aa r X i v : . [ m a t h . N T ] F e b AN INTRODUCTION TO EISENSTEIN MEASURES
E. E. EISCHEN
Abstract.
This paper provides an introduction to Eisenstein measures, a powerfultool for constructing certain p -adic L -functions. First seen in Serre’s realization of p -adic Dedekind zeta functions associated to totally real fields, Eisenstein measuresprovide a way to extend the style of congruences Kummer observed for values of theRiemann zeta function (so-called Kummer congruences ) to certain other L -functions.In addition to tracing key developments, we discuss some challenges that arise in moregeneral settings, concluding with some that remain open. Introduction
In the mid 1800s, Kummer proved that the values of the Riemann zeta function atnegative odd numbers satisfy striking congruences modulo powers of any prime number p . More precisely, he proved that if k and k ′ are positive even integers not divisible by p −
1, then for all positive integers d , ( − p k − ) ζ ( − k ) ≡ ( − p k ′ − ) ζ ( − k ′ ) mod p d , whenever k ≡ k ′ mod ϕ ( p d ) , with ϕ denoting Euler’s totient function [Kum51]. (ByEuler’s work a century earlier, for all positive integers k , ζ ( − k ) is the rational num-ber (− ) k + B k k for all positive integers k , where B k denotes the k -th Bernoulli number,defined as the coefficients in the Taylor series expansion te t e t − = ∑ ∞ n = B n t n n ! .) Kummer’smotivation for studying these congruences stemmed from his interest in determiningwhen a prime is (what we now call) regular , i.e. does not divide the class number ofthe cyclotomic field Q ( ζ p ) with ζ p ∈ C × a primitive p -th root of unity, in which caseKummer could prove Fermat’s Last Theorem for exponents divisible by p . As part ofhis investigations, Kummer had shown that the regularity is equivalent to a condition onthe values of the Riemann zeta function. More precisely, Kummer showed that a prime p is regular if and only if if does not divide the numerator of the Bernoulli numbers B , B , . . . , B p − [Kum50a, Kum50b].After Kummer’s exciting discoveries, this topic then lay nearly dormant for a century.Even though Hensel (who had been one of Kummer’s graduate students) introduced the p -adic numbers soon after Kummer’s death [Hen05], the first formulation of a p -adiczeta function (a p -adic analytic function that interpolates the values of a modified zetafunction at certain points, essentially encoding Kummer’s congruences) occurred onlyin the 1960s, as a result of work of Kubota and Leopoldt [KL64]. These functions, too,play a key role in the structure of cyclotomic fields, on a deeper level than Kummer hadoriginally developed (or presumably even envisioned, given mathematical developmentsduring the century following Kummer’s life). Partially supported by NSF Grants DMS-1559609 and DMS-1751281.
In the 1960s, Iwasawa linked the behavior of Galois modules over towers of cyclotomicfields to p -adic zeta-functions, forming the foundations of Iwasawa theory , a p -adic theoryfor studying families of arithmetic data [Iwa69a, Iwa69b]. For example, the subgroupΓ ∶= Gal ( Q ∞ / Q ) ≅ Z p of Gal ( Q ( µ p ∞ )/ Q ) acts on the p -part of the ideal class group ofthe p m -th cyclotomic extension for each m ≥
1. The inverse limit (under a norm map)of these groups is a module over the Iwasawa algebra Λ ∶= Λ Γ , Z p = Z p [[ Γ ]] ≅ Z p [[ T ]] .The main conjecture of Iwasawa theory, proved in [MW84], says a realization θ ∈ Λ ofa p -adic L -function generates the characteristic ideal of this Λ-module. Thus, the p -adic L -function controls substantial structural information about this collection of classgroups.Iwasawa’s conjectures were further generalized. In particular, R. Greenberg predictedthe existence of more general p -adic L -functions ( p -adic analytic functions that can berealized as elements of certain Iwasawa algebras and whose values encode analogues ofKummer’s congruences for more general L -functions) and their meaning in the context ofcertain Galois modules. In particular, the Greenberg–Iwasawa main conjectures [Gre89,Gre91, Gre94] predict that for a wide class of ordinary Galois representations ρ , thereis a p -adic L -function L ρ interpolating values of an L -function associated to ρ ⊗ χ as χ varies over certain Hecke characters and that L ρ can be realized as the generator ofthe characteristic ideal of a certain Λ-module (a Selmer group), where Λ ∶= O[[ Γ K ]] ,with Γ K the Galois group of a compositum of Z p -extensions of K and O an appropriate p -adic ring. In other words, the main conjectures predict p -adic L -functions govern thestructure of Selmer groups as Galois modules.Given L -functions’ starring role in the main conjectures of Iwasawa theory (and theirconjectured existence, including not only in Greenberg’s conjectures, but also in, forexample, [CPR89, Coa89]), it is natural to ask: Question 1.
Given an L -function whose values at certain points are known to be alge-braic, how might we construct a p -adic L -function encoding congruences between valuesof (a suitably modifed at p ) version of that L -function? The main goal of this paper is to introduce particular tools,
Eisenstein measures , whichhave proved to be especially useful for constructing p -adic L -functions during the pasthalf-century, at least under certain conditions. Even putting aside the challenge of tryingto prove main conjectures in Iwasawa theory, it is generally hard to answer Question 1.One might first look to Kummer or to Kubota–Leopoldt (who actually considered p -adic Dirichlet L -functions) for answers, in the hope that earlier techniques could begeneralized. This would, however, require extending congruences coming from Bernoullipolynomials to other settings, and unfortunately, we do not generally have realizations ofvalues of L -functions in terms of similarly convenient polynomials. While there has beensome successful work in that direction (see, e.g., work of Barsky [Bar78] and P. Cassou-Nogues [CN79], who employed formulas of Shintani that have recently been furtherexplored in work of Charollois–Dasgupta [CD14]), one of the most powerful tools forconstructing p -adic L -functions in increasing generality during the past half-centurycomes from the theory of p -adic modular forms .In the early 1970s, Serre produced the first p -adic families of Eisenstein series (thefirst instances of Eisenstein measures , which arose as part of his development of the
ISENSTEIN MEASURES 3 theory of p -adic modular forms) and used them (together with Iwasawa’s constructionof the p -adic zeta function as an element of an Iwasawa algebra) to construct p -adicDedekind zeta functions associated to totally real number fields [Ser73]. Because modularforms are special cases of automorphic forms and because the behavior of L -functionsis closely tied to the behavior of automorphic forms (at least, in certain settings), thisapproach seemed more amenable to generalization. Indeed, its promise was immediatelyrealized, including by Coates–Sinnott [CS74], Deligne–Ribet [DR80], and Katz [Kat78],who employed Eisenstein measures in constructions of p -adic L -functions associated toHecke characters for quadratic real fields, real number fields, and CM fields (with Katzproving the CM case only for half of all primes, a restriction that stood for over fourdecades until work of Andreatta–Iovita in 2019 [AI19]), respectively.Given that Eisenstein series govern key properties of certain L -functions (not only al-gebraicity, but also functional equations and meromorphic continuation), it is perhapsnot surprising that they play key roles in our context as well. Thus, another importantquestion becomes: Question 2.
