푓푘). archimedean embeddings. The relevant 푝-adic 퐿-functions (푝−푘) were separately constructed by Deligne and Ribet, Cassou- Having proven that (푓푘) ∣ char(푋∞ ), Mazur and Wiles apply a consequence of analytic class number for- Noguès, and Barsky. Wiles proved a main conjecture for mula to obtain the main conjecture. That is, Iwasawa had odd eigenspaces of the unramified Iwasawa module 푋∞ shown in 1972 that over the cyclotomic ℤ푝-extension 퐹∞ of an abelian extension 퐹 of a totally real field using Galois representations (푝−3)/2 (푝−3)/2 (푝−2푗) attached to Hilbert modular forms [Wi]. Greenberg has ∑ 휆(푋∞ ) = ∑ deg 푓2푗. 푗=1 푗=1 conjectured that the even eigenspaces of 푋∞ are finite. The existence of an Euler system in abelian extensions From this, it follows that one divisibility for all odd eigen- of totally real fields was conjectured by Stark and Rubin, spaces implies the other. but it is still unproven. This is related to Hilbert’s twelfth Later work of Wiles gave a more streamlined perspec- problem, or Kronecker’s Jugendtraum, of an explicit class tive, casting the proof in terms of the theory of families field theory over totally real fields. In contrast, abelian ex- of ordinary modular forms of Hida. That is, Wiles em- tensions of imaginary quadratic fields contain analogues ployed the residual representation of a Galois representa- of cyclotomic units called elliptic units that do form an tion 휌ℱ ∶ 퐺ℚ → GL2(핀푓) attached to a 푝-adically contin- Euler system. In this case, there is an explicit form of class uously varying family ℱ of ordinary cuspidal eigenforms field theory arising from the theory of complex multipli- congruent to a family of Eisenstein series, where 핀푓 is a fication (CM) of elliptic curves that we’ll discuss. Elliptic nite local Λ-algebra. units are generated by values at torsion points of a theta function that is a meromorphic function on a CM elliptic The Method of Euler Systems curve. Work of Thaine and Kolyvagin led to a new and more ex- In 1977, Coates and Wiles proved an analogue of Iwa- plicit, though technically complex, approach to the main sawa’s theorem on local units modulo cyclotomic units for

JANUARY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 23 an imaginary quadratic field 퐾 with ℎ퐾 = 1 [CW]. Choos- The group 퐸(ℚ) has a canonical action of 퐺ℚ via the ing a split prime 푝 with 푝풪퐾 = 픭픭,̄ their theorem states action on coordinates. A point of 퐸(ℂ) ≅ (ℝ/ℤ)2 is said that the quotient of the local units at 픭 modulo elliptic to be 푛-torsion if it has order dividing 푛. The group 퐸[푛] units up a ℤ푝-extension of 퐾 ramified only at 픭 is isomor- of all 푛-torsion points is a subgroup of 퐸(ℚ) isomorphic phic to the quotient of the one-variable Iwasawa algebra by to (ℤ/푛ℤ)2. a power series corresponding to a 푝-adic 퐿-function con- A rational elliptic curve 퐸 has a ring 풪 of endomor- structed by Katz. phisms consisting of nonzero ℚ-rational maps 퐸 → 퐸 The compositum of ℤ푝-extensions of an imaginary qua- taking ∞ to itself. Among these are the morphisms given 2 dratic field 퐾 has Galois group ℤ푝. Its Iwasawa algebra is a by multiplication by integers using the group law of 퐸. If power series ring in two variables over ℤ푝. The unramified 풪 ≠ ℤ, then 풪 is a finite index subring of the integer ring 2 Iwasawa module 푋∞ over the corresponding ℤ푝-extension of an imaginary quadratic field. In this case, we say that 퐸 of an abelian extension of 퐾 is conjecturally small enough has CM by 풪. to have unit characteristic ideal. The main conjecture here The cohomology of a group 퐺 with coefficients in a compares 푋∞ and the quotient of global units by elliptic module 푀 for the group ring ℤ[퐺] is a sequence of abelian units. It was proven by Rubin using the method of