MPS Conference on The Eisenstein Ideal and Galois Representations: Looking Forward after 50 Years

This conference marked the fifty-year anniversary of two of the most important and influential papers in modern number theory, Modular curves and the Eisenstein ideal by Barry Mazur and A modular construction of unramified 𝑝-extensions of Q(𝜇_𝑝) by Ken Ribet. These papers have a shared theme of Eisenstein congruences, and the ideas initiated in them have led to major advances in number theory. The goal of the conference was to bring together experts in the vast of array topics that have been touched by these papers with the intention that, although these topics have spread out into different areas of expertise, their shared heritage suggests that ideas from one domain may spark new ideas in another.

Rational points on modular curves. One of the main results of Mazur’s paper is a classification of the possible rational torsion subgroups of elliptic curves over Q. This result can be stated moduli-theoretically as an answer to the question of for which integers 𝑁 the modular curve 𝑋₁(𝑁) has a non-cuspidal rational point: it is exactly those 𝑁 for which 𝑋₁(𝑁) has genus zero. There are analogous questions for modular curves with other level structures, such as 𝑋₀(𝑁), 𝑋_split(𝑁), and 𝑋_ns(𝑁); these curves can have non-cuspidal rational points, called isolated points even if they have positive genus, and the question becomes classifying isolated points. Some instances of isolated points have been studied by Mazur and Ogg, where they “explain” these point in terms of CM theory or using the geometry of the curves involved (like the existence of an involution). These questions about rational points on modular curves, in turn, relate back to elliptic curves and the problem of classifying the possible images of the adelic Galois representations afforded by their torsion points. These classification problems include Serre’s uniformity question and Mazur’s Program B, and are considered central open problems in Diophantine geometry. Likewise, speaker Sweeting presented results characterizing the torsion property of the class of the Ceresa cycle of an algebraic curve.

At the conference, speakers Hashimoto, Kim, Balakrishnan, and Bourdon reported on some of the cutting-edge results on these questions. A new program to give systematic geometric “explanations” of isolated rational points on modular curves, in the spirit of Mazur and Ogg, was discussed, as well as a conjectural classification of 𝑗-invariants of isolated points. For particular modular curves, such as 𝑋⁺ns(13), a conjectural answer for the rational points has long been known, but has resisted proof until recently. Kim’s non-abelian Chabauty method is a powerful tool for understanding rational points on curves. Both theoretical and computational refinements of this method were discussed, yielding a promising approach to computing rational points on a variety of modular curves.

Mazur’s method for analyzing rational points on modular curves used Eisenstein congruences. The idea is that the Jacobian of the modular curve is related to cusp forms whereas the cuspidal rational points on the modular curve are related to Eisenstein series. The interplay between these two is manifest in the cuspidal subgroup of the Jacobian, and understanding this subgroup—which controls the congruences between cusp forms and the Eisenstein series—is a key step in Mazur’s method of Eisenstein descent. For Mazur’s problem, it was sufficient to study modular curves of prime level, but it was so clear, even at the time, that the case of a general level 𝑁 would be of great arithmetic interest that Mazur made the unusual notational choice of calling his prime ‘𝑁’ (much levity was made at the conference about this choice). Speakers Yoo, Lang, Hsu, and Pozzi presented results extending aspects of Mazur’s results into further levels and other kinds of generalization. Novel techniques presented included methods of modular representation theory and Galois deformation theory.

Using Eisenstein congruences to study Galois cohomology. The main result of Ribet’s paper relates special values of the Riemann zeta function to Galois cohomology of cyclotomic fields. If a prime number 𝑝 divides the value 𝜁(1 − 𝑘), then the Eisenstein series of weight 𝑘 is congruent to a cusp form 𝑓 modulo 𝑝 and the mod-𝑝 Galois representation of 𝑓 realizes an extension class in Galois cohomology. This process for exhibiting classes in Galois cohomology is now known as Ribet’s method and has been used by Mazur–Wiles to prove the Iwasawa main conjecture and by Skinner–Urban to prove one divisibility in the main conjecture for modular forms. It remains one of the most important tools for studying conjectures, like the Birch–Swinnerton–Dyer conjecture, Bloch–Kato conjecture, and Iwasawa main conjectures, that relate special values of 𝐿-functions to Galois cohomology. Another tool in Iwasawa theory, complementary to the role of Ribet’s lemma, is the theory of Euler systems: instead of producing classes in Galois cohomology, the Euler-system method proves upper bounds on the size of Galois cohomology groups.