How might we construct p -adic families of Eisenstein series or, more specif-ically, p -adic Eisenstein measures? Constructing p -adic Eisenstein measures is generally hard. Were it not for the presti-gious journal in which all those first papers following Serre’s introduction of Eisensteinmeasures were published or the accomplished mathematicians whose names are attachedto these results, the reader could not be blamed for thinking these results sound incre-mental. Instead, though, this should be viewed as evidence that seemingly small tweaksto the data to which L -functions are attached can lead to significant technical challengesin constructing the corresponding p -adic L -functions.For most readers of this paper, implicit in Question 2 is that we want Eisenstein seriesthat can be directly related to values of L -functions. It is worth noting, though, thatinterest in Question 2 extends beyond number theory. At least in the cases of modularforms and automorphic forms on unitary groups of signature ( , n ) , p -adic families ofEisenstein series also are of interest in homotopy theory [Hop02, Hop95, Beh09].Returning to Question (1), we note the favorable fact that (at least, as it appears tothis author) all known constructions of p -adic L -functions seem to rely on building onthe specific techniques employed in the proof of the algebraicity of the values of the C -valued L -function in question. Thus, if you know a proof of algebraicity, then you areat least in possession of clues to the techniques needed to construct p -adic L -functions.For example, Serre’s development of the theory of p -adic modular forms and its use forconstructing p -adic zeta functions built on the approaches of Klingen and Siegel (whowere, in turn, building on ideas of Hecke, as recounted in [BCG20, Section 1.3] and[IO06]) for studying algebraicity of values of zeta functions by exploiting properties ofFourier coefficients of modular forms [Kli62, Sie69, Sie70]. Likewise, Katz’s constructionof p -adic L -functions associated to Hecke characters of CM fields employs Damerell’sformula, which was first used by Shimura to prove algebraicity. In a similar spirit, Hida’sconstruction of p -adic Rankin–Selberg L -functions attached to modular forms builds onShimura’s proof of algebraicity via the Rankin–Selberg convolution [Shi76, Hid85]. In amore recent instance, the construction of p -adic L -functions for unitary groups due to E. E. EISCHEN the author, Harris, Li, and Skinner [EHLS20] employs the doubling method (a pull-backmethod of the sort used by Shimura to prove algebraicity, e.g. in [Shi00], and whichspecializes in the case of rank 1 unitary groups to Damerell’s formula).1.1.
Organization of this paper.
Now that we have established some history and mo-tivation for studying Eisenstein series, we spend the remainder introducing their math-ematical features. Section 2 introduces distributions and measures from several view-points, each of which is useful for different aspects of working with p -adic L -functions.Section 3 then discusses the first example of an Eisenstein measure, produced by Serreas a tool for constructing p -adic Dedekind zeta functions associated to totally real fields.This development inspired efforts to construct Eisenstein measures valued in the spaceof p -adic Hilbert modular forms, tools for constructing p -adic L -functions attached tocertain Hecke characters, as discussed in Section 4. We conclude with a discussion of gen-eralizations to other L -functions (Section 5), as well as some of the significant challengesencountered as one tries to construct useful Eisenstein measures.1.2. Acknowledgements.
I would like to thank the local organizers of Iwasawa 2019,Denis Benois and Pierre Parent, for inviting me to give the four lectures that eventuallyled to this paper, as well as for their patience as I wrote it. I would also like to thankthem, along with the scientific organizers of Iwasawa 2019 (Henri Darmon, Ming-LunHsieh, Masato Kurihara, Otmar Venjakob, and Sarah Zerbes), for organizing an exciting,educational workshop. The many excellent discussions I had with participants aftereach of my lectures influenced my approach to explaining the material in this paper. Iwould especially like to thank Chi-Yun Hsu and Sheng-Chi Shih for taking careful notesin my lectures and sharing them with me. In addition, I would like to thank PierreCharollois for alerting me to some interesting aspects of the history of the approach ofusing constant terms of modular forms to study zeta functions. I am also grateful to thereferee for providing helpful feedback.2. p -adic distributions and measures Motivated by Iwasawa’s and Greenberg’s conjectures about the Galois theoretic role of p -adic L -functions, our goal is to find an element in an Iwasawa algebra whose valuesat certain characters encode congruences between certain (modified) L -functions. Dis-tributions and measures will provide a convenient tool for realizing p -adic L -functionsinside Iwasawa algebras.For a more detailed treatment of distributions and measures, we especially recommend[MSD74, §
7] and [Was97, § § Conventions and preliminaries.
Throughout this paper, we fix a prime number p . For convenience, we assume p is odd. We denote by C p the completion of an algebraicclosure of Q p . We call a ring O a p -adic ring if it is complete and separated with respectto the p -adic topology, i.e. O ≅ lim ←Ð O/ p n O . Given a p -adic ring O and a profinite group G = lim ←Ð n G / G n with each G n a finite index subgroup of G , we define the Iwasawa algebraΛ G ∶= O[[ G ]] ∶= lim ←Ð n O [ G / G n ] . ISENSTEIN MEASURES 5
Following the usual conventions in Iwasawa theory, we defineΓ ∶= + p Z p Λ ∶= Z p [[ Γ ]] ≅ Z p [[ T ]] . We also denote by µ p − the multiplicative subgroup of order p − Z × p . Given a numberfield F , we denote by F ( p ∞ ) the maximal abelian unramified away from p extension of F .In addition, throughout this section, let T be a locally compact totally disconnectedtopological space, and let W be an abelian group. (For example, T could be the Galoisgroup of a Z p -extension and W could be the ring of integers in a finite extension of Q p .)We denote by Step ( T ) the group of Z -valued functions on T that are locally constant ofcompact support. For any compact open subset U ⊆ T , we denote by χ U ∈ Step ( T ) thecharacteristic function of U .2.2. Distributions.
We begin by introducing distributions, which include measures asa special case.
Definition 2.1. A distribution on T with values in W is a homomorphism µ ∶ Step ( T ) → W. We set the notation ∫ T ϕ ( t ) dµ ∶= µ ( ϕ ) for each ϕ ∈ Step ( T ) .The space of W -valued distributions on T is thenDist ( T, W ) ∶= Hom ( Step ( T ) , W ) . Observe that we have a bijection between Dist ( T, W ) and the set A( T, W ) of finitelyadditive W -valued functions on compact open subsets of T . By abuse of notation, given µ ∈ Dist ( T, W ) , we also denote by µ ∈ A( T, w ) the corresponding element under thisbijection. More precisely, given a compact open subset U ⊂ T , µ ( U ) ∶= ∫ U dµ ∶= µ ( χ U ) = ∫ T χ U dµ. Distributions on (pro)finite sets.
Observe that if T is finite, then since T is totallydisconnected, Dist ( T, W ) is identified with the abelian group of W -valued functions on T . So if X is the inverse limit of a collection of finite sets X i , i ∈ I a directed poset, suchthat whenever i ≥ j , we have a surjection π ij ∶ X i ↠ X j and whenever i ≥ j ≥ k , π jk ○ π ij = π ik , then we can reformulate the notion of W -valueddistribution on X as a collection of W -valued maps µ j ∶ X j → W such that µ j ( x ) = ∑ { y ∣ π ij ( y )= x } µ i ( y ) E. E. EISCHEN for all i ≥ j and all x ∈ X j . So we haveDist ( X, W ) = lim ←Ð n Dist ( X n , W ) . Measures.
We now suppose that W is a finite-dimensional Banach space over anextension K of Q p , as this case will be particularly interesting to us. Definition 2.2. A W -valued measure on T is a bounded W -valued distribution on T . Definition 2.3.
If a measure µ takes values in a subgroup A ⊆ W , then we call µ an A -valued measure .Given topological spaces X and Y , we denote by C( X, Y ) the space of continuous mapsfrom X to Y . Observe that if T is compact and W is a finite-dimensional K -Banachspace, then there is a bijection { W -valued measures on T } ↔ { bounded homomorphisms of K -Banach spaces C( T, K ) → W } . Likewise, if O is a p -adically complete ring, then we have bijections {O -valued measures on Y } ↔ {O -linear maps C( Y, O) → O} ↔ { Z p -linear maps C ( Y, Z p ) → O} . More generally, given an O -valued measure µ on Y and a homomorphism ϕ ∶ O → O ′ ,with O ′ also a p -adic ring, we get an O ′ -valued measure µ ′ on Y defined by ∫ Y f dµ ′ ∶= ϕ ( ∫ Y f dµ ) for all f ∈ C( Y, Z p ) .2.3.1. Measures on profinite groups.
