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Home , Class number formula, Iwasawa theory

Iwasawa theory: a climb up the tower Romyar Sharifi

Iwasawa theory is an area of that of algebraic numbers forms a subfield Q of C known emerged from the foundational work of Kenkichi as an algebraic closure of Q. Inside Q sits a subring Iwasawa in the late 1950s and onward. It studies the Z of algebraic integers, consisting of the roots of growth of arithmetic objects, such as class groups, monic polynomials with integer coefficients. in towers of number fields. Its key observation is The field automorphisms of Q form a huge that a part of this growth exhibits a remarkable reg- group called the absolute GQ = ularity, which it aims to describe in terms of values Gal(Q/Q). These automorphisms permute the of meromorphic functions known as L-functions, roots of each rational polynomial, and consequently such as the . this action preserves the algebraic integers. Through such descriptions, Iwasawa theory un- We’ll use F to denote a number field. The inte- veils intricate links between algebraic, geometric, ger ring O of F is the subring of algebraic integers and analytic objects of an arithmetic nature. The F in F . It is a PID if and only if it’s a UFD, but unlike existence of such links is a common theme in many O = , it need not in general be either. Rather, central areas within arithmetic geometry. So it is Q Z O is what is known as a Dedekind domain. As that Iwasawa theory has found itself a subject of F such, it has the property that every nonzero ideal continued great interest. This year’s Arizona Win- factors uniquely up to ordering into a product of ter School attracted nearly 300 students hoping to prime ideals. This property provides a replacement learn about it! for unique factorization of elements. A “prime” of The literature on Iwasawa theory is vast and of- F is a nonzero prime ideal of O . ten technical, but the underlying ideas are possess- F ing of an undeniable beauty. I hope to convey some A fractional ideal of F is a nonzero, finitely gen- of this, while explaining the original questions of erated OF -submodule of F . In particular, nonzero Iwasawa theory and giving a sense of the directions ideals of OF are fractional ideals, but so for instance O 1 I in which the area is heading. are all F -multiples of 2 . The set F of fractional ideals is an abelian group under multiplication with identity OF . The class group ClF of F is the quo- tient of IF by its subgroup of principal fractional ideals generated by nonzero elements of F . The To understand Iwasawa theory requires some knowl- group ClF is trivial if and only if OF is a PID. edge of the background out of which it arose. We at- Remarkably, ClF is finite. Its order hF is known tempt to chart a course, beginning with a whirlwind as the class number. Like the class group itself, it is tour of the elements of algebraic number theory. We the subject of many open questions. For instance, make particular note of two algebraic objects, the work of K. Heegner, A. Baker, and H. Stark in the class group and the unit group of a number field, 1950s and 60s solved a problem of Gauss by show- that play central roles for us. ing that there are exactly nine imaginary quadratic√ Algebraic numbers are the roots inside the com- fields√ with class√ number one: Q(i), Q( −2), plex numbers of nonzero polynomials in a single Q( −3), Q( −7), .... On the other hand, Gauss’ variable with rational coefficients. They lie in finite conjecture that there are infinitely many such real field extensions of Q called number fields. The set quadratic fields is still open. Romyar Sharifi is professor of mathematics at UCLA. His email address is [email protected]. This material is based upon work supported by the National Science Foundation under Grant Nos. 1661658 and 1801963.

1 A number field F can be viewed as a subfield at s = 1 and satisfies a functional equation relating of C in multiple ways. That is, any σ ∈ GQ gives ζF (s) and ζF (1 − s) that involves factors coming an isomorphism σ : F → σ(F ), and σ(F ) is a sub- from the archimedean embeddings. The Dedekind field of C as well, so precomposition with σ yields a zeta function of Q is the ubiquitous Riemann zeta different “archimedean” embedding of F in C. We function ζ(s). may then place a metric on F by restricting the usual The functional equation tells us that ζF (s) van- × distance function. An embedding of F in C is real ishes to order the rank r of OF at s = 0. The if it has image in R and complex if it is not real, in leading term in its Taylor expansion is given by the which case it has dense image in C. The numbers analytic of real and complex-conjugate pairs of complex em- (r) beddings are respectively denoted r1(F ) and r2(F ). ζ (0) h R × F F F The unit group O of invertible elements in OF = − . F r! wF under multiplication is deeply intertwined with the class group ClF . In fact, these groups are the ker- where wF is the number of roots of unity in F . × nel and cokernel of the map F → IF taking an This provides an important instance of a meromor- element to its principal fractional ideal. Dirichlet’s phic function intertwining the unit and class groups. × unit theorem says that OF is a direct product of the More broadly, it’s a first fundamental example of group of roots of unity in F and a free abelian group the links between analytic and algebraic objects of of rank r = r1(F )+r2(F )−1. This is proven using arithmetic. logarithms of absolute values of units with respect To a prime p of F , we can attach a p-adic met- to archimedean embeddings. The regulator RF of ric under which elements are closer together if their F is a nonzero real number defined as a determinant difference lies in a higher power of p. We may of a matrix formed out of such logarithms. √ complete F with respect to this metric to obtain a The prototypical√ example is F = Q( −5), complete “local” field Fp that has a markedly non- for which OF = Z[ −5]. One has r1(F ) = 0, Euclidean topology. It has a compact valuation ring r (F ) = 1 O× = {±1} 2 , and F . We have two factor-√ Op equal to the open and closed unit ball about 0. izations of 6 into irreducible elements of Z[ −5] For example, the p-adic metric dp on Q is de- −n that don’t differ up to units: fined by dp(x, y) = p for the largest n ∈ Z such n √ √ that x − y ∈ p Z. The completion of Q with re- 6 = 2 · 3 = (1 + −5)(1 − −5). spect to dp is called the p-adic numbers Qp, which has valuation ring the p-adic integers Zp. Alterna- The unique factorization into primes that resolves n this for our purposes is tively, Zp is the of the rings Z/p Z √ √ √ under the reduction maps between them. 2 (6) = (2, 1 + −5) (3, 1 + −5)(3, 1 − −5). Every prime p of F is maximal in OF with fi- √ nite residue field OF /p. The Galois group G of The class number of Q( −5) is 2, so the class a finite Galois extension E/F acts transitively on group is generated by√ the class of any nonprincipal√ the set of primes q of E containing p. The stabi- ideal, such as (2, 1 + −5) or (3, 1 ± −5). lizer of q in G is called its decomposition group Gq Ramification in an extension of number fields is and is isomorphic to Gal(Eq/Fp). The extension akin to the phenomenon of branching in branched OE/q of OF /p has cyclic Galois group generated covers in topology. As we’ve√ implicitly noted,√ the Np 2 by x 7→ x . This element lifts to an element of Gq factorization of (2) in Z[ −5] is (2, 1 + −5) . called a Frobenius element at q. The lift is unique The square is telling us that the same prime is occur- if p is unramified in E/F . It is independent of q if ring (at least) twice in the factorization: this is the G is also abelian, in which case we denote it by ϕp. branching. Whenever this happens in an extension The Hilbert class field HF of F is the largest un- of number fields, we say that the prime of the base ramified of F . Here, “unramified” field ramifies in the extension. Only finitely many means that no prime ramifies and no real embedding primes ramify in an extension of number fields. becomes complex. The Artin map taking the class The of F is the unique of a prime p to ϕp provides an isomorphism meromorphic continuation to C of the series ∼ X −s ClF −→ Gal(HF /F ). ζF (s) = (Na) , √ a⊂OF For example,√ the Hilbert class field of Q( −5) where a runs over the nonzero ideals of OF and is Q( 5, i), and the Artin isomorphism√ tells us Na is the index of a in OF . It has a simple pole whether or not a prime p of Z[ −5] is principal