Ribet’s method, Bloch–Kato conjectures, and Iwasawa theory were well-represented at the conference, including in the talks by Khare, Deo, Hsieh, Grossi, Sharifi, Kakde, Dasgupta, Skinner, and Urban. There was an interpretation of the Tamagawa factors in the Bloch–Kato conjecture in terms of a newly-defined local congruence module. Applications of Ribet’s method in a variety of settings beyond Eisenstein congruences, including CM congruences, Saito–Kurokawa congruences, and Bianchi congruences, were discussed. A novel program for constructing new Euler systems using Eisenstein congruences was presented. Because Eisenstein congruences and Euler systems are typically seen as orthogonal tools, one for proving lower bounds and one for proving upper bounds, these results were particularly striking and surprising.

The conference days were organized into themes, which participants praised as an inspiring feature of the conference. Nonetheless, there were some talks that crossed boundaries and defied tight theming. For instance, Darmon and Vonk both spoke about Stark–Heegner points and rational points on non-split Cartan modular curves. Merel and Lecouturier presented results and conjectures regarding analogs of Mazur’s winding element associated to an Artin motive and a leading-term formula that resembles something like a Bloch–Kato conjecture. Emerton spoke about applications of the newly formed categorical Langlands program to Eisenstein cohomology. The influence of the two celebrated papers on these topics is vast and not confined to some simple set of themes, illustrating the way that the subject has expanded in the past fifty years.

Given the originally of the ideas and research presented in the lectures, and the vibrancy of the mathematical conversations had alongside them, it was evident that the subject is thriving fifty years on. The variety of exciting questions and conjectures presented promise that there is much to look forward to in the next fifty years!

Jennifer Balakrishnan
Boston University

Rational Points on Modular Curves via Quadratic Chabauty
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In this talk, Jennifer Balakrishnan discusses how to compute rational points on modular curves using the quadratic Chabauty method, illustrating techniques with various examples and commenting on the computational challenges. This is based on joint work with Alexander Betts, Netan Dogra, Daniel Hast, Aashraya Jha, Steffen Mueller, Jan Tuitman, and Jan Vonk.
 

Abbey Bourdon
Wake Forest University

Isolated Points on Modular Curves

Let \(C\) be a curve defined over a number field \(k\). We say a closed point on \(C\) is isolated if it is not part of an infinite family of points of the same degree which are parametrized by a geometric object — either the projective line or a positive rank abelian subvariety of the curve’s Jacobian. If \(C\) is the modular curve \(X_1(N)\) or \(X_0(N)\), then characterizing isolated points is a key obstruction to classifying all points of a fixed degree. In this talk, Abbey Bourdon focuses on the collection of “isolated \(j\)-invariants” associated with a family of modular curves, which are the values obtained by mapping isolated points to the \(j\)-line, and explores how recent finiteness questions concerning isolated \(j\)-invariants relate to other conjectures in the field.
 

Henri Darmon
McGill University and Hebrew University

The Winding Kernel and Richaud–Degert Orders of Class Number One
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The classification of rational points on modular curves undertaken in Mazur’s great paper on the Eisenstein Ideal resonates with the class number one problem for imaginary quadratic fields. This is because quadratic imaginary orders of class number one give rise to non-trivial rational points on modular curves (like \(X_0(163)\)), which any sufficiently precise and general classification must necessarily account for. Heegner resolved the long-standing Gauss class number one problem by determining the integer points on a specific affine modular curve, associated to the normaliser of a non-split Cartan subgroup of level \(6\). The family of non-split Cartan modular curves remains tantalizingly just beyond the reach of the methods pioneered in the Eisenstein ideal paper, and the large supply of rational CM points with which these curves are endowed hints at the source of the new difficulties that arise for this family. Mazur’s Eisenstein quotient of \(J_0(N)\) is a quotient of the so-called winding quotient \(J_w(N)\), which is itself a quotient of the minus part \(J_0(N)^-\) for the action of the Atkin-Lehner involution at \(N\). The winding kernel is the kernel of the natural map \(J_0(N)^- \rightarrow J_w(N)\). This talk will explain how the winding kernel, when it is non-trivial, can be combined with the conjectural theory of Stark-Heegner points to give a conditional solution of the class number one problem for real quadratic orders of Richaud-Degert type (with discriminants of the form \((nv)^2+4n\)), in the spirit of Heegner’s method for imaginary orders. This is joint work with Elias Caeiro.
 