Our main case of interest is the case where T is aprofinite group. We write T = lim ←Ð j T / T j with the subgroups T j the ones that are openfor the topology on T . (So the groups T j are the finite index, normal subgroups of T .)Then for any p -adic ring O , we have an isomorphism of O -modules ψ ∶ Dist ( T, O) ∼ → Λ T = O[[ T ]] (2.1) µ ↦ α µ ∶= ⎛⎝ ∑ T / T j µ j ( g ) g ⎞⎠ j ≥ , (2.2)where µ j is as in Section 2.2.1.Note that each f ∈ C( T, O) can be extended O -linearly to a function on the group ring O[ T ] , via f ( ∑ g ∈ T a g g ) = ∑ g ∈ T a g f ( g ) (2.3)for each finite sum ∑ g ∈ T a g g ∈ O[ T ] with a g ∈ O . (Since O[ T ] is a group ring, a g = g .) Also, note that O[ T ] is a subring of O[[ T ]] , via O[ T ] ↪ O[[ T ]] ∑ g ∈ T a g g ↦ ⎛⎝ ∑ T / T j a g ( g mod T j )⎞⎠ j ≥ Since O[ T ] is dense in O[[ T ]] , we extend the map in Equation (2.3) continuously to O[[ T ]] . (The rings O[ T / T j ] are endowed with the product topology coming from O , andso the same is true for O[[ T ]] .) Of particular interest is the case where T is generatedby a topological generator γ (e.g. γ = + p in T = + p Z p ) and f ∶ T → O × is a grouphomomorphism, in which case the elements of O[[ T ]] can be identified with power series ∑ j a j γ j , and f ( ∑ j a j γ j ) = ∑ j a j f ( γ ) j . Similarly, if T = Γ × ⋯ × Γ d with Γ i = + p Z p , i = , . . . , d , with generators γ i = + p Z p , respectively, then each element of O[[ T ]] can beexpressed as a power series in γ , . . . , γ d , and for h = ∑ n ,...,n d ≥ a n ,...,n d γ n ⋯ γ n d d ∈ O[[ T ]] , f ( h ) = ∑ n ,...,n d ≥ a n ,...,n d f ( γ ) n ⋯ f ( γ d ) n d ∈ O[[ T ]] .The inverse map ψ − is given by µ α ← [ α, where for each f ∈ C( T, O) µ α ( f ) ∶= f ( α ) . If O is flat over Z p , then each element α ∈ Λ T corresponding to a measure µ α is com-pletely determined by ∫ T χdµ α , where χ varies over finite order characters with valuesin extensions of Q p (see Proposition 2.13).2.3.2. First examples.
Let K be a finite extension of Q p , and let T be an infinite profinitegroup. Example 2.4.
It is a simple exercise to show that the K -valued Haar distributions µ Haar (i.e. translation invariant distributions, so µ Haar ( U + y ) = µ Haar ( U ) for all y ∈ T and compact open subsets U ⊂ T ) on T are not measures if T is a pro- p group (but aremeasures if T is a pro- ℓ group with ℓ ≠ p ). Example 2.5.
Fix an element g ∈ T . The Dirac distribution δ g defined by δ g ( U ) ∶= ⎧⎪⎪⎨⎪⎪⎩ g ∈ U , for all compact open subsets of T , is a measure on T . Under the isomorphism ψ in (2.1), µ g corresponds to the element g ∈ Λ T .2.4. Bernoulli distributions and Dirichlet L -functions. We now briefly introducea measure that produces a p -adic Dirichlet L -function. More details are available in[Was97, § χ be a Dirichlet character, let L ( s, χ ) be the associatedDirichlet L -function. Then for all positive integers n , L ( − n, χ ) = − B n,χ n , where the numbers B n,χ are the generalized Bernoulli numbers, i.e. the numbers definedby f ∑ a = χ ( a ) te at e ft − = ∞ ∑ n = B n,χ t n n ! , E. E. EISCHEN with f denoting the conductor of χ . We also define a modified Dirichlet L -function L ( p ) ( − n, χ ) = ( − χ ( p ) p n − ) L ( − n, χ ) . If χ is the trivial character, so f =
1, then B n,χ = B n , where B n denotes the n -th Bernoulli number, and L ( χ, − n ) = ζ ( − n ) is theRiemann zeta function studied by Kummer in the mid-1800s, as discussed in Section 1. Remark . As above, let n be a positive integer. Note that when χ is odd and n ispositive, L ( − n, χ ) =
0. Likewise, when χ is even and n is odd, L ( − n, χ ) =
0, unless χ is the trivial character and n = ζ ( ) = − ). This can be seenfrom the functional equation for L ( s, χ ) , as explained in, e.g., [Was97, Chapter 4].Let ω ∶ Z × p → µ p − ⊆ Z × p denote the Teichm¨uller character (so ω ( a ) ≡ a mod p for each a ∈ Z × p ). Kummer’scongruences are a special case of the following: Theorem 2.7 ([KL64]) . Let χ be a Dirichlet character. Then there exists a p -adicmeromorphic (analytic, if χ ≠ ) function L p ( − n, χ ) on { x ∈ C p ∣ ∣ s ∣ p < p p − p − } such that L p ( − n, χ ) = ( − χω − n ( p ) p n − ) − B n,χω − n n = L ( p ) ( − n, χω − n ) for all positive integers n . This is the first example of a p -adic L -function , i.e. a p -adic analytic function whosevalues at certain points agree with values of (suitably modified) C -valued L -functions. Remark . Fix an integer n such that 0 < n < p −
1. Then for all n ≡ n mod p − ω n = ω n and L p ( − n, ω n ) = ζ ( p ) ( − n ) . Thus, we easily locate the values studied by Kummer among those p -adically interpolatedby L p . Furthermore, by expressing L p ( s, χ ) in terms of a Z p -valued measure on Z × p (for example, by setting d = n , let B n ( X ) denote the n -th Bernoulli polynomial, i.e.the polynomial defined by te Xt e t − = ∞ ∑ n = B n ( X ) t n n ! . So B n ( ) = B n B n ( ) = ⎧⎪⎪⎨⎪⎪⎩ B n if n ≠ B + n = ISENSTEIN MEASURES 9 If χ is a Dirichlet character, f is the conductor of χ , and F is a positive integer divisibleby f , then B n,χ = F n − F ∑ a = χ ( a ) B n ( aF ) . We also have B k ( X ) = ∑ ki = ( ki ) B i X k − i and B k ( − X ) = ( − ) k B k ( X ) for all nonnegativeintegers k .For each positive integer i , we define Y i ∶= i Z / Z , and for all positive integers j ∣ i , we define π ij ∶ Y i → Y j y ↦ ij × y. Definition 2.9.
The k -th Bernoulli distribution is the distribution φ = ( φ i ) i ≥ on Y ∶= lim ←Ð i Y i defined for each positive integer i by φ i ( ai ) ∶= i k − B k ({ ai }) . While the Bernoulli distribution is not a measure, we modify it to obtain a measure on X ∶= lim ←Ð n X n , where X n ∶= ( Z / dp n + ) × and d is a fixed integer, as follows. Fix c ∈ Z suchthat gcd ( c, dp ) =
1. For x n ∈ X n , we define E c ( x n ) ∶= E c, ( x n ) = B ({ x n dp n + }) − B ({ c − x n dp n + }) + c − , where {} denotes the fractional part of a number. Then, as further discussed in theproof of [Was97, Theorem 12.2], E c is a Z p -valued measure, and furthermore, letting ⟨ ⋅ ⟩ denote the projection onto 1 + p Z p , we have ∫ ( Z / dp Z ) × ×( + p Z p ) χω − ⟨⟩ s dE c = − ( − χ ( c )⟨ c ⟩ s + ) L p ( s, χ ) , (2.4)for all Dirichlet characters χ of conductor dp m with m a nonnegative integer and s ∈ Z p .2.5. Some convenient spaces for defining measures.