2 via the sign in ϕp(i) = ±i. Class field theory con- Kummer proved the following, which amounts cerns “reciprocity maps” generalizing the Artin iso- to a special case of the class number formula. morphism by relaxing the ramification conditions. These in turn can be used to prove reciprocity laws Theorem (Kummer). A prime p is irregular if and generalizing Gauss’ law of quadratic reciprocity. only if p divides the numerator of Bk for some pos- itive even k < p.

Cyclotomic fields In particular, 691 is irregular as it divides B12. Here’s a table of irregular primes p < 150 and the Iwasawa theory has its origins in the study of the indices n of the Bernoulli numbers Bn they divide: arithmetic of cyclotomic fields, a classical area of number theory that dates back to attempts at proving p 37 59 67 101 103 131 149 Fermat’s last theorem in the mid-1800’s. This is a n 32 44 58 68 24 22 130 fascinating subject in its own right, not least for the connections it reveals between Bernoulli numbers The prime 157 divides both B62 and B110. The in- and the structure of cyclotomic class groups. dex of irregularity ip of p is the number of Bernoulli For a positive integer N, the Nth cyclotomic numbers Bk with k < p even that p divides. Its val- field Q(ζN ) is given by adjoining the primitive Nth ues up to an increasingly large bound are expected 2πi/N 1 ζN = e to Q. The Kronecker- fit a Poisson distribution with parmeter 2 . The lat- Weber theorem states that every finite abelian ex- est in a long history of computations is due to Hart, tension of the rationals is contained in a cyclotomic Harvey, and Ong for p < 231 = 2147483648. In field. If N > 1, the prime ideals that ramify in this range, ip attains a maximum value of 9. Q(ζ2N ) are exactly those dividing N. The integer Regularity of p is equivalent to the triviality of ring of Q(ζN ) is Z[ζN ], so every cyclotomic integer the Sylow p-subgroup A of ClQ(ζp). We call A the is a Z-linear combination of roots of unity. p-part of the class group. So, what more does the Each element σ of Gal(Q(ζN )/Q) carries ζN fact an odd p divides a particular to another primitive Nth root of unity, which has tell us about A? The answer is found in the action i the form ζN for some i prime to N. The map taking of ∼ × σ to the unit i modulo N provides an isomorphism ∆ = Gal(Q(ζp)/Q) = (Z/pZ) ∼ × Gal( (ζN )/ ) −→ ( /N ) on A induced by the ∆-action on field elements. Q Q Z Z × For each δ ∈ (Z/pZ) , there is a unique ele- × known as the Nth cyclotomic character. ment ω(δ) ∈ Zp of order dividing p−1 that reduces For x, y, and z satisfying the Fermat equation to δ. The resulting homomorphism in odd prime exponent p, we have a factorization × × ω :(Z/pZ) → Zp p−1 p p Y i p x + y = (x + ζpy) = z is a splitting of the reduction modulo p map. The i=0 group A breaks up as a direct sum in [ζ ]. Using this, Kummer proved in an 1850 pa- M Z p A = A(i) per that if p is regular, which is to say that p - hQ(ζp), then Fermat’s last theorem holds in exponent p. (Its i∈Z/(p−1)Z i use lies in the fact that if p is regular, then (x + ζpy) cannot be the pth power of a nonprincipal ideal.) of subgroups

It’s known that there are infinitely many irregular (i) i primes but not that there are infinitely many regular A = {a ∈ A | δ(a) = ω(δ) a for all δ ∈ ∆} primes, though over 60% of primes up to any given number are expected to be regular. for integers i modulo p − 1. For obvious reasons, For k ≥ 0, the kth Bernoulli number Bk ∈ Q is these are often called ∆-eigenspaces of A. It is such x eigenspaces that we shall seek to study, beginning the kth derivative at 0 of the function ex−1 . One has Bk = 0 for odd k ≥ 3. Here’s a table of Bernoulli with the following important theorem. numbers for positive even indices: Theorem (Herbrand-Ribet). If k is a positive even k 2 4 6 8 10 12 14 integer with k < p, then p divides Bk if and only if 1 1 1 1 5 691 7 A(p−k) 6= 0. Bk 6 − 30 42 − 30 66 − 2730 6