Samit Dasgupta
Duke University

On the Factorization of Katz p-adic L-series
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In 1980, Gross proved that if \(\chi\) is a Hecke character of \(\mathbb{Q}\), and \(\chi_K\) is its restriction to a Hecke character of a quadratic imaginary field \(K\), then the \(p\)-adic \(L\)-series \(L_p(\chi_K, s)\) factors as the product of two Kubota–Leopoldt \(p\)-adic \(L\)-series. His proof relies on special value formulae for these \(L\)-series in terms of elliptic and circular units. A natural generalization of Gross’s formula to arbitrary CM fields \(K\) was formulated by Colmez. However, the corresponding formulae involving units remain conjectural, as instances of the \(p\)-adic Stark conjecture. In this talk, Samit Dasgupta describes work in progress, joint with Bergdall, Dimitrov, and Kakde, which provides a new “unit-free” approach to Gross’s theorem. This method realizes the factorization formula as arising from the intersection of two \(p\)-adic families of modular forms: an Eisenstein family and a theta family. The hope is to extend this strategy to arbitrary CM fields \(K\), at least under the assumption that there is a unique prime above \(p\) in the maximal totally real subfield of \(K\).
 

Shaunak Deo
Indian Institute of Science

Greenberg’s Question for Siegel Modular Forms

A famous question of Greenberg (which was also formulated independently by Coleman) asks the following: Suppose \(p\) is a prime and \(f\) is a \(p\)-ordinary modular eigenform such that the restriction of the \(p\)-adic Galois representation attached to \(f\) to the local Galois group at \(p\) splits into a direct sum of two characters. Then does f have complex multiplication? In this talk, Shaunak Deo explores an analogue of this question in the setting of Siegel modular forms of genus 2. This talk is based on a joint work with Bharathwaj Palvannan.
 

Matthew Emerton
University of Chicago

Describing Eisenstein Cohomology via Categorical Langlands

In this talk, Matthew Emerton discusses a conjecture that describes the cohomology of Shimura varieties (and more general congruence locally symmetric spaces) in terms of Galois-theoretic data, working in the context of the categorical Langlands program, with Emerton emphasizing the aspects of the conjecture which relate to Eisenstein eigenspaces in cohomology. This is part of joint work with Xinwen Zhu, and with Dougal Davis and Kari Vilonen.
 

Giada Grossi
Université Sorbonne Paris Nord and the Institute for Advanced Study

Iwasawa Theory in the Residually Eisenstein Case

In this talk, Giada Grossi overviews (not so recent) results about the cyclotomic and anticyclotomic Iwasawa main conjectures for elliptic curves (and more generally cusp forms) which are residually Eisenstein at p, together with their consequences for the p-part of the Birch and Swinnerton–Dyer conjecture. If time permits, some possible generalisations to the Rankin Selberg setting will be discussed.

Sachi Hashimoto
Brown University

Rational Points on Modular Curves
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An open problem is to classify all rational points on modular curves, in other words, classify all Galois images of elliptic curves over the rationals. This is part of “Mazurs Program B”. In this talk, Sachi Hashimoto will discuss questions related to and inspired by Mazur’s program B. In particular, Hashimoto will discuss how to parameterize the set of all rational points on modular curves, conditional on a conjecture of Zywina. This parameterization is then used to answer the following question: to what extent do the rational points on modular curves come from the intrinsic geometry of the curves? This is joint work with Maarten Derickx, Filip Najman, and Ari Shnidman.
 

Ming-Lun Hsieh
National Taiwan University

Yoshida Congruence and the Rankin-Selberg Convolution
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In this talk, Ming-Lun Hsieh reports on work in progress applying Yoshida congruences to obtain lower bounds for Selmer groups associated with Rankin–Selberg convolutions. After studying congruences between Yoshida lifts attached to two Hida families of elliptic modular forms and Hida families of Siegel modular forms on \(GSp(4)\), Hsieh uses an explicit pullback formula of Furusawa to construct a nontrivial Yoshida congruence modulo the \(p\)-adic Rankin–Selberg \(L\)-functions. Ribet’s lattice construction is then applied to produce nontrivial elements in the Selmer group associated with the Rankin–Selberg convolution. This is joint work with Zheng Liu.
 