When constructing more gen-eral measures, in particular Eisenstein measures, it will be convenient to establish someparticular subsets of characters on which it is sufficient to define a measure in order forthat measure to be uniquely determined. More precisely, we have the following.For each of the following two lemmas, let O be a p -adic ring. Lemma 2.10. An O -valued measure on Γ = + p Z p is completely determined by itsvalues on characters of finite order. Lemma 2.11. An O -valued measure on Γ = + p Z p is completely determined by itsvalues on ⟨⟩ k ∶ Γ → Γ ⊂ Z p for any infinite set of k ∈ Z p . Proof of Lemmas 2.10 and 2.11.
Note that
O[[ Γ ]] is isomorphic to the power series ring O[[ T ]] (which, as explained in [Was97, Section 7.1], follows from the isomorphisms O[ Γ n ] ≅ O[ T ]/ (( + T ) p n − ) , γ ↔ + T where Γ n = Γ / Γ p n ). The proofs of bothlemmas then follow from the Weierstrass Preparation Theorem, which tells us that if0 ≠ f ( T ) ∈ O[[ T ]] , then f ( T ) = π r P ( T ) U ( T ) , with π a non-unit in O , r a nonnegativeinteger, P ( T ) a monic polynomial whose non-leading coefficients are all divisible by π ,and U ( T ) ∈ O[[ T ]] × [Bou98, Chapter VII Section 4]. Consequently, each nonzero ele-ment of O[[ T ]] has only finitely many zeroes β ∈ C p with ∣ β ∣ p < O[[ T ]] × cannot have any zeroes withabsolute value < µ be an O -valued measure on Γ, and let f µ ∈ O[[ T ]] be the power seriescorresponding to µ as in Section 2.3.1. Then µ ( χ ) = f µ ( χ ( γ ) − ) . So if µ vanishes atinfinitely many finite order characters or infinitely many ⟨⟩ k , then µ is identically 0. (cid:3) More generally, we have the abstract
Kummer congruences , a generalization of the styleof congruences established by Kummer.
Theorem 2.12 (Abstract Kummer congruences) . Let Y be a compact, totally discon-nected space, let O be a p -adic ring that is flat over Z p , and let I be some indexing set.Let { f i } i ∈ I ⊆ C( Y, O) be such that the O [ p ] -span of the functions f i is uniformly densein C ( Y, O [ p ]) . Let { a i } i ∈ I ⊆ O . Then there exists an O -valued p -adic measure µ on Y such that ∫ Y f i = a i for all i ∈ I if and only if the elements a i satisfy the abstract Kummer congruences , i.e.: Given { b i } i ∈ I ⊂ O [ p ] such that b i = i ∈ I , together with anonnegative integer n such that ∑ i ∈ I b i f i ( y ) ∈ p n O for all y ∈ Y , we have ∑ i ∈ I b i a i ∈ p n O . Proof.
This is [Kat78, Proposition (4.0.6)], which is proved in loc. cit. (cid:3)
When working with a profinite abelian group, the following consequence is particularlyconvenient for constructing Eisenstein measures in general.
Proposition 2.13 (First half of Proposition (4.1.2) of [Kat78]) . Let G be a profiniteabelian group. Let O be a p -adically complete ring that is flat over Z p , and supposethat R contains a primitive n -th root of unity for all n such that G contains a subgroupof index n . Let µ be an O -valued p -adic measure on G , and let χ be a continuoushomomorphism from G to O × . Then µ is completely determined by the values ∫ G χ χdµ as χ ranges over finite order characters of G . Dictionary between several approaches to defining p -adic measures. We concludethis section with Figure 1, which summarizes the connections between several formula-tions of the definition of an O -valued p -adic measure given above, each of which is usefulfor different aspects of constructing p -adic L -functions. In the figure, O is a p -adic ring,and G = lim ←Ð i G i is a profinite p -adic group, with transition maps π ij ∶ G i → G j whenever i ≥ j and π jk ○ π ij = π ik for all i ≥ j ≥ k . ISENSTEIN MEASURES 11 O -linear functionals on Step ( G ) µ Õ××××Ö bounded O -modulehomomorphisms C ( G, O ) → O lim i f i ↦ lim i µ ( f i ) for all f i ∈ Step ( G )Õ××××Ö finitely additive O -valuedfunctions on compact opensubsets of G ⊔ i U i ↦ ∑ i µ ( χ U i ) , where χ U i isthe characteristic function of theopen set U i Õ××××Ö collections of maps µ j ∶ G j → O such that µ j ( g ) = ∑ { y ∣ π ij ( h )= g } µ i ( h ) for all i ≥ j and all g ∈ G j ( µ j ) j , where µ j ( g ) = µ ( χ U g ) , with U g = {( h i ) i ∣ π ij ( h i ) = g }Õ××××Ö elements ofΛ = O [[ G ]] = lim ←Ð j O [ G j ] ( ∑ g ∈ G j µ j ( g ) g ) j Figure 1.
Dictionary between several formulations of p -adic measures3. A first look at p -adic Eisenstein measures We are primarily interested in measures as a vehicle for obtaining p -adic L -functions in-side an Iwasawa algebra. While Bernoulli numbers were useful constructing the measurein Equation (2.4), they do not necessarily generalize to many other L -functions of inter-est. It turns out that p -adic modular forms provide a convenient tool for constructing p -adic L -functions in much more generality, while also producing p -adic Dedekind zetafunctions associated to totally real fields. Remark . Because of their links with L -functions, we will be particularly interestedin Eisenstein measures , measures whose values on certain sets of characters (like thosein Section 2.5) are Eisenstein series.In this section, we briefly introduce p -adic modular forms, following Serre’s approach.For more details, see [Ser73]. We denote by v p the valuation on Q p such that v p ( p ) = f ( q ) = ∑ ∞ n = a n q n ∈ Q p [[ q ]] , we define v p ( f ) ∶ = inf n v p ( a n ) . So v p ( f ) ≥ m if and only if f ≡ p m and v p ( f ) ≥ f ∈ Z p [[ q ]] . Let { f i } ⊆ Q p [[ q ]] . We write f i → f and say “The sequence f , f , . . . converges to f ” if v p ( f i − f ) → ∞ , i.e. the coefficients of f i converge uniformly to those of f as i → ∞ . Wealso write f ≡ g mod p m if v p ( f − g ) ≥ m. Example 3.2.
Let k ≥ k Eisensteinseries G k whose Fourier expansion is given by G k ( z ) = ζ ( − k ) + ∑ n ≥ σ k − ( n ) q n where q = e πiz and σ k − ( n ) = ∑ d ∣ n d k − . In the 1800s, Kummer proved that if p − ∤ k ,then ζ ( − k ) is p -integral, as well as that if k ≡ k ′ mod p −
1, then ζ ( − k ) ≡ ζ ( − k ′ ) mod p [Kum51]. So if we also apply Fermat’s little theorem to the non-constant coefficients, wesee that G k ≡ G k ′ mod p whenever k ≡ k ′ / ≡ p − Congruences mod p m . Recall that, for convenience, we assume p is odd. Thereader who is curious about p = Theorem 3.3 (TH´EOR`EME 1 of [Ser73]) . Let m ∈ Z ≥ , and let f, g ∈ Q [[ q ]] be modularforms of weights k, k ′ , respectively, with v p ( f − g ) ≥ v p ( f ) + m . If f ≠ , then k ≡ k ′ mod ( p − ) p m − . Ultimately, we want not just congruences but p -adic measures, which leads us to Section3.2.3.2. p -adic modular forms. Let X m = Z /( p − ) p m − Z , and let X = lim ←Ð m X m = Z p × Z /( p − ) Z . We identify X with the space of Z × p -valued characters of Z × p = ( Z / p Z ) × × ( + p Z p ) , i.e. X = Z p × Z /( p − ) Z k ↔ ( s, u ) corresponds to the Z × p -valued character of Z × p defined by a ↦ a k ∶ = ⟨ a ⟩ s ω u ( a ) . ISENSTEIN MEASURES 13
Definition 3.4 (Serre) . A p -adic modular form is a power series f = ∑ n ≥ a n q n ∈ Q p [[ q ]] such that there exists a sequence of modular forms f , f , . . . such that f i → f .As a consequence of Theorem 3.3, we see that a nonzero p -adic modular form f = lim i f i has weight k = lim i k i ∈ X , where k i denotes the weight of f i . A p -adic limit of p -adicmodular forms is again a p -adic modular form f , and if f ≠
0, the weights again convergeas in Theorem 3.3.