Euler groups 퐻푖(퐺, 푀) that allows one to study the group action systems in 1991 [Ru]. If 푝 splits in 퐾, one can instead re- using the tools of homological algebra. For Galois groups, place 푋∞ with a larger Iwasawa module that allows rami- the closely related theory of Galois cohomology is a cru- fication at exactly one of the two primes over 푝. This gives cial tool in Iwasawa theory. Many theorems of class field an equivalent form involving a two-variable Katz 푝-adic theory can be phrased in terms of duality in Galois coho- 퐿-function. mology, and abelian groups of arithmetic interest can be In 1994, Hida and Tilouine gave an alternate proof of encoded in cohomology. the specialization of this conjecture to an anticyclotomic For a number field 퐹, the group 퐸(퐹)/푛퐸(퐹) is con- ℤ푝-extension using an approach closer to that of Mazur– tained in 퐻1(Gal(ℚ/퐹), 퐸[푛]). In fact, it is contained in Wiles. Their work extends to analogues of imaginary qua- a smaller subgroup of cohomology classes that are unram- dratic fields over totally real fields known as CM fields for ified at all but finitely many primes and partially vanish which one once again has no known Euler system to em- locally at the remaining primes. The direct limit over 푛 ploy. More recently, progress has been made on a divisi- of these subgroups of 퐻1(Gal(ℚ/퐹), 퐸[푛]) is the Selmer bility in a general main conjecture over CM fields, in par- group Sel퐸(퐹) we wish to study. ticular by Hsieh. The most important thing to know about the Selmer group is that it intertwines the Mordell–Weil group with Elliptic Curves and BSD a mysterious, conjecturally finite group called the Shafar- Elliptic curves over number fields provide a next step be- evich–Tate group X퐸(퐹) of 퐸. That is, there is an exact yond the theory of multiplicative groups that we’ve been sequence: in effect describing. In this setting, the arithmetic objects X that replace class groups are known as Selmer groups. We 0 → 퐸(퐹) ⊗ ℚ/ℤ → Sel퐸(퐹) → 퐸(퐹) → 0. run quickly through the arithmetic theory of elliptic curves, This is analogous to what happens with cohomology with from the basic theory to deep results and a famous conjec- coefficients in roots of unity: in that case, the unit andclass ture, before we begin passing up towers. groups get wrapped up together. A complex elliptic curve 퐸 is defined by an equation The intertwining of 퐸(ℚ) and X퐸(ℚ) is also reflected 푦2 = 푥3 + 푎푥 + 푏 in analytic formulas. One can construct an 퐿-series for 퐸 픽 푝 3 2 from the data of the number of 푝-points of mod with 푎, 푏 ∈ ℂ and 4푎 + 27푏 ≠ 0 to ensure smooth- reductions of a minimal equation for 퐸. It has analytic ness. More precisely, it is the projective curve of genus one continuation to ℂ by the modularity of rational elliptic defined by the homogenization of the above polynomial. curves proven by Wiles, Taylor–Wiles, and Breuil–Conrad– Effectively, this means adding a single point ∞ at infinity. Diamond–Taylor, which tells one that 퐸 has an associated The set 퐸(ℂ) of points of 퐸 with complex coordinates cuspidal eigenform with the same 퐿-series. has an abelian group law with ∞ as its identity. It is given The Birch and Swinnerton–Dyer conjecture, or BSD, was by drawing a line between two points 푃 and 푄 and declar- formulated in 1965 and is one of the Clay Math Institute’s ing the third point of the line in 퐸(ℂ) to be −푃 − 푄, tak- Millennium Problems. ing multiplicity into account. We’ll assume that 푎, 푏 ∈ ℚ, which is to say that 퐸 is rational. The Mordell–Weil group Conjecture (Birch and Swinnerton–Dyer). The order of van- 퐸(퐹) of points with coordinates in a number field 퐹 is ishing of the 퐿-function 퐿(퐸, 푠) at 푠 = 1 is equal to the rank then a finitely generated subgroup of 퐸(ℂ). of 퐸(ℚ).