3 (p−k) n Herbrand proved that A = 0 unless p | Bk with Fn/F cyclic of degree p . The Galois group in 1932. proved the converse in a 1976 Γ = Gal(F∞/F ) of the tower is the inverse limit of ∼ n paper [Ri]. The proof of Herbrand’s theorem re- the groups Γn = Gal(Fn/F ) = Z/p Z and as such lies on a direct construction of an annihilator of the is isomorphic to the additive group of Zp. It is thus class group in Z[∆] due to Stickelberger. Ribet’s a compact group under the p-adic topology. Ev- proof is more delicate, involving a congruence be- ery number field F has a cyclotomic Zp-extension tween modular forms that occurs when p | Bk and defined as the unique Zp-extension of F inside the its consequences for a Galois representation. We’ll union of the fields F (ζpn ). explain his method later. If F has more than one Zp-extension, it has There is also a coarser decomposition of A as infinitely many. Yet, the Galois group of the com- + − ± A ⊕ A , where A is the subgroup of elements positum of all Zp-extensions still has finite Zp-rank on which complex conjugation acts as ±1. Then t ≥ r2(F ) + 1. Leopoldt conjectured this to be an A± is the direct sum of the ∆-eigenspaces of A for equality in 1962. This notoriously difficult conjec- even/odd i. The Herbrand-Ribet theorem concerns ture can be phrased as the equality of the Z-rank of − − × A , but Kummer had already shown that A = 0 the unit group OF and the Zp-rank of the closure implies A = 0. of its image inside the direct sum of its local com- The order of A+ is the highest power of p divid- pletions at primes over p. Leopoldt’s conjecture is + + ing the class number h of the fixed field Q(ζp) of known for abelian extensions of Q and imaginary complex conjugation. In 1920, H. Vandiver redis- quadratic fields by 1967 work of Armand Brumer. covered and later popularized a conjecture of Kum- Let’s fix a Zp-extension F∞ of F in our dis- mer’s that A+ = 0. Hart, Harvey, and Ong have cussion. In 1959, Iwasawa proved a result on the 31 verified this conjecture for p < 2 . growth of the orders of the p-parts An of the class Vandiver’s conjecture can be rephrased as a groups of the fields Fn [Iw1]. question about units. That is, the group of cyclo- Theorem (Iwasawa). There exist nonnegative inte- tomic units in Z[ζp] is generated by gers λ and µ and an integer ν such that n j−1 |A | = pp µ+nλ+ν 1 + ζp + ··· + ζp for 1 < j < p. n for all sufficiently large n. The class number formula implies that the index of × + As a p-group, A is a module over . Since this subgroup of the unit group Z[ζp] is h . Van- n Zp diver’s conjecture asserts that p does not divide this it also has a commuting Γn-action, An is a module index. for the group ring Zp[Γn], which consists of finite By a reflection principle of H.-W. Leopoldt, an formal sums of elements of Γn with Zp-coefficients. (k) (p−k) A A → even eigenspace A vanishes if A = 0, while We can compare the n via norm maps n+1 A for every n ≥ 0, as well as via maps A → A A(k) = 0 implies that A(p−k) is cyclic. The proof of n n n+1 induced by the inclusion of F in F . These maps this uses class field theory and a general duality be- n n+1 are compatible with the action of [Γ ] on both tween Galois and field elements known as Kummer Zp n+1 sides, with the action on A arising through the theory. n restriction map Γn+1 → Γn. The Λ = lim [Γ ] ←− Zp n Classical Iwasawa theory n then acts on both the inverse limit lim A and the Iwasawa theory concerns the growth of arithmetic ←−n n direct limit A = lim A . The Iwasawa algebra objects in towers of number fields. More precisely, ∞ −→n n it concerns the growth of p-parts of class groups, is a completion of the usual group ring Zp[Γ], and and more general objects called Selmer groups, in as such it is a compact topological ring. Modules towers of number fields of p-power degree. This over Λ are also known as Iwasawa modules. growth exhibits a certain regularity that in good cir- The Artin isomorphism identifies An with the cumstances can be partially described by a p-adic Galois group Gal(Ln/Fn) of the maximal unram- variant of a complex-valued L-function. ified abelian p-extension (i.e., of p-power degree) The simplest sort of tower consists of a sequence Ln of Fn. These maps are compatible with norms Fn of fields on class groups and restriction on Galois groups. The inverse limit of Artin isomorphisms identi- ∞ fies lim A with the Galois group [ ←−n n F = F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ F∞ = Fn n=0 X∞ = Gal(L∞/F∞),