Corollary 3.5 (COROLLAIRE 1 of [Ser73]) . Let f = ∑ n ≥ a n q n be a p -adic modularform of weight k ∈ X . Suppose the image of k in X m + is nonzero. Then v p ( a ) + m ≥ inf n ≥ v p ( a n ) .Proof. We briefly recall Serre’s proof. If a =
0, then the corollary is immediate. Supposenow that a ≠
0. Let g = a , so g is a modular form of weight k ′ =
0, and v p ( f − g ) = inf n ≥ v p ( a n ) . Also, since the image of k in X m + is nonzero, k / ≡ k ′ in X m + . So by Theorem 3.3, v p ( f − g ) < v p ( g ) + m + . Consequently, v p ( a ) + m + > inf n ≥ v p ( a n ) , so v p ( a ) + m ≥ inf n ≥ v p ( a n ) . (cid:3) Corollary 3.6 (COROLLAIRE 2 of [Ser73]) . Consider p -adic modular forms f ( i ) =∑ ∞ n = a ( i ) n q m of weights k ( i ) , for i = , , . . . , respectively. Suppose that both of the followinghold: lim Ð→ i a ( i ) n = a n ∈ Q p for all n ≥ Ð→ k ( i ) = k ∈ X, with k ≠ . Then a ( i ) converges p -adically to an element a ∈ Q p , and f = ∑ ∞ n = a n q n is a p -adicmodular form of weight k . Example 3.7 (Application to G k ) . Applying Corollary 3.6 to a sequence of Eisensteinseries G k i , i = , , . . . , with k i ≥ i and such that k i → ∞ in thearchimedean metric and also k i → k ∈ X , we obtain a p -adic modular form (in fact, a p -adic Eisenstein series , i.e. a p -adic limit of Eisenstein series) G ∗ k ∶ = G ( p ) k ∶ = lim Ð→ i G k i = ζ ∗ ( − k ) + ∑ n ≥ σ ∗ k − ( n ) q n with σ ∗ k − ( n ) ∶ = σ ( p ) k − ( n ) ∶ = ∑ d ∣ np ∤ d d k − and ζ ∗ ( − k ) ∶ = lim i → ∞ ζ ( − k i ) . Since the p -adic number ζ ∗ ( − k ) is a p -adic limit of values of the Riemann zeta function,it is natural to ask about its relationship to values of the Kubota–Leopoldt p -adic zetafunction. This is given in Theorem 3.8 below. More generally, a consequence of Theorem3.10 is the construction of p -adic Dedekind zeta functions as elements of Λ (i.e. as p -adicmeasures).We say that an element of k = ( s, u ) ∈ X = Z p × Z /( p − ) Z is even if k ∈ X (equivalently,since we are assuming p is odd, u ∈ Z /( p − ) Z ). Otherwise, we say ( s, u ) is odd . Theorem 3.8 (TH´EOR`EME 3 of [Ser73]) . If ( s, u ) ≠ is odd, then ζ ∗ ( s, u ) = L p ( s, ω − u ) ,where L p denotes the Kubota–Leopoldt p -adic zeta function from Theorem 2.7.Proof. We recall Serre’s proof. If ζ ′ denotes the function ( s, u ) ↦ L p ( s, ω − u ) , then ζ ′ is the Kubota–Leopoldt p -adic zeta function, and ζ ′ ( − k ) = ( − p k − ) ζ ( − k ) for each positive even integer k .If k ∈ X (so 1 − k is odd), k i → k in X , k i → ∞ in the archimedean topology, then ζ ′ ( − k ) = lim i → ∞ ζ ′ ( − k i ) = lim i → ∞ ( − p k i − ) ζ ( − k i ) = lim i → ∞ ζ ( − k i ) = ζ ∗ ( − k ) . (cid:3) For any fixed even u ≠ Z /( p − ) Z , we can also prove that the function s ↦ ζ ∗ ( − s, − u ) arises as an element of Λ ∶ = Λ Γ , without reference to the work of Kubota–Leopoldt(and also without reference to the work of Iwasawa, who proved this as well). First, wenote that for each positive integer m ≢ p −
1, the rational numbers ζ ( − m ) = ( − ) m + B m m were already known in the mid-1800s to be p -integral, thanks to the vonStaudt–Clausen theorem [vS40, Cla40] and a result of von Staudt on numerators ofBernoulli numbers ([vS45], which was later rediscovered and misattributed, as discussedon [Gir90, p. 136]). So the p -adic limits ζ ∗ ( − s, − u ) are elements of Z p whenever u ≠ Z /( p − ) Z . (Alternatively, Corollary 3.5 shows that because all the higher orderFourier coefficients of the Eisenstein series G ∗( s,u ) are p -integral, so is the constant termof G ∗( s,u ) .) Applying Corollary 3.6 to the p -adic Eisenstein series G ∗( s,u ) from Example3.7, we obtain congruences for the constant terms ζ ∗ ( − s, − u ) . So applying Lemma2.11 and Theorem 2.12, we see that for fixed even u ≠ Z /( p − ) Z , ζ ∗ ( − s, − u ) canbe obtained as an element of Γ. Via different methods, Iwasawa’s work also addressedthe case where u = Theorem 3.9.
Let Λ ∶ = Λ Γ , and fix an even u ∈ Z /( p − ) Z . Then:(1) If u ≠ , then the function ⟨⟩ s ↦ ζ ∗ ( − s, − u ) is an element of Λ ∶ = Λ Γ .(2) The function s ↦ ζ ∗ ( − s, ) − is an element Λ ≅ Z p [[ T ]] , and moreover, is ofthe form T g ( T ) with g ( T ) invertible in Z p [[ T ]] . For n ≥ n th Fourier coefficient of the p -adic Eisenstein series G ∗ k , with k = ( s, u ) , isof the form σ ∗ k − ( n ) = ∑ d ∣ np ∤ d d − ω ( d ) k ⟨ d ⟩ k = ∑ d ∣ np ∤ d d − ω ( d ) u ⟨ d ⟩ s , which gives an element of ISENSTEIN MEASURES 15
Λ, when we fix u . Consequently, for fixed u ≠
0, the coefficients of G ∗( s,u ) can be viewed aselements of Λ (by Theorem 3.9, Part (1)), and furthermore, for u =
0, the coefficients ofthe normalized Eisenstein series E ∗ s ∶ = ( ζ ∗ ( − s, )/ ) − G ∗( s, ) can be viewed as elementsof Λ (by Theorem 3.9, Part (2)).More generally, we have the following result. Theorem 3.10 (TH´EOR`EME 17 and TH´EOR`EME 18 of [Ser73]) . Let f s be a p -adicmodular form of weight k ( s ) = ( sr, u ) ≠ for some fixed r and u . Suppose the function ⟨⟩ s ↦ a n ( f s ) is in Λ ∶ = Λ Γ for all n ≥ .(1) If u ≠ in Z /( p − ) Z , then the same is true for n = .(2) If u = in Z /( p − ) Z , then ⟨⟩ s ↦ ζ ∗ ( − rs, ) − a ( f s ) is in Λ .Proof. Serre’s proof of each part of this theorem involves a careful analysis of the element f ′ s ∶ = f s E ∗− rs ∈ Λ, which is of weight ( , u ) and has the same constant term as f s . Sincewe will not need the details in this paper, we do not elaborate here and instead refer thereader to [Ser73, proofs of Theorems 17 and 18].We note, however, that we can also give an alternate proof for Part (1), i.e. when u ≠
0, using the results developed thus far in the present paper: If a n ( f s ) is in Λ forall n ≥
1, then Corollaries 3.5 (with m =
0) and 3.6 guarantee the constant terms meetthe conditions necessary to apply Lemma 2.11 and Theorem 2.12, so we can realize theconstant term a ( f s ) as an element of Λ. (cid:3) Serre uses Theorem 3.10 to obtain a p -adic Dedekind zeta function ζ ∗ K , for K a totallyreal number field, as an element of Λ (where ζ ∗ K is defined analogously to ζ ∗ and occursas the constant term of an Eisenstein series).4. Hilbert modular forms and L -functions attached to Hecke characters Serre’s use of p -adic families of Eisenstein series to construct p -adic zeta functions inspiredconstructions in other contexts. We now summarize a generalization to the space of p -adic Hilbert modular forms, where realizations of Eisenstein measures enabled theconstruction of p -adic L -functions attached to Hecke characters of totally real or CMfields.4.1. The strategy of Deligne–Ribet.