24 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 1 BSD has a refined form that links the leading termof We mention a bit of what’s known for elliptic curves 퐿(퐸, 푠) in its Taylor expansion about 푠 = 1 to the orders with good supersingular (i.e., nonordinary) reduction. For of the torsion subgroup 퐸(ℚ)tor of Mordell–Weil and of 푝 ≥ 5, there are in this case not one but two Selmer groups X퐸(ℚ). The conjectural BSD formula has the form constructed by Kobayashi and two 푝-adic 퐿-functions con- (푟) structed by Pollack. The corresponding main conjecture 퐿 (퐸, 1) |X퐸(ℚ)|Ω퐸푅퐸 ∏ℓ 푐ℓ = , (1) was proven by Rubin and Pollack for CM curves in 2004. 푟! |퐸(ℚ) |2 tor In this case, 푝 does not split in 퐾, and the main conjecture where 푟 is the order of vanishing, 푅퐸 is a regulator related is closely related to Rubin’s main conjecture without 퐿- to the heights of rational points, the quantity Ω퐸 is a real functions for imaginary quadratic fields. The main conjec- period, and each 푐ℓ for a prime ℓ is the number of com- ture for non-CM curves has recently been proven by Wan ponents of a certain mod ℓ reduction of 퐸, all but finitely under a hypothesis on the congruence subgroup. Sprung many being 1. has additionally treated the prime 3, in particular employ- Much of the progress on BSD to date employs Iwasawa ing work of Kim and Lei in the formulation. theory. For instance, the first major result that serves as The main conjecture for elliptic curves implies a 푝-adic theoretical evidence for BSD was due to Coates–Wiles. As analogue of BSD of Mazur–Tate–Teitelbaum that relates a consequence of their theorem on local units modulo el- the rank of 퐸(ℚ) to the order of vanishing of 퐿푝(퐸, 푠) at 푟 = 0 퐸 liptic units, they proved that if and has CM by a 1. As 퐿(퐸, 푠) is complex analytic and 퐿푝(퐸, 푠) is 푝-adic subring of 풪퐾 for 퐾 with ℎ퐾 = 1, then 퐸(ℚ) is finite. analytic, the derivatives are not clearly related, and neither form of BSD obviously implies the other in the case of pos- Iwasawa Theory of Elliptic Curves itive rank. We turn to the question of how Selmer groups of a rational The order 푟 of vanishing of 퐿(퐸, 푠) at 푠 = 1 is known elliptic curve 퐸 grow in the cyclotomic ℤ푝-extension ℚ∞ as the analytic rank of 퐸, and the actual rank of 퐸(ℚ) is of ℚ. This amounts to studying the finitely generated Λ- known as the algebraic rank. Kolyvagin used an Euler sys- module tem of Heegner points and a theorem of Gross and Zagier 픛퐸 = Hom(Sel퐸(ℚ∞), ℚ푝/ℤ푝) to prove that if the analytic rank of 퐸 is 푟 ≤ 1, then the al- that is the Pontryagin dual of the direct limit gebraic rank is 푟 and X퐸(ℚ) is finite. Both the converse to ∞ Sel퐸(ℚ∞)[푝 ] of 푝-power torsion subgroups of the this and the BSD formula follow from the main conjecture Selmer groups Sel퐸(ℚ푛). for 푟 = 0. Recent work of Skinner and of Zhang implies a The elliptic curve 퐸 has good reduction at 푝 if its mod converse for 푟 = 1 under mild hypotheses, and for 푟 = 1 푝 reduction 퐸푝 is nonsingular, and it is then ordinary if 퐸푝 significant progress has been made towards the BSD for- has a point of order 푝. For such elliptic curves, Mazur and mula as well, particularly in work of Jetchev, Skinner, and Swinnerton–Dyer constructed a 푝-adic 퐿-function 퐿푝(퐸, 푠) Wan. interpolating values of 퐿(퐸, 푠) up to certain Euler factors, and again it is determined by a power series ℒ퐸. Mazur Recent Directions then formulated the following main conjecture. Iwasawa theory extends to study the growth of other arith- Conjecture (Main conjecture for elliptic curves). Suppose metic objects attached to Galois representations in towers that 퐸 has good ordinary reduction at 푝 and that 퐸(ℚ)[푝] is of number fields. Beginning in the late 1980s, Greenberg an irreducible 퐺ℚ-representation. Then 픛퐸 is Λ-torsion and proposed main conjectures for a whole host of ordinary motivic Galois representations, and even continuously 푝- char(픛 ) = (ℒ ). 퐸 퐸 adically varying families thereof. Since then, main conjec- For elliptic curves with CM, this is equivalent to the tures have been extended to more general towers, nonordi- main conjecture for imaginary quadratic fields and split nary families, finer-grained analogues, and even character- primes 푝 proven by Rubin. The general divisibility istic 푝 base fields. Recent developments have seen consid- char(픛퐸) ∣ (ℒ퐸) was proven by Kato via the method of erable progress on methods of proof of both divisibilities. Euler systems [Ka]. Kato’s Euler system is constructed us- Here’s a sampling. ing cohomological products formed from pairs of Siegel The early part of the new millennium saw the devel- units on a modular curve parameterizing 퐸, first studied opment of Iwasawa theory over towers of number fields by Beilinson. The other divisibility was proven under fairly with Galois groups that are isomorphic to subgroups of mild hypotheses by Skinner and Urban using Galois repre- GL푛(ℤ푝) for some 푛. The breakthrough came in a paper sentations attached to automorphic forms on the unitary of Coates, Fukaya, Kato, Sujatha, and Venjakob contain- group GU(2, 2) [SU]. With no analytic class number for- ing a noncommutative main conjecture for elliptic curves. mula that can be used in this setting, one needs both meth- Invariants playing the roles of characteristic ideals and 푝- ods. adic 퐿-functions lie in a first 퐾-group of a localization of