Catherine Hsu
Swarthmore College

Eisenstein Congruences at Squarefree Level

The question of the existence of Eisenstein congruences has been approached with complementary methods: on the one hand, through the study of the geometry of modular curves, pioneered by Mazur and Ribet, and on the other hand, via modularity lifting methods. In this talk, Catherine Hsu will review known results for modular forms of weight 2, squarefree level N, and trivial nebentype and then present some generalized results for varying nebentype.
 

Mahesh Kakde
Indian Institute of Science

Ritter–Weiss Modules and the Brumer–Stark Conjecture

In this talk, Mahesh Kakde will talk about the Tate sequence and its refinement due to Ritter–Weiss. Kakde will then talk about its role in the proof of Brumer–Stark conjecture. This is part of a joint work with Samit Dasgupta and Jesse Silliman.
 

Chandrashekhar Khare
University of California Los Angeles

The Commutative Algebra of Congruence Ideals and Applications to Number Theory
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In his proof of Fermat’s Last Theorem, Wiles deployed a commutative algebra technique, namely a numerical criterion to show that a map between rings is an isomorphism of complete intersections. In recent work with Srikanth Iyengar and Jeffrey Manning we generalize the numerical criterion to “higher codimension”. A critical ingredient is a notion of congruence module in higher codimension: this has turned out to be a key definition whose utility extends beyond the role it plays in the numerical criterion. This leads to considering congruence ideals of local deformation rings: these are local counterparts of the much studied congruence ideals of Hecke algebras and global deformation rings associated to a newform f of weight k and level N. In ongoing work with Fred Diamond, we use our notion of local congruence ideals to prove cases of the Bloch-Kato conjecture for the p-part of the value at s=1 of the degree 3 L-function, L(s,Ad_f) for primes p that divide Nk!. These were primes that were excluded in the earlier work of Diamond, Flach and Guo in 2004. I will report on these developments.
 

Minhyong Kim
University of Edinburgh

Mixed Tate Motives and Integral Points on the Projective Line Minus Three Points Over Cyclotomic Fields
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The non-abelian method of Chabauty computes and bounds rational or integral points on curves over number fields via moduli spaces of torsors for unipotent fundamental groups, the so-called Selmer schemes. The original moduli spaces were p-adic spaces of sheaves of Q_p-schemes in the etale topology. Since then, rational versions of these moduli spaces via Tannakian fundamental groups for mixed Tate motives were constructed by M. Hadian, then studied and refined by many others. In this talk, Minhyong Kim will outline this construction and their application to the study of integral points on the projective line minus three points over cyclomotic fields. This is joint work with Xiang Li and Martin Luedtke.
 

Jaclyn Lang
Temple University

Eisenstein Congruences at Prime-Square Level

In Mazur’s celebrated Eisenstein ideal paper, he studies congruences between prime-level cusp forms and the unique weight-2 Eisenstein series of the same level. He shows that (if p is at least 5) such mod-p congruences exist if and only if the level N is congruent to 1 modulo p. In this talk, Jaclyn Lang considers Eisenstein–cuspidal congruences in weight 2 and level \(N^2\), still under the condition that N = 1 mod p. In this case, recent work with Pollack and Wake shows that the relevant level-\(N^2\) Hecke algebra is a free module over an appropriate inertia-at-N pseudodeformation ring. This structure turns out to be surprisingly powerful. One can recover Mazur’s existence theorem that there exists a mod-p Eisenstein–cuspidal congruence in weight 2 and prime level N when N = 1 mod p. It also allows one to deduce the relevant \(R=T\) theorem (for an appropriate pseudodeformation ring R) from the corresponding theorem in prime level, which is due to Wake and Wang–Erickson. Finally, one can recover the results of Merel and Lecouturier that characterize the rank of Mazur’s Hecke algebra in terms of the order of vanishing of a certain zeta element in cases when that rank is at most 3, and Lang explains some of the ideas that go into the aforementioned results. In addition to the joint work with Pollack and Wake, some of these results are joint with Palvannan and Müller.
 

Emmanuel Lecouturier
Westlake University

Artin Motives and the Eisenstein Ideal II: Specific Cases

In this talk, Emmanuel Lecouturier considers specific cases of Artin motives, some for which the winding element introduced is understood, and then examines general predictions in those cases. In particular, Lecouturier discusses an analogue of the Harris–Venkatesh conjecture where, instead of the adjoint of a weight one modular form, the adjoint of an algebraic Maass form is considered. Whereas in the former case, the winding element belongs to the first power of the Eisenstein ideal in the appropriate Hecke module, in the latter case one expects it to belong to the third power. This is joint work with L. Merel.
 