Our goal now is to introduce the strategy ofDeligne–Ribet from [DR80] to p -adically interpolate values of L ( s, ρ ) for ρ a finite orderHecke character of a totally real field K unramified away from p . Note that for negativeintegers s , L ( s, ρ ) lies in the field extension Q ( ρ ) obtained by adjoining all values of ρ to Q . Following the conventions established in Section 2.1, for any number field F , wedenote by F ( p ∞ ) the maximal abelian extension of F that is unramified away from p . Theorem 4.1 (Main Theorem (8.2) of [DR80]) . Fix a totally real field K and a prime-to- p ideal A of K . Then there exists a Z p -valued p -adic measure µ A on G ∶ = Gal ( K ( p ∞ ) / K ) such that for all positive integers k and finite order characters ρ on G , ∫ G ρ ⋅ N k dµ α = ( − ρ ( A ) N A k + ) L ( p ) ( − k, ρ ) , where N denotes the norm and L ( p ) ( − k, ρ ) = ∏ p ∣ p ( − ρ ( p ) N ( p ) k ) L ( − k, ρ ) . To prove Theorem 4.1, Deligne and Ribet work in the space of Hilbert modular forms.As one might expect from Serre’s approach to constructing the p -adic zeta function, onestep toward proving Theorem 4.1 is the construction of Eisenstein series (this time, inthe space of Hilbert modular forms) of weight k with L ( − k, ρ ) as the constant term,for each positive integer k . Similarly to the Eisenstein series G ∗ k , it is easy to see thenon-constant terms of the Eisenstein series in [DR80] satisfy congruences as the weight k varies. Now that we are in the setting of Hilbert modular forms, though, we need anew approach to proving that the constant terms satisfy congruences. This requires thetheory of p -adic Hilbert modular forms and q -expansion principles, which require moregeometry than the discussion thus far.4.1.1. Ingredients from the theory of p -adic Hilbert modular forms. We briefly delve intothe setup of p -adic Hilbert modular forms, the space where the families of Eisensteinseries from [Kat78, DR80] live. For more details, see [Hid04, Chapter 4], [Kat78, ChapterI], or [Gor02].We can give a formulation of Hilbert modular forms as sections of line bundles overa moduli space M of Hilbert–Blumenthal abelian varieties (with additional structure).More precisely, fix a totally real number field K of degree g over Q , a fractional ideal c of K , and an integer N ≥ p . Let O K denote the ring of integers in K , and let d − denote its inverse different. There is a scheme M ∶ = M ( N, c ) over Spec ( O K ) classifyingtriples ( X, i, λ ) , consisting of an abelian scheme X of relative dimension g togetherwith an action of O K on it, a level structure i ∶ d − ⊗ Z µ N ↪ X , and a c -polarization λ ∶ X ∨ ∼ → X ⊗ O K c (where X ∨ denotes the dual abelian scheme to X ).We denote by π ∶ A univ → M the universal object, and we define ω ∶ = π ∗ Ω A univ / M . Thespace of Hilbert modular forms of weight ( k ( σ )) σ ∶ K ↪ R is identified with H ( M , ⊠ σ ω ( k σ )) .Note that ω ( k σ ) is a subsheaf of Sym k ( ω ) , where k = ∑ σ k ( σ ) . Note that there exists asmooth toroidal compactification M of M that includes the cusps of M , and the uni-versal abelian scheme A univ extends to the universal semi-abelian scheme over M . Alsonote that when [ K ∶ R ] > , K¨ocher’s principle guarantees that a Hilbert modular formover M extends holomorphically to the cusps. As explained in [DR80, Example 5.3],[Hid04, Section 4.1] (see also [Kat78, Section 1.1] on algebraic q -expansions), when wework over a Q -algebra R (for example, R = C ), the cusps are in bijection with fractionalideals A of K (which we will call the “cusp corresponding to A ”). Remark . While we shall not need this fact here (as we are working in settings specificto Katz and Deligne–Ribet), it is worth noting that as discussed in [AIP16, p. 2-3], thenotion of “Hilbert modular form” for F ≠ Q varies slightly depending on where in theliterature one looks. More precisely, the moduli problem represented by M = M ( N, c ) corresponds to the group G ∗ = G × Res K / Q G m G m , where G = Res K / Q GL , G → Res K / Q G m is the determinant morphism, and G m → Res K / Q G m is the diagonal embedding. Onthe other hand, there are also approaches to eigenforms on the group G , but the moduliproblem for G is not representable. For further discussion about the relationship betweenautomorphic forms on these two spaces, see [AIP16, p. 2-3].Let W denote the ring of Witt vectors associated to an algebraic closure of Z / p Z , andlet W m = W / p m W . We identify W with the ring of integers in the maximal unramified ISENSTEIN MEASURES 17 extension of Q p inside an algebraic closure of Q p . We fix an embedding K ↪ ¯ Q p . Theimage of O K under this embedding lies in W .The space of p -adic Hilbert modular forms is defined over the ordinary locus M ord (inside of M × Spec O K Spec W ), which can be described as the nonvanishing locus of a liftof the Hasse invariant, like in [Hid04, Section 4.1.7]. More precisely, the space of p -adicHilbert modular forms is realized as follows. We build an Igusa tower over M ord (as in,e.g, [Hid04, Section 8.1.1]). For each pair of positive integers n, m , Ig n,m is defined tobe a cover of M ord × W W m classifying ordinary Hilbert–Blumenthal abelian varieties A together with level p n -structure µ p n ↪ A [ p n ] . So we have canonical maps Ig n,m → Ig n ′ ,m for all n ′ ≥ n (and likewise for m ′ ≥ m ), giving us a tower of schemes. Following thenotation of [Hid04, Section 8.1.1], we set V n,m ∶ = H ( Ig n,m , O Ig n,m ) V ∞ ,m ∶ = lim Ð→ n V n,m . Following [Kat78, Section 1.9] (or the more general discussion from [Hid04, Section8.1.1]), the space of p -adic Hilbert modular forms is then V ∶ = V ∞ , ∞ ∶ = lim ←Ð m V ∞ ,m . We identify V ∞ , ∞ with the ring of global sections of the structure sheaf of a formalscheme parametrizing Hilbert–Blumental abelian varieties with p ∞ -level structure.An advantage of this construction is that is provides a canonical map from the spaceof Hilbert modular forms to the space V of p -adic Hilbert modular forms (as in, e.g.,[Kat78, Theorem (1.10.15)]). Remark . More generally, this construction can be modified to produce p -adic auto-morphic forms in other cases, such as in the setting of Shimura varieties of PEL type.For a detailed treatment, see [Hid04, CEF +
16, EM21].4.1.2. q -expansion principles. Like in Serre’s construction, Deligne and Ribet’s approachalso relies substantially on properties of q -expansions of Eisenstein series. So we nowwill need some q -expansion principles, i.e. theorems that explain to what degree Hilbertmodular forms are determined by their q -expansions. In Proposition 4.4 and Theorem4.6, we choose the level structure so that the reduction of M is connected. (Alternatively,we could modify the statements of Proposition 4.4 and Theorem 4.5 to take a q -expansionat a cusp on each connected component.) Proposition 4.4 (algebraic q -expansion principle for Hilbert modular forms) . Let f bea Hilbert modular form defined over a ring R .(1) If the algebraic q -expansion of f vanishes at some cusp, then f = .(2) Let R ⊆ R be a ring. If the q -expansion of f at some cusp has coefficients in R , then f is defined over R . The proof of Statement (1) relies on the irreducibility results in [Rib75, Rap78], andStatement (2) can be proved as a consequence of (1) (similarly to the proof of [Kat73,Corollary 1.6.2]).