4 S where L∞ = n Ln. As an inverse limit of finite isomorphism p-groups, X∞ is said to be pro-p, and L∞ is the r s M M maximal unramified abelian pro-p extension of F∞. M → Λ/(f ki ) ⊕ Λ/(pmj ), Γ A i The -action on the inverse limit of the n is i=1 j=1 identified via the Artin isomorphisms with the con- jugation action of Γ on X∞. For this, one lifts γ ∈ Γ where r, s ≥ 0, each fi is a monic irreducible poly- deg fi to γ˜ ∈ Gal(L∞/F ) and allows γ to act on σ ∈ X∞ nomial in Zp[T ] satisfying fi ≡ T mod p, and by each ki and mj is a positive integer. γ : σ 7→ γσ˜ γ˜−1. In the notation the theorem, we set This gives X∞ the structure of a Λ-module, and we r s X X refer to X∞ as the unramified Iwasawa module. In λ(M) = ki deg fi and µ(M) = mj. that each An is finite, X∞ is finitely generated and i=1 j=1 torsion over Λ. The Λ-module X∞ is compact, while A∞ is dis- We can also associate to M its characteristic ideal crete. To obtain a compact Λ-module from A∞, we char(M) in Λ. That is, given the above pseudo- ∨ take its Pontryagin dual A∞ = Hom(A∞, Qp/Zp), isomorphism, we define which is again finitely generated and torsion. Its r ! structure is very closely related to that of X∞. µ(M) Y ki char(M) = p fi . In good circumstances, such as when F∞ is the i=1 cyclotomic Zp-extension of F = Q(ζp), we can re- Qr ki cover An from X∞ as the largest quotient of X∞ The polynomial f is the usual characteristic n i=1 i upon which Γp acts trivially. Crucial to this is the polynomial of T acting on the finite-dimensional fact that Γ is a pro-p group. Because of this, Λ is a Qp-vector space M ⊗Zp Qp. local ring, and one can employ Nakayama’s lemma. In the case of the unramified Iwasawa module This stands in stark contrast to the case of finite Ga- X∞, the quantities λ = λ(X∞) and µ = µ(X∞) lois extensions of prime-to-p degree, for which one are those in Iwasawa’s growth formula. In fact, we ∨ ∨ has far less control over the growth of p-parts of class also have λ = λ(A∞) and µ = µ(A∞), with the ∨ groups. Nevertheless, Larry Washington showed in characteristic ideals of X∞ and A∞ differing by the 1979 that the p-parts eventually stop growing in the change of variables T 7→ (1 + T )−1 − 1. cyclotomic Z`-extension for ` 6= p of an abelian Iwasawa conjectured that µ = 0 if F∞ is the extension of Q. cyclotomic Zp-extension of F . Bruce Ferrero and As observed by Jean-Pierre Serre, the Iwasawa Washington proved this when F/Q is abelian [FW]. algebra Λ is isomorphic to a power series ring Zp T In this case, Iwasawa also showed that the (−1)- − in a single variable T . For this, we fix a topologicalJ K eigenspace X∞ for the action of complex conjuga- generator γ of Γ, which is to say an element gener- tion has no finite Λ-submodule. ating a dense subgroup, or equivalently, an element For F∞ the cyclotomic Zp-extension of F = that restricts to a generator of each Γn. There is Q(µp), the Iwasawa module X∞ has an action of then a unique continuous isomorphism of compact ∆ = Gal(F/Q) that commutes with its Λ-action, Zp-algebras that takes γ −1 to T . We shall use such so we can again break up X∞ as a direct sum an isomorphism to identify Λ and Zp T . of ∆-eigenspaces. In the computations of Hart- The structure theory of finitely generatedJ K mod- Harvey-Ong, not a single ∆-eigenspace of X∞ has (i) ules over Λ mimics the theory of finitely generated λ-invariant greater than 1. Such an eigenspace X∞ modules over a PID if one treats Λ-modules as be- is nonzero if and only if A(i) is nonzero, so these ing defined up to finite submodules and quotient computations imply that λ ≤ 9 for p < 231. modules. The idea is that Λ becomes a PID upon localization at any principal prime ideal, and there are no nonzero finite modules over the localization. p-adic L-functions A homomorphism f : M → N of finitely gen- erated Λ-modules is called a pseudo-isomorphism For any positive integer n, the Riemann zeta func- if it has finite kernel and cokernel. The notion of tion satisfies pseudo-isomorphism gives an equivalence relation B ζ(1 − n) = − n . on any set of finitely generated, torsion Λ-modules. n

Theorem (Iwasawa, Serre). For any finitely gen- The prime p divides the denominator of Bn in low- erated, torsion Λ-module M, there is a pseudo- est form if and only if p − 1 divides n by an 1840

5 result of Klausen and Von-Staudt. Let’s assume The main conjecture implies that the highest k p − 1 does not divide n, since otherwise it’s known power of p dividing Lp(ω , s) is also the order of A(1) = A(p−n) 0 (p−k) that the relevant eigenspace is . the largest quotient of X∞ upon which T acts as For m ≡ n mod p−1, Kummer showed in 1851 (1 + p)s − 1. Taking s = 0, the main conjecture that gives the order of A(p−k); it does not, however, tell ζ(1 − m) ≡ ζ(1 − n) mod p. us the isomorphism class (though refinements do This explains why our criterion for regularity re- exist). quires only indices n < p. Even better, if m ≡ The main conjecture can also be formulated in p X = n mod pj−1(p − 1) for some j ≥ 1, then we have terms of the -ramified Iwasawa module ∞ Gal(M∞/F∞) for M∞ the union of the maximal abelian p-extensions of the F ramified only at the (1−pm−1)ζ(1−m) ≡ (1−pn−1)ζ(1−n) mod pj. n unique prime (1 − ζpn+1 ) over p. This version of the main conjecture asserts that Fixing an even integer k, these congruences im- ply the existence of a continuous Zp-valued function char(X(k)) = (g ), k ∞ k Lp(ω , s) of a p-adic variable s ∈ Zp satisfying where gk is given by the change of variables k n−1 −1 Lp(ω , 1 − n) = (1 − p )ζ(1 − n) gk(T ) = fk((1 + p)(1 + T ) − 1). The equiva- lence follows from an Iwasawa-theoretic version of for all n ≡ k mod p − 1. Here as before, ω denotes Kummer duality. × × the p-adic character ω :(Z/pZ) → Zp . In [Iw2], Iwasawa proved his main conjecture ; A(k) = 0 The p-adic L-functions Lp(ω ;;) were con- assuming that . In this case, the equality structed by Kubota and Leopoldt. Their values at of characteristic ideals becomes an isomorphism other nonnegative integers are similarly given by (p−k) ∼ special values of Dirichlet L-functions of complex- X∞ = Λ/(fk). valued characters of ∆. The Riemann zeta function is the Dirichlet L-function for the trivial character. It’s worth understanding how Iwasawa’s argument goes, as through it one obtains a form of the main conjecture free from L-functions asserting an equal- The Iwasawa main conjecture ity of characteristic ideals of Iwasawa modules com- ing from unit and class groups. The Iwasawa main conjecture describes the charac- Iwasawa studied the image of the cyclotomic (p−k) units inside the local units of the completion at the teristic ideal of X∞ for an odd prime p and an prime over p, working up the tower of completions even integer k in terms of the Kubota-Leopoldt p- at p of the fields Fn by considering sequences of el- k (1) adic L-function Lp(ω , s). As X∞ is trivial since ements compatible under norm maps. The Iwasawa (1) A = 0, let us suppose that k 6≡ 0 mod p − 1. module U∞ of norm compatible sequences of local k Iwasawa showed that Lp(ω , s) is determined units contains submodules E∞ and C∞ generated by on s ∈ Zp by a unique power series fk ∈ Λ satisfy- the sequences of global units and cyclotomic units, ing respectively. s k Class field theory provides an exact sequence fk((1 + p) − 1) = Lp(ω , s). Here, we’ve taken the variable T to correspond to 0 → E∞/C∞ → U∞/C∞ → X∞ → X∞ → 0. γ − 1 for the topological generator γ of Γ that raises This in turn yields an exact sequence with each all roots of unity of p-power order to the power 1+p, k and (1+p)s is the limit of the sequence of (1+p)sn module replaced by its ω -eigenspace under ∆. If n (k) for sn ∈ satisfying sn ≡ s mod p p. X∞ = 0, then the class number formula can be Z Z (k) (k) The following conjecture of Iwasawa’s, formu- used to see that E∞ = C∞ as well. The four-term lated in the late 1960s, was given a proof by Barry exact sequence therefore reduces to an isomorphism Mazur and in a 1984 paper [MW]. (k) (k) ∼ (k) U∞ /C∞ = X∞ . Theorem (Iwasawa main conjecture, Mazur-Wiles). (k) ∼ For any even integer k 6≡ 0 mod p − 1, we have Iwasawa then obtains that X∞ = Λ/(gk) from the following unconditional theorem, which amounts to (p−k) char(X∞ ) = (fk). a p-adic regulator computation on cyclotomic units.

