Number Theory Seminar

We plan to study Eisenstein series and related arithmetic topics during this semester. To begin with, we will go through basic definitions and properties of Eisenstein series and make explicit computations for the GL_2 case. Then we move on to some arithmetic results where Eisenstein series play a role. Hida theory in particular cases is also involved and we will see how Eisenstein congruence are utilized in arithmetic problems through most basic examples. At this point, we are looking forward to understand the followings: They are all applications for GL_1. We are also open to applications for other groups to discuss at the end of the semester.


Time: Monday 4:15 pm to 5:30 pm
Location: Math 520

Schedule (tentative)

Date Title Speaker Reference Notes
9/19 Basic definitions and properties for Eisenstein series (1) Vivian Yu [Sh] note
9/26 Basic definitions and properties for Eisenstein series (2) David Marcil [BL] note
10/3 Ribet's converse to Herbrand theorem Yu-Sheng Lee [Wa], [Ri] note
10/10 Eisenstein series on GL_2 Hung Chiang [Bu], [JL] note
10/17 Hida theory for GL_2 over totally real fields (1) Haodong Yao [Wi1]
10/24 Hida theory for GL_2 over totally real fields (2) Haodong Yao [Wi1]
10/31 Deligne-Ribet p-adic L-functions David Marcil [DR]
11/7 Holiday break - - -
11/14 Review of Iwasawa theory and main conjecture Hung Chiang [Wa]
11/21 Iwasawa main conjecture for Q (1) Baiqing Zhu [Wi2]
11/28 Iwasawa main conjecture for Q (2) Baiqing Zhu [Wi2]
12/05 Iwasawa main conjecture for Q (3) Yu-Sheng Lee [Wi2]
12/12 Alternative proof of the Iwasawa Main Conjecture over Q Vivian Yu

Abstract of each talk


We review general theory of Eisenstein series on the upper half plane as an introduction, where we will see L-values of Dirichlet characters occur in the constant term of q-expansions. Then we follow Shahidi's book to introduce reductive groups and necessary notions to define Eisenstein series in general. We will compute constant terms of Eisenstein series in some fundamental cases and discuss the functional equation among Eisenstein series, provided the meromorphic continuation.


For this talk, we will study the analytic continuation of Eisenstein series on an arbitrary reductive group using the modern approach of Bernstein and Lapid. They use a general principal that provides meromorphic continuation to solutions of "special" analytic family of systems of linear equations (ASLE) over a complex manifold. We will discuss the concept of cuspidal exponents of an automorphic form along standard parabolic subgroups and use them to show that Eisenstein series are unique solutions to such an ASLE. We will briefly mention why the latter is "special" but omit the proof. The talk will go over the general theory and discuss the special case of SL(2) to provide a concrete example.


We discuss the relation between Eisenstein congruences and Ribet's converse to the Herbrand-Ribet theorem. As a precursor to Wiles' proof of the Iwasawa main conjecture, we emphsize the role of congruence modules as the bridge between L-values and Selmer groups.


We define Eisenstein series on GL_2 in terms of Godement-Jaquet sections instead of flat sections. With these sections, we are able to obtain explicit formulas for all Fourier coefficients and establish meromorphic continuation and functional equations. We will briefly review the local new form theory and compute Fourier expansions of holomorphic Eisenstein series in the GL_2 case with new or ordinary section at each finite places. The computation will show the occurence of L-values at the constant term and will recover the well-known basis of classical Eisenstein series.


Hung will finish the computation of Fourier expansion of some Eisenstein series. Haodong will discuss Hida theory for GL_2 over a totally real field following the first section of [Wi1].


Haodong will continue the discussion on Hida theory for GL_2 over totally real fields, which are involved in the proof of Iwasawa main conjecture.


Let K be a totally real field. Given a Schwartz function \epsilon on the class group of K, one can construct its usual L-function L(s, \epsilon). Originally defined for Re (s) large enough, it can be continued meromorphically (with at most a simple pole at s = 1) and satisfies some functional equation. In our talk, we will focus instead on its p-adic interpolation. Following the construction of Deligne and Ribet, we will identify special L-values at negative integers as constant terms of q-expansions of some quite general Hilbert-Blumenthal Eisenstein series. Then, using a p-adic q-expansion principal, we will deduce Z_p-integral properties of the constant term (our L-values) out of Z_p-integral conditions on the non-constant terms (depending mostly on \epsilon). Then, we will constuct a p-adic measure using this integrality to finally obtain the so-called Deligne-Ribet p-adic L-function.


David will finish the construction of Deligne-Ribet p-adic L-function. Hung will define Iwasawa algebra for GL_1 over Q, explain the growth of p-part of ideal class group in a Z_p-extension, and formulate the main conjecture for GL_1 over Q.


Baiqing will formulate and prove the Iwasawa main conjecture of GL_1 over Q.


Baiqing will finish the proof of Iwasawa main conjecture of GL_1 over Q.


In the same vein of the previously discussed proof of the Herbrand-Ribet theorem, using the Eisenstein congruence from last week, we will prove a divisibility result, at height one primes not containing p, between the Fitting ideal of the Selmer groups and the p-adic L-functions. While the lattice construction technique that we will use doesn’t apply to those height one primes providing the trivial zeros directly, we will follow Wiles’ paper and sketch how we can deal with them.


Previously, we have discussed the proof of Iwasawa Main Conjecture following Mazur and Wiles using deep techniques from algebraic geometry. In this talk, we will discuss an alternative proof by Rubin using Kolyvagin’s Euler System.


[Bu] Bump, Daniel. Automorphic forms and representations. No. 55. Cambridge university press, 1998.
[BL] Bernstein, Joseph, and Erez Lapid. "On the meromorphic continuation of Eisenstein series." arXiv preprint arXiv:1911.02342 (2019).
[DR] Deligne, Pierre, and Kenneth A. Ribet. "Values of abelian L-functions at negative integers over totally real fields." Inventiones mathematicae 59.3 (1980): 227-286.
[JL] Jacquet, Hervé, and Robert P. Langlands. Automorphic Forms on GL (2): Part 1. Vol. 114. Springer, 2006.
[Ri] Ribet, Kenneth A. "A modular construction of unramified p-extensions of Q(μp)." Invent. math 34.3 (1976): 151-162.
[Sh] Shahidi, Freydoon. Eisenstein series and automorphic L-functions. Vol. 58. American Mathematical Soc., 2010.
[Wa] Washington, Lawrence C. Introduction to cyclotomic fields. Vol. 83. Springer Science & Business Media, 1997.
[Wi1] Wiles, Andrew. "On ordinary λ-adic representations associated to modular forms." Inventiones mathematicae 94.3 (1988): 529-573.
[Wi2] Wiles, Andrew. "The Iwasawa conjecture for totally real fields." Annals of mathematics 131.3 (1990): 493-540.