The Fourier coefficients of the Eisenstein series needed for studying L -values of totallyreal Hecke characters have coefficients in the ring of integers O of a number field. Soas a consequence of Proposition 4.4(2) and the fact that the algebraic and analytic q -expansions of a Hilbert modular form agree (by [Kat78, Equation (1.7.6)]), we have thatour Eisenstein series are actually defined over O .In order to construct the p -adic L -functions, we will also need a p -adic q -expansionprinciple for Hilbert modular forms. Theorem 4.5 ( p -adic q -expansion principle for Hilbert modular forms, (5.13) of [DR80]) . If f ∈ V and the q -expansion of f vanishes at some cusp, then f = . Furthermore, if R is flat over Z p , then the R -submodule V R of p -adic modular forms defined over R consists of the elements f ∈ V R ⊗ Z p Q p whose q -expansion coefficients lie in R , and ifthe q -expansion of a p -adic modular form f at some cusp has coefficients in R , then thesame is true at all the cusps. As an important consequence of Theorem 4.5, we obtain Corollary 4.6.
Corollary 4.6 (Corollary (5.14) of [DR80]) . Let f ∈ V R ⊗ Q p , and suppose that at somecusp, all the q -expansion coefficients, aside possibly from the constant term, of f lie in R . Then the difference between the constant terms of the q -expansions of f at any twocusps also lies in R .Proof. Let v ∈ V R ⊗ Q p be such that at some cusp, all the q -expansion coefficients, asidepossibly from the constant term, of f lie in R . Let a be the constant term of f at thatcusp. Then a is a weight 0 modular form, and all the coefficients of f − a ∈ V R ⊗ Q p liein R . So by Theorem 4.5, f − a ∈ V R and all the q -expansion coefficients of f − a , inparticular its constant term, at any other cusp lie in R . So the difference between anytwo constant terms of f lies in R . (cid:3) As an immediate corollary of Corollary 4.6, we obtain:
Corollary 4.7.
If the abstract Kummer congruences hold for all the non-constant termsof the q -expansions of a family of p -adic modular forms f at some cusp, then they alsohold for the difference between the constant terms at two cusps. To prove Theorem 4.1, it then suffices to realize L ( p ) ( − k, ρ ) in the constant term of a q -expansion of an Eisenstein series E k,ρ and observe that the constant term of E k,ρ at acusp corresponding to a fractional ideal A is ρ ( A ) N ( A ) k L ( p ) ( − k, ρ ) , which is provedin [DR80, Theorem (6.1))].4.2. The case where χ is a Hecke character of a CM field. Given that we justconsidered the case of Hecke characters of totally real fields, it is natural now to moveto CM fields K . Fix a CM type Σ for K , i.e. a set of [ K ∶ Q ]/ K ↪ C such that exactly one representative from each pair of complex conjugate embeddings { σ, ¯ σ } lies in Σ. In [Kat78], Katz considered the case where χ ∶ K × / A × K → C is a Heckecharacter of type A , i.e. χ is of the form χ = χ fin ∏ σ ∈ Σ ( σ ) k ( ¯ σσ ) d ( σ ) , ISENSTEIN MEASURES 19 with k a positive integer, d ( σ ) a nonnegative integer for all σ ∈ Σ, and χ fin a finite ordercharacter. Building on ideas of Eisenstein, the study of the algebraicity properties of thevalues L ( , χ ) was initiated Damerell and later extended and completed by Goldstein–Schappacher [GS81, GS83], Shimura [Shi75], and Weil [Wei99]. (A summary of thehistorical development is in [HS85, § χ of type A as above, the values L ( , χ ) can be expressed (in what is known as Damerell’s formula ) as finite sums of values of Eisenstein series in the space of Hilbertmodular forms. Thus, it is natural to try to construct Eisenstein measures suited to thisapplication and adapt the techniques introduced thus far.Indeed, this is what Katz did in [Kat78], but there are several new challenges Katz hadto solve in this setting, which also helped uncover paths toward generalizations. Becausethese challenges also arise more broadly, we continue the discussion in Section 5.5.
Generalizations and challenges
Various constructions of automorphic L -functions are closely tied to Eisenstein series.This includes Damerell’s formula, the Rankin–Selberg method, and pullback methods likethe doubling method. Each of these methods was used to prove algebraicity of certainvalues of the corresponding automorphic L -functions. Given the developments discussedthus far, it is therefore natural to try to construct Eisenstein measures valued in appro-priate spaces of p -adic automorphic forms and use those to construct p -adic L -functions.Those familiar with any of these methods might recall, though, that the Eisenstein se-ries occurring in the constructions of the L -functions can be quite intricate (and likewisefor computations of the Fourier coefficients), and furthermore, the L -functions are notsimply realized as constant terms of these particular Eisenstein series.In addition, on the p -adic side, the slightest modification to input can have drasticgeometric consequences. For example, changing a prime from split to inert can lead tothe entire ordinary locus employed in the definition of p -adic modular forms describedabove to disappear in certain settings. In another direction, working with the full rangeof Hecke characters from Section 4.2 requires considering Eisenstein series that are notholomorphic.Extending the approach of constructing Eisenstein measures to produce p -adic L -functionsattached to Hecke characters of CM fields, as well as those considered in higher rankgeneralizations like [Eis15, Eis14, EHLS20], involves working in a setting where: ● The approach of using constant terms (from [Ser73, DR80]) no longer applies,due to the fact that for Eisenstein series occurring in particular formulas for L -functions, the Fourier expansions of those Eisenstein series at cusps where it isconvenient to work lack constant terms. For example, the Fourier expansions ofthe particular Eisenstein series employed in the formulas in [Kat78] turn out tolack constant terms at the cusps where they computed, as seen in, e.g., [Kat78,Theorem (3.2.3)]. (That said, if one has a convenient way to compute and studythe Fourier coefficients at a cusp where the constant term is nonzero, then thisissue disappears.) ● The Eisenstein series are substantially more complicated to construct. ● The constructions of the L -functions require considering values of C ∞ (not nec-essarily holomorphic) Eisenstein series. ● The points in the ordinary locus needed in the construction of the L -functionsmight be empty.Moving beyond Hecke characters to Rankin–Selberg L -functions and L -functions asso-ciated to automorphic forms (e.g. through the doubling method), we also must contentwith the following: ● L -functions might be represented not as finite sums of values of Eisenstein series,but instead as integrals of cusp form(s) against restrictions of Eisenstein seriesto certain spaces (e.g. as in the doubling method)5.1. Strategies of Katz, and beyond.