6 Theorem (Iwasawa). There is an isomorphism of There are Hecke operators Tn for each n ≥ 1 Λ-modules that act on modular forms by summing over the ac- Γ( 1 0 )Γ (k) (k) ∼ tion of representatives of the double coset 0 n U∞ /C∞ = Λ/(gk). as a union of right cosets. A modular form is an As characteristic ideals are multiplicative in eigenform if it is a simultaneous eigenform for all exact sequences of finitely generated, torsion Λ- Hecke operators. If an eigenform is normalized a (f) = 1 T (f) = a (f)f modules, this also tells us unconditionally that the so that 1 , then n n for all n ≥ 1 a (f) main conjecture is equivalent to the equality . The Fourier coefficients n of a nor- malized eigenform f are algebraic numbers that are (k) (k) (k) n ≥ 1 char(E∞ /C∞ ) = char(X∞ ) integral for , and the coefficient field they generate is a number field. in which no L-functions appear. Eisenstein series form a class of modular forms that are not cusp forms. For instance, for a positive Modular forms even integer k ≥ 4, we have an Eisenstein series

∞ One might say that the theme of the work of Ribet Bk X X E = − + dk−1qn and Mazur-Wiles is that the study of the geometry k 2k of varieties over Q can be used to solve arithmetic n=1 d|n questions. Specifically, their work makes use of modular curves and congruences between modu- which is an eigenform of weight k for Γ = SL2(Z) lar forms. The Galois representations attached to itself. When an odd prime p divides ζ(1 − k) = modular forms are two-dimensional, presenting a −Bk/k, the constant term of Ek is zero modulo p. natural next class of objects to study beyond the one- In this case, the reduction of Ek modulo p may be dimensional abelian characters of class field theory. lifted to a cuspidal eigenform with coefficients in We embark upon another brief tour. the ring of integers of a number field. For exam- The group of matrices in GL2(R) with positive ple, E12 is congruent modulo 691Z q to the unique determinant acts by Möbius transformations on the normalized cusp form J K upper half-plane H of complex numbers with posi- tive imaginary part. A modular curve is a quotient ∞ Y of H by the action of a subgroup of SL2(Z) that is q (1 − qn)24 determined by congruences among its entries. This n=1 quotient can be compactified by adding in the equiv- alence classes of the cusps, which are the rational of weight 12 for SL ( ). numbers and infinity. 2 Z f A modular form f is a holomorphic function To a normalized cuspidal eigenform of weight k ≥ 2 p of that transforms under a congruence subgroup , work of Shimura and Deligne attached a - H ρ : G → GL (K ) Γ (in the standard notation, but not to be confused adic Galois representation f Q 2 f , K with the Galois group appearing in Iwasawa theory) with f the field obtained by adjoining to Qp the f in a manner prescribed by its “weight” k, and which Fourier coefficients of . Equivalently, it is a two- K V is bounded and holomorphic at the cusps. Specifi- dimensional f -vector space f with a commuting G ρ cally, if a b  ∈ Γ, then Q-action. This representation f is irreducible, it c d is odd (i.e., it has determinant −1 on complex con- az + b jugation), and it has the property that the trace of f = (cz + d)kf(z). cz + d ρf (ϕ`) for a Frobenius element ϕ` of an unramified prime over a rational prime ` is equal to a`(f). A modular form is a cusp form if it is zero at all Inside Vf , there is a rank two module Lf for cusps. the valuation ring Of of Kf that is preserved by ( 1 1 ) ∈ Γ f(z + 1) = f(z) f If 0 1 , then , so has a the GQ-action. Roughly, this says that ρf can Fourier expansion about the cusp at ∞ of the form be viewed as taking values in GL2(Of ). So, it ∞ makes sense to reduce ρf modulo the maximal ideal X n of O and talk about the resulting representation f = an(f)q f ρ¯ : G → GL ( ) n=0 f Q 2 Fq over the residue field Fq. This residual representation is unique up to isomorphism 2πiz with q = e for z ∈ H. If f is a cusp form, then if and only if it is irreducible. Otherwise, its iso- a0(f) = 0. morphism class depends upon the choice of Lf .