JANUARY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 25 the noncommutative Iwasawa algebra. Soon after, Fukaya Fukaya and Kato proved a major result in this direction in and Kato formulated a remarkably general noncommuta- which the derivative of the 푝-adic 퐿-function plays a cru- tive main conjecture that relates closely to the equivariant cial and potentially unavoidable intermediate role. Their Tamagawa number conjecture of Burns and Flach which in result implies the conjecture for 푝 < 231. turn generalized a conjecture of Bloch and Kato on special Fukaya, Kato, and I expect this to be a special case of values of 퐿-functions. a general phenomenon of the geometry and topology of At the start of this decade, a noncommutative main con- locally symmetric spaces of higher dimension informing jecture for totally real fields was proven through work of the arithmetic of Galois representations attached to lower- Ritter and Weiss, Burns, and Kakde, assuming Iwasawa’s dimensional automorphic forms. This begs the question conjecture on the vanishing of the 휇-invariant. The key of its elliptic curve analogue, which is but one of a wealth step is the verification of congruences among Deligne– of intriguing possibilities for future directions in Iwasawa Ribet 푝-adic 퐿-functions over different totally real fields theory. to reduce to the main conjecture proven by Wiles. The work of Skinner–Urban has inspired a rush of new References theorems on divisibilities in main conjectures, with recent [CW] Coates J, Wiles A, On the conjecture of Birch and progress on the construction of Galois representations at- Swinnerton–Dyer, Invent. Math., (39): 223–251, 1977. MR0463176 tached to automorphic forms in the sense of the Langlands [FW] Ferrero B, Washington L, The Iwasawa invariant 휇푝 van- program providing a boon for the area. The construction ishes for abelian number fields, Ann. of Math., (109): 377– of 푝-adic 퐿-functions that interpolate special values of com- 395, 1979. MR528968 plex 퐿-functions is a whole industry unto itself and often [Iw1] Iwasawa K, On Γ-extensions of algebraic number fields, proceeds via distributions formed out of modular symbols Bull. Amer. Math. Soc., (65): 183–226, 1959. MR0124316 or computations of local integrals. [Iw2] Iwasawa K, On 푝-adic 퐿-functions, Ann. of Math., (89): Euler systems have long had a reputation as difficult to 198–205, 1969. MR0269627 construct that has only recently begun to soften. An ap- [Ka] Kato K, 푝-adic Hodge theory and values of zeta func- 푝 proach to construct them via geometric methods starting tions of modular forms, Cohomologies -adiques et ap- plications arithmétiques, III, Astérisque, (295): 117–290, from cycles on varieties was initiated in work of Bertolini, 2004. MR2104361 Darmon, Prasanna, and Rotger and carried further in work [MW] Mazur B, Wiles A, Class fields of abelian extensions of of Loeffler, Zerbes, Lei, Kings, and Skinner. One typically ℚ, Invent. Math., (76): 179–330, 1984. MR742853 computes complex and 푝-adic regulators to prove nonva- [Ri] Ribet K, A modular construction of unramified 푝- nishing of Euler systems and relate them with (푝-adic) 퐿- extensions of ℚ(휇푝), Invent. Math., (34) 151–162, 1976. functions. MR0419403 [Ru] Rubin K, The “main conjectures” of Iwasawa theory for Beyond Mazur–Wiles imaginary quadratic fields, Invent. Math., (103): 25–68, 1991. MR1079839 I end with a brief discussion of a deeper relationship be- [SU] Skinner C, Urban E, The Iwasawa main conjectures for tween the geometry of modular curves and the arithmetic GL2, Invent. Math., (195): 1–227, 2014. MR3148103 of cyclotomic fields. In a 2011paper, I formulated a conjec- [Wi] Wiles A, The Iwasawa conjecture for totally real fields, ture relating relative homology classes {훼 → 훽} of paths Ann. of Math., (131): 493–540, 1990. MR1053488 between cusps on a modular curve, taken here to be 푋1(푝), and cup products 푥 ∪ 푦 of cyclotomic units in a Galois cohomology group that agrees with 퐴− modulo 푝. In fact, there is a simple explicit map 푎 푏 { → } ↦ (1 − 휁푐) ∪ (1 − 휁푑) 푐 푑 푝 푝 taking one set of elements to the other (for 푎푑 − 푏푐 = 1 and 푝 ∤ 푐푑). On a certain Eisenstein quotient of the plus part of homology, I conjectured this to provide an inverse to a canonical version of the map that appeared in the proof of Ribet’s theorem. Up the cyclotomic tower, this yields what can be viewed Credits as a refinement of the Iwasawa main conjecture. That is,it Opening photo is courtesy of Getty Images. provides not only an equality of characteristic ideals, but Author photo is by Joshua M. Shelton. an isomorphism given by a recipe on special elements.

26 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 1