Loïc Merel
Université Paris Cité

Artin Motives and the Eisenstein Ideal I: The Winding Element

In 1978, with the purpose of partial results on the Birch and Swinnerton–Dyer conjecture, Mazur twisted his Eisenstein quotient by an odd Dirichlet character and studied the corresponding L-function. Consider a similar twist by any Artin motive ρ over Q. It leads to the introduction of what we call the winding element of ρ, which belongs to a certain Hecke module, and whose localization at the Eisenstein primes would carry information on ρ of global nature, similar to those provided by the Stark conjectures. In the special instance when ρ is attached to the adjoint of a weight one modular form, all this connects to the Harris–Venkatesh conjecture, whose understanding was our initial motivation. This is joint work with E. Lecouturier.
 

Alice Pozzi
University of Bristol

Non-Holomorphic Eisenstein Series and Obstructed Modularity Liftings

Since the celebrated proof of Fermat’s Last Theorem, the Taylor–Wiles method has seen many generalisations—yet many aspects fail to adapt at Eisenstein primes. In this talk, Alice Pozzi focuses on the setting of weight 2 modular forms of squarefree level. Pozzi explains how a missing “degree” Hecke eigensystem, related to the absence of a holomorphic Eisenstein series of level 1, creates an obstruction to a naive formulation of modularity lifting in this setting. Pozzi formulates an “obstructed” conjecture and proves it in some cases. This is joint work in preparation with Amie Bray, Cathy Hsu, Óscar Rivero, Nike Vatsal, and Carl Wang-Erickson.
 

Romyar Sharifi
University of California, Los Angeles

Eisenstein Cocycles for Imaginary Quadratic Fields
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In this talk, Romyar Sharifi discusses the construction of maps from the first homology groups of Bianchi spaces for an imaginary quadratic field F to second K-groups of ray class fields of F. These maps are “Eisenstein” in the sense that they factor through the quotient by the action of an Eisenstein ideal way from the level. They are direct analogues of known explicit maps in the setting of modular curves and cyclotomic fields. Sharifi intends to motivate this through the lens of current work on “artificial complexes” that yields explicit formulas in terms of Steinberg symbols of (artificial) elliptic units.
 

Christopher Skinner
Princeton University

Eisenstein series and Iwasawa theory

TBA
 

Naomi Sweeting
Princeton University

Nontrivial Ceresa and Modified Diagonal Classes via Semistable Reduction

The Ceresa class is a canonical, cohomologically trivial algebraic cycle on the Jacobian of a curve. Modified diagonal classes are closely related, and almost equivalent, algebraic cycles defined on the triple product of the curve with itself. These canonical cycles are known to be non-torsion in the Chow group for sufficiently general curves, but the proof is not explicit. It is therefore interesting to ask for which curves these cycles are non-torsion. In this talk, Naomi Sweeting reports on joint work in progress with Ari Shnidman in calculating the ramification of an l-adic Abel–Jacobi image of a modified diagonal class for any semistable curve over a finite extension of Z_p. The calculation is in terms of purely combinatorial data on the special fiber. As an application, Sweeting is able to give new and explicit conditions under which a curve over a number field has nonzero or nontorsion Ceresa class.
 

Eric Urban
Columbia University

Eisenstein Congruences and Euler Systems

TBA
 

Jan Vonk
Leiden University

Non-Split Cartan Modular Curves

In this talk, Jan Vonk discusses some recent results on the arithmetic geometry of non-split Cartan modular curves, related to their arithmetic geometry at primes of bad reduction, and ranks of their Jacobians. This is based on various joint works with Henri Darmon, Alice Pozzi, Jonathan Love, and Elie Studnia, as well as works of Alex Braat and Antigona Pajaziti.
 

Hwajong Yoo
Seoul National University

Kernels of Eisenstein Primes
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Let \(\ell\) be a prime at least 5 and \(N\) a squarefree integer not divisible by \(\ell\). A foundational result due to Mazur states that if \(N\) is a prime, the kernel of an Eisenstein prime containing \(\ell\) on modular Jacobian \(J_0(N)\) is always two-dimensional. In this talk, Hwajong Yoo explores the breakdown of this behavior for composite levels. Under a mild assumption, computing the dimension of this kernel explicitly determines its structure. This is joint work with Ken Ribet.

Watch a playlist of all presentations from this meeting here.