7 The method of Ribet-Mazur-Wiles From this, it follows that one divisibility for all odd eigenspaces implies the other. Ribet and Mazur-Wiles employed congruences be- Later work of Wiles gave a more streamlined tween cusp forms and Eisenstein series to construct perspective, casting the proof in terms of the theory unramified abelian extensions of cyclotomic fields. of families of ordinary modular forms of Haruzo As we’ve noted, if p | Bk for an even k < p, then Hida. That is, Wiles employed the residual rep- there exists a cuspidal eigenform f congruent to Ek resentation of a Galois representation ρF : GQ → modulo the maximal ideal of Of . The fixed field of GL2(If ) attached to a p-adically continuously vary- the kernel of the Galois representation ρf is rami- ing family F of ordinary cuspidal eigenforms con- fied only at the prime p. Ribet used this to construct gruent to a family of Eisenstein series, where If is an unramified abelian p-extension of F = Q(ζp) on a finite local Λ-algebra. which ∆ acts through ωp−k. By class field theory, if the extension is nontrivial, then A(p−k) is nonzero, which is Ribet’s converse to Herbrand’s theorem. The method of Euler systems The method works by playing two facts forced Work of Francisco Thaine and Victor Kolyvagin led by the congruence off of each other. The first is that to a new and more explicit, though technically com- f is ordinary in the sense that a (f) is a p-adic unit. p plex, approach to the main conjecture, which Karl For ordinary forms, there exists a basis of V such f Rubin completed to a full proof. The method uses that ρ restricted to a decomposition group at p is f what Kolyvagin termed an , the first upper triangular as a map to GL (O ). If needed, 2 f example of which consists of cyclotomic units in one can rescale so that the map φ: G → to Q Fq abelian extensions of . This system of elements the residue field given by the reduction of the Q Fq is used to bound the order of an even eigenspace lower-left hand corner of ρ modulo the maximal f of the p-part of the class group by the order of an ideal of O is nonzero on the larger group G . f Q eigenspace of the quotient of the global units by the The second fact is that the residual Galois rep- cyclotomic units. resentation ρ¯ is reducible. The above basis can f The key property is a norm compatibility from actually be chosen so that ρ¯f has the form Q(ζN`) to Q(ζN ) for p dividing N and a prime `. ωk−1(σ) 0 Explicitly, if ` N, one has ρ¯ (σ) = - f φ(σ) 1 1 − ζN N (1 − ζN`) = . on σ ∈ GQ, viewing ω as a character of GQ. The re- Q(ζN`)/Q(ζN ) `−1 1 − ζN striction of φ to Gal(Q/F ) is then a homomorphism that is unramified at the prime over p by construc- With this relation, one applies a Galois-theoretic tion, so it factors through the Galois group of a non- derivative construction to elements 1−ζN` for good trivial unramified abelian p-extension H of F that choices of primes ` congruent to ±1 modulo a suffi- is Galois over Q. The group ∆ acts on Gal(H/F ) ciently high power of p to obtain field elements that compatibly with conjugation of matrices, which is p−k k−1 −1 are powers of chosen non-principal ideals. to say that it acts by ω = (ω ) . Up the cyclotomic tower, the method of Euler Mazur and Wiles generalized and refined Ri- systems shows that bet’s method in the context of Iwasawa theory in order to prove the main conjecture and its gener- (k) (k) (k) char(X∞ ) | char(E∞ /C∞ ) alization to arbitrary abelian extensions of Q. By studying the Galois actions on Jacobians of modular for even k ∈ Z. This is equivalent to the opposite curves, they construct an unramified abelian exten- divisibility to that of Mazur-Wiles, and again the sion of the field F∞ of all p-power roots of unity analytic class number formula yields equality. (p−k) with Galois group a Λ-module quotient of X∞ that has characteristic ideal (fk). (p−k) Totally real and CM fields Having proven that (fk) | char(X∞ ), Mazur and Wiles apply a consequence of analytic class The Iwasawa main conjecture generalizes directly number formula to obtain the main conjecture. That from Q to totally real fields, those number fields is, Iwasawa had shown in 1972 that with only real archimedean embeddings. The rele- (p−3)/2 (p−3)/2 vant p-adic L-functions were separately constructed X (p−2j) X λ(X∞ ) = deg f2j. by P. Deligne and Ribet, P. Cassou-Noguès, and j=1 j=1 D. Barsky. Wiles proved a main conjecture for

8 odd eigenspaces of the unramified Iwasawa mod- Elliptic curves and BSD ule X∞ over the cyclotomic Zp-extension F∞ of an abelian extension F of a totally real field using Elliptic curves over number fields provide a next Galois representations attached to Hilbert modular step beyond the theory of multiplicative groups that forms [Wi]. has conjectured that we’ve been in effect describing. In this setting, the even eigenspaces of X∞ are finite. the arithmetic objects that replace class groups are The existence of an Euler system in abelian ex- known as Selmer groups. We run quickly through tensions of totally real fields was conjectured by the arithmetic theory of elliptic curves, from the ba- Harold Stark and Rubin, but it still unproven. This sic theory to deep results and a famous conjecture, is related to Hilbert’s 12th problem, or Kronecker’s before we begin passing up towers. Jugendtraum, of an explicit class field theory over A complex elliptic curve E is defined by an totally real fields. In contrast, abelian extensions equation 2 3 of imaginary quadratic fields contain analogues of y = x + ax + b cyclotomic units called elliptic units that do form with a, b ∈ and 4a3 +27b2 6= 0 to ensure smooth- an Euler system. In this case, there is an explicit C ness. More precisely, it is the projective curve of form of class field theory arising from the theory genus one defined by the homogenization of the of complex multiplication (CM) of elliptic curves above polynomial. Effectively, this means adding a that we’ll discuss. Elliptic units are generated by single point ∞ at infinity. values at torsion points of a theta function that is a The set E( ) of points of E with complex co- meromorphic function on a CM elliptic curve. C ordinates has an abelian group law with ∞ as its In 1977, John Coates and Wiles proved an ana- identity. It is given by drawing a line between two logue of Iwasawa’s theorem on local units modulo points P and Q and declaring the third point of the K cyclotomic units for an imaginary quadratic field line in E(C) to be −P − Q, taking multiplicity into h = 1 p with K [CW]. Choosing a split prime with account. We’ll assume that a, b ∈ Q, which is to say pOK = pp¯, their theorem states that the quotient of that E is rational. The Mordell-Weil group E(F ) of the local units at p modulo elliptic units up a Zp- points with coordinates in a number field F is then K p extension of ramified only at is isomorphic to a finitely generated subgroup of E(C). the quotient of the one-variable Iwasawa algebra by The group E(Q) has a canonical action of a power series corresponding to a p-adic L-function GQ via the action on coordinates. A point of constructed by Nick Katz. ∼ 2 E(C) = (R/Z) is said to be n-torsion if it has order The compositum of Zp-extensions of an imagi- dividing n. The group E[n] of all n-torsion points 2 2 nary quadratic field K has Galois group Zp. Its Iwa- is a subgroup of E(Q) isomorphic to (Z/nZ) . sawa algebra is a power series ring in two variables A rational elliptic curve E has a ring O of endo- over Zp. The unramified Iwasawa module X∞ over morphisms consisting of nonzero -rational maps 2 Q the corresponding Zp-extension of an abelian exten- E → E taking ∞ to itself. Among these are the sion of K is conjecturally small enough to have unit morphisms given by multiplication by integers us- characteristic ideal. The main conjecture here com- ing the group law of E. If O 6= Z, then O is a pares X∞ and the quotient of global units by elliptic finite index subring of the integer ring of an imagi- units. It was proven by Rubin using the method of nary quadratic field. In this case, we say that E has Euler systems in 1991 [Ru]. If p splits in K, one complex multiplication, or CM, by O. can instead replace X∞ with a larger Iwasawa mod- The cohomology of a group G with coefficients ule that allows ramification at exactly one of the in a module M for its group ring is a sequence of two primes over p. This gives an equivalent form abelian groups Hi(G, M) that allows one to study involving a two-variable Katz p-adic L-function. the group action using the tools of homological al- In 1994, Hida and Jacques Tilouine gave an al- gebra. For Galois groups, the closely related theory ternate proof of the specialization of this conjec- of Galois cohomology is a crucial tool in Iwasawa ture to an anticyclotomic Zp-extension using an ap- theory. Many theorems of class field theory can proach closer to that of Mazur-Wiles. Their work be phrased in terms of duality in Galois cohomol- extends to analogues of imaginary quadratic fields ogy, and abelian groups of arithmetic interest can over totally real fields known as CM fields for which be encoded in cohomology. one once again has no known Euler system to em- For a number field F , the group E(F )/nE(F ) 1 ploy. More recently, progress has been made on a is contained in H (Gal(Q/F ),E[n]). In fact, it divisibility in a general main conjecture over CM is contained in a smaller subgroup of cohomology fields, in particular by Ming-Lun Hsieh. classes that are unramified at all but finitely many