As noted in Section 4.2, Damerell’s formulaexpresses values of the L -functions associated to Hecke characters of CM fields in termsof finite sums of values of Eisenstein series from the space of Hilbert modular forms. Inhis construction of p -adic L -functions for CM fields [Kat78], Katz exploits the fact thatthe Eisenstein series get evaluated only at CM points, i.e. Hilbert–Blumenthal abelianvarieties with complex multiplication. He constructs an Eisenstein measure and thenconstructs a p -adic measure at each of these CM points A by evaluating the Eisensteinseries in the image of his Eisenstein measure at A .Constructing the Eisenstein series and measure is considerably more involved than inthe examples mentioned so far, though, and it is the subject of [Kat78, Chapter IIIand Section 4.2]. Part of Katz’s strategy is to introduce a partial Fourier transform ([Kat78, Section 3.1]), which allows him to construct an Eisenstein series amenable tocomputations for L -functions but which also has q -expansion coefficients that satisfycongruences (so that he can employ the q -expansion principles from above). The keypoint with the partial Fourier transform is to take the Fourier transform of appropriatedata that interpolates well to produce the Eisenstein series and then exploit the closerelationship between the Fourier transform and the Fourier transform of the Fouriertransform, namely that the Fourier transform of the Fourier transform of t ↦ f ( t ) is t ↦ f ( − t ) . Hence we get an Eisenstein measure whose coefficients interpolate well.To handle the C ∞ Eisenstein series that occur in the construction of L -functions for CMfields, Katz must consider certain differential operators. The C ∞ Eisenstein series in theconstruction can be obtained by applying the Maass–Shimura differential operators toholomorphic Eisenstein series. Katz exploits the Hodge theory of Hilbert–Blumenthalabelian varieties to construct p -adic analogues (built out of the Gauss–Manin connectionand Kodaira–Spencer morphism) of those differential operators [Kat78, Chapter II]. On q -expansions, these operators are a generalization of the operator q ddq , and they preserveinterpolation properties of the Hilbert modular forms to which they are applied.These techniques for constructing Eisenstein measures have since been extended to thePEL setting. For example, differential operators on p -adic automorphic forms on unitarygroups are the subject of [Eis12, EFMV18] (which also builds on [Har81]), and they wereused as a starting point in the construction of Eisenstein measures taking values in thespace of p -adic automorphic forms on unitary groups in [Eis15, Eis14], which were inturn employed in the constructions of p -adic L -functions in [EHLS20, EW16]. ISENSTEIN MEASURES 21
Like in Section 4.1.1, Katz’s construction is over the ordinary locus. This introduces aserious obstacle, namely that there are no ordinary CM points, if p is inert. Given thatDamerell’s formula is a sum over CM points, this means Katz’s approach did not address p inert.Over four decades passed before an approach to p inert was introduced. In [AI19],Andreatta and Iovita explain how to adapt Katz’s approach to the case of quadraticimaginary fields with p inert. In separate work [Kri18], Kriz also introduced an approachfor inert p . Parts of [AI19] are also being extended to the case of CM fields in [Ayc21,Gra20]. The idea of Andreatta and Iovita is to work instead with overconvergent p -adicmodular forms and modify the approach to handling the differential operators. WhereasKatz exploits Dwork’s unit root splitting that exists over the ordinary locus, Andreattaand Iovita build an operator from the Gauss–Manin connection and then take pairingsthat do not require projecting modulo a unit root splitting.5.2. Working with pairings and pullback methods.
Katz’s approach to construct-ing Eisenstein measures provides a starting point for other cases, in particular auto-morphic forms in the PEL setting. Since we are often faced with representations of L -functions not as a finite sum but rather as an integral of an Eisenstein series against cuspform(s), we now briefly explain the key ideas for adapting such a representation to the p -adic setting. We discuss this strategy in the context of the Rankin–Selberg zeta function,where it was first developed (by Hida in [Hid85]), but it has also since been extended tovarious settings, including, among others, in [Hid91, Pan03, Liu20, LR20, EHLS20].The Rankin–Selberg product of a weight k holomorphic cusp form f = ∑ n ≥ a n q n and aweight ℓ ≤ k holomorphic modular form g = ∑ n ≥ b n q n is a zeta series D ( s, f, g ) = ∞ ∑ n = a n b n n s . Shimura and Rankin proved in [Shi76, Ran52] that D ( k − − r, f, g ) = cπ l ⟨ ˜ f , gδ ( r ) λ E ⟩ , where E denotes a particular weight λ ∶ = k − ℓ − r Eisenstein series, ˜ f ( z ) ∶ = f ( − ¯ z ) , δ ( r ) λ isa Maass–Shimura operator that raises the weight of a modular form of weight λ by 2 r (so δ ( r ) λ ∶ = ∂ λ + r − ○ ∂ λ + ○ ∂ λ with δ λ ∶ = πi ( λ iy + ∂∂z ) ), c = Γ ( k − ℓ − r ) Γ ( k − − r ) Γ ( k − ℓ − r ) ( − ) r k − N ∏ p ∣ N ( + p − ) (with N the level of the modular forms), and ⟨ , ⟩ denotes the Petersson inner product.As a consequence, Shimura proved in [Shi76, Theorem 2] that π − k D ( m, f, g )⟨ f, f ⟩ is algebraic for all integers k, ℓ, m satisfying ℓ < k and k + ℓ − < m < k .In [Hid85], Hida constructed p -adic Rankin–Selberg zeta functions by building on Shimura’sapproach to studying algebraicity. In particular, the idea to interpret the Rankin–Selbergzeta function in terms of the Peterssen pairing plays a key role, and this remains true inextensions to higher rank groups (including in the discussions of algebraicity in [Har81]and in extensions to the p -adic case in PEL settings involving the doubling method in[Liu20, LR20, EHLS20]). The idea is to reinterpret the linear Petersson pairing ⟨ h , h ⟩ as a functional ℓ h ( h ) . This suggests identifying a space of modular forms with its dualspace, which in turn leads to use of the associated Hecke algebra. This is the point thatallows Hida to integrate Eisenstein measures (generally coming from familiar families ofEisenstein series, at least in the case of modular forms) into the construction of p -adic L -functions. While this approach makes sense in higher rank (e.g. in the context of thedoubling method), putting it into practice is nontrivial for various reasons, includinggeometric issues (like those mentioned above) and new properties of the Hecke algebrathat must be taken into account.5.3. Some remaining challenges and future directions.
Putting aside the biggergoal of proving the Greenberg–Iwasawa main conjectures, challenges still remain for pro-ducing p -adic L -functions. Even in settings where we have constructions of L -functionsclosely tied to the behavior of Eisenstein series and we anticipate the existence of Eisen-stein measures, actually carrying out the construction can be nontrivial. We concludeby highlighting three categories of challenges and suggest some future directions towardresolving them:(1) As noted in Section 5.2, the pairings that arise from integral representations of L -functions can be useful for p -adic interpolation, but one often has to deal withsignificant technical challenges. Properties of Hecke algebras (and the ordinary Hecke algbras where one often works in practice) can present obstacles. Forexample, a Gorenstein property is often useful, but not necessarily known, in thiscontext. Pilloni’s higher Hida theory seems to present a promising and naturalalternative framework for interpreting these pairings [Pil20, LPSZ19, BP20].(2) As noted in Section 5.1, cases where the prime p does not split can lead toconsiderable geometric challenges, which have been recently addressed in lowrank in [AI19]. For unitary groups, work on differential operators in [EM21,dSG16, dSG19] and Hecke operators in [BR19] addresses some challenges thatarise when the ordinary locus is empty, but work remains in the inert case (evenjust for constructing appropriate Eisenstein series for the Eisenstein measure) toconstruct the full p -adic L -functions. In one of the most promising directions,the work in [Ayc21, Gra20] suggests the possibility of extending the techniquesof [AI19] to the PEL setting, but again, details of the Eisenstein measures wouldstill need to be worked out by adjusting the choices of local data that feed intothe partial Fourier transforms in [Eis15].(3) At a more fundamental level, before one can construct p -adic L -functions viathe method of Eisenstein measures, one needs a representation of the L -functionin terms of Eisenstein series. Such a representation, though, is insufficient un-less we also can reinterpret it algebraically. For example, given the success inadapting the doubling method to the p -adic setting in [Liu20, EW16, EHLS20],it is natural to try to adapt the twisted doubling representation of L -functions(i.e. for producing L -functions associated to a twist of a cuspidal automorphicrepresentation by a representation of GL n for some n ) in [CFGK19] to the p -adicsetting. As of yet, though, we do not have an appropriate interpretation in termsof algebraic geometry or another familiar algebraic tool, and without an algebraicinterpretation, are unlikely to see a path toward a p -adic realization. There iscurrently active work to produce integral representations of various L -functions. ISENSTEIN MEASURES 23
It will be interesting to see which ones become suitable for proving algebraicityresults, either in terms of the techniques described above or in terms of those yetto be discovered.
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E. Eischen, Department of Mathematics, University of Oregon, Fenton Hall, Eugene, OR97403-1222, USA
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