9 primes and partially vanish locally at the remaining Iwasawa theory of elliptic curves primes. The direct limit over n of these subgroups of 1 H (Gal(Q/F ),E[n]) is the SelE(F ) We turn to the question of how Selmer groups of we wish to study. a rational elliptic curve E grow in the cyclotomic The most important thing to know about the Zp-extension Q∞ of Q. This amounts to studying Selmer group is that it intertwines the Mordell-Weil the finitely generated Λ-module group with a mysterious, conjecturally finite group X = Hom(Sel ( ), / ) called the Shafarevich-Tate group XE(F ) of E. E E Q∞ Qp Zp That is, there is an exact sequence: that is the Pontryagin dual of the direct limit ∞ SelE(Q∞)[p ] of p-power torsion subgroups of the 0 → E(F ) ⊗ / → SelE(F ) → XE(F ) → 0. Q Z Selmer groups SelE(Qn). The elliptic curve E has good reduction at p This is analogous to what happens with cohomol- if its mod p reduction Ep is nonsingular, and it is ogy with coefficients in roots of unity: in that case, then ordinary if Ep has a point of order p. For such the unit and class groups get wrapped up together. elliptic curves, Mazur and Swinnerton-Dyer con- The intertwining of E(Q) and XE(Q) is also structed a p-adic L-function Lp(E, s) interpolating reflected in analytic formulas. One can construct an values of L(E, s) up to certain Euler factors, and L-series for E from the data of the number of Fp- again it is determined by a power series LE. Mazur points of mod p reductions of a minimal equation then formulated the following main conjecture. for E. It has analytic continuation to C by the mod- ularity of rational elliptic curves proven by Wiles, Conjecture (Main conjecture for elliptic curves). Taylor-Wiles, and Breuil-Conrad-Diamond-Taylor, Suppose that E has good ordinary reduction which tells one that E has an associated cuspidal at p and that E(Q)[p] is an irreducible GQ- eigenform with the same L-series. representation. Then XE is Λ-torsion and The Birch and Swinnerton-Dyer conjecture, or char(X ) = (L ). BSD, was formulated in 1965 and is one of the Clay E E Math Institute’s Millennium Problems. For elliptic curves with complex multiplication, this is equivalent to the main conjecture for imag- Conjecture (Birch and Swinnerton-Dyer). The or- inary quadratic fields and split primes p proven by der of vanishing of the L-function L(E, s) at s = 1 Rubin. The general divisibility char(XE) | (LE) is equal to the rank of E(Q). was proven by Kazuya Kato via the method of Euler systems [Ka]. Kato’s Euler system is constructed The BSD conjecture has a refined form that links using cohomological products formed from pairs of the leading term of L(E, s) in its Taylor expansion Siegel units on a modular curve parameterizing E, about s = 1 to the orders of the torsion subgroup first studied by A. Beilinson. The other divisibility E(Q)tor of Mordell-Weil and of XE(Q). The con- was proven under fairly mild hypotheses by Chris jectural BSD formula has the form Skinner and using Galois representa- tions attached to automorphic forms on the unitary (r) Q group GU(2, 2) [SU]. With no analytic class num- L (E, 1) |XE(Q)|ΩERE ` c` = 2 , (1) ber formula that can be used in this setting, one r! |E(Q)tor| needs both methods. We mention a bit of what’s known for elliptic where r is the order of vanishing, RE is a regulator curves with good supersingular (i.e., non-ordinary) related to the heights of rational points, the quantity reduction. For p ≥ 5, there are in this case not ΩE is a real period, and each c` for a prime ` is the one but two Selmer groups constructed by Shinichi number of components of a certain mod ` reduction Kobayashi, and two p-adic L-functions constructed of E, all but finitely many being 1. by Rob Pollack. The corresponding main conjec- Much of the progress on BSD to date employs ture was proven by Rubin and Pollack for CM curves Iwasawa theory. For instance, the first major result in 2004. In this case, p does not split in K, and that serves as theoretical evidence for BSD was due the main conjecture is closely related to Rubin’s to Coates-Wiles. As a consequence of their theorem main conjecture without L-functions for imaginary on local units modulo elliptic units, they proved that quadratic fields. The main conjecture for non-CM if r = 0 and E has CM by a subring of OK for K curves has recently been proven by Xin Wan under a with hK = 1, then E(Q) is finite. hypothesis on the congruence subgroup. F. Sprung

10 has additionally treated the prime 3, in particular main conjecture for totally real fields was proven employing work of B.D. Kim and A. Lei in the for- through work of Ritter and Weiss, Burns, and mulation. Kakde, assuming Iwasawa’s conjecture on the van- The main conjecture for elliptic curves implies a ishing of the µ-invariant. The key step is the verifi- p-adic analogue of BSD of Mazur-Tate-Teitelbaum cation of congruences among Deligne-Ribet p-adic that relates the rank of E(Q) to the order of vanish- L-functions over different totally real fields to re- ing of Lp(E, s) at 1. As L(E, s) is complex analytic duce to the main conjecture proven by Wiles. and Lp(E, s) is p-adic analytic, the derivatives are The work of Skinner-Urban has inspired a rush not clearly related, and neither form of BSD obvi- of new theorems on divisibilities in main con- ously implies the other in the case of positive rank. jectures, with recent progress on the construction The order r of vanishing of L(E, s) at s = 1 of Galois representations attached to automorphic is known as the analytic rank of E, and the actual forms in the sense of the pro- rank of E(Q) is known as the algebraic rank. Koly- viding a boon for the area. The construction of vagin used an Euler system of Heegner points and a p-adic L-functions that interpolate special values of theorem of B. Gross and D. Zagier to prove that if complex L-functions is a whole industry unto itself the analytic rank of E is r ≤ 1, then the algebraic and often proceeds via distributions formed out of rank is r and XE(Q) is finite. Both the converse modular symbols or computations of local integrals. to this and the BSD formula follow from the main Euler systems have long had a reputation as dif- conjecture for r = 0. Recent work of Skinner and of ficult to construct that has only recently begun to Wei Zhang implies a converse for r = 1 under mild soften. An approach to construct them via geomet- hypotheses, and for r = 1 significant progress has ric methods starting from cycles on varieties was been made towards the BSD formula as well, par- initiated in work of Bertolini, Darmon, Prasanna, ticularly in work of D. Jetchev, Skinner, and Wan. and Rotger and carried further in work of Loef- fler, Zerbes, Lei, Kings, and Skinner. One typically computes complex and p-adic regulators to prove Recent directions nonvanishing of Euler systems and relate them with (p-adic) L-functions. Iwasawa theory extends to study the growth of other arithmetic objects attached to Galois representa- Beyond Mazur-Wiles tions in towers of number fields. Beginning in the late 1980s, Greenberg proposed main conjectures I end with a brief discussion of a deeper relation- for a whole host of ordinary motivic Galois repre- ship between the geometry of modular curves and sentations, and even continuously p-adically vary- the arithmetic of cyclotomic fields. In a 2011 paper, ing families thereof. Since then, main conjectures I formulated a conjecture relating relative homology have been extended to more general towers, to non- classes {α → β} of paths between cusps on a mod- ordinary families, to finer-grained analogues, and ular curve, taken here to be X (p), and cup products even to characteristic p base fields. Recent develop- 1 x ∪ y of cyclotomic units in a Galois cohomology ments have seen considerable progress on methods group that agrees with A− modulo p. In fact, there of proof of both divisibilities. Here’s a sampling. is a simple explicit map The early part of the new millennium saw the de- velopment of Iwasawa theory over towers of number a b  → 7→ (1 − ζc) ∪ (1 − ζd) fields with Galois groups that are isomorphic to sub- c d p p groups of GLn(Zp) for some n. The breakthrough came in a paper of Coates, Fukaya, Kato, Sujatha, taking one set of elements to the other (for ad−bc = and Venjakob containing a noncommutative main 1 and p - cd). On a certain Eisenstein quotient of the conjecture for elliptic curves. Invariants playing the plus part of homology, I conjectured this to provide roles of characteristic ideals and p-adic L-functions an inverse to a canonical version of the map that lie in a first K-group of a localization of the non- appeared in the proof of Ribet’s theorem. commutative Iwasawa algebra. Soon after, Fukaya Up the cyclotomic tower, this yields what can and Kato formulated a remarkably general noncom- be viewed as a refinement of the Iwasawa main con- mutative main conjecture that relates closely to the jecture. That is, it provides not only an equality equivariant Tamagawa number conjecture of Burns of characteristic ideals, but an isomorphism given and Flach which in turn generalized a conjecture of by a recipe on special elements. Fukaya and Kato Bloch and Kato on special values of L-functions. proved a major result in this direction in which the At the start of this decade, a noncommutative derivative of the p-adic L-function plays a crucial

11 and potentially unavoidable intermediate role. Their [Iw2] K. Iwasawa, On p-adic L-functions, Ann. of Math. 89 result implies the conjecture for p < 231. (1969), 198–205. MR0269627 Fukaya, Kato, and I expect this to be a spe- cial case of a general phenomenon of the geometry [Ka] K. Kato, p-adic Hodge theory and values of zeta func- tions of modular forms, Cohomologies p-adiques et and topology of locally symmetric spaces of higher applications arithmétiques, III, Astérisque 295 (2004), dimension informing the arithmetic of Galois repre- 117–290. MR2104361 sentations attached to lower-dimensional automor- phic forms. This begs the question of its elliptic [MW] B. Mazur, A. Wiles, Class fields of abelian extensions curve analogue, which is but one of a wealth of of Q, Invent. Math. 76 (1984), 179–330. MR0742853 intriguing possibilities for future directions in Iwa- sawa theory. [Ri] K. Ribet, A modular construction of unramified p- extensions of Q(µp), Invent. Math. 34 (1976), 151– 162. MR0419403 References [Ru] K. Rubin, The “main conjectures” of Iwasawa the- [CW] J. Coates, A. Wiles, On the conjecture of Birch and ory for imaginary quadratic fields, Invent. Math. 103 Swinnerton-Dyer, Invent. Math. 39 (1977), 223–251. (1991), 25–68. MR1079839 MR0463176 [SU] C. Skinner, E. Urban, The Iwasawa main conjec- [FW] B. Ferrero, L. Washington, The Iwasawa invariant tures for GL2, Invent. Math. 195 (2014), 1–227. µp vanishes for abelian number fields, Ann. of Math. MR3148103 109 (1979), 377–395. MR0528968 [Iw1] K. Iwasawa, On Γ-extensions of algebraic number [Wi] A. Wiles, The Iwasawa conjecture for totally fields, Bull. Amer. Math. Soc. 65 (1959), 183–226. real fields, Ann. of Math. 131 (1990), 493–540. MR0124316 MR1053488

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