Algebra and Number Theory Seminar

In this talk, we will discuss various notions of multiplicity that are objects of study within commutative algebra and its interaction with algebraic geometry. These include the usual Hilbert-Samual multiplicity, intersection multiplicities from the work of Serre, Hilbert-Kunz multiplicity, j-multiplicity, and mixed multiplicities. I aim to provide several examples, classical properties, and important applications of these multiplicities. If time permits, I will mention some of my research work in these areas.

For an imaginary quadratic field, there are two natural Zp\mathbb{Z}_p-extensions, the cyclotomic and the anticyclotomic. We'll start with a brief description of Iwasawa theory for the cyclotomic extensions, and then describe some computations for anticyclotomic Zp\mathbb{Z}_p extensions, especially the fields and their class numbers. This is joint work with LC Washington.

In this talk I will give an overview of Robert Gold's paper "The Nontriviality of Certain Z_l-extensions", and suggest a way we might generalize his results.

In this talk I will give an overview of Robert Gold's paper "The Nontriviality of Certain Z_l-extensions", and suggest a way we might generalize his results.

Hilbert’s Tenth Problem asks whether there is an algorithm that can decide whether a Diophantine (integral coefficients) equation has integral solutions. Matiyasevich in 1970 showed the answer in general was no. Denef and Lipshitz generalized Matiyasevich’s theorem to ’Diophantine’ equations with coefficients in rings of integers of some number fields, and conjectured that their result should hold for any number field. We present some new cases of their conjecture, building crucially on the connection with elliptic curves and concomitant work by Poonen, Shlapentokh, and Garcia-Fritz–Pasten. This is joint work with D. Kundu and A. Lei.

Building on a result of Swanson, Cutkosky--Herzog--Trung and Kodiyalam described the surprisingly predictable asymptotic behavior of Castelnuovo--Mumford regularity for powers of ideals on a projective space: given an ideal I in a polynomial ring over a field, there exist integers d and e such that for n sufficiently large the regularity of I^n satisfies reg(I^n) = dn + e.

Through a medley of examples we will see why asking the same question about I, an ideal in the total coordinate ring S of a smooth projective toric variety X is interesting. After that I will summarize the ideas and methods we used to bound the region reg(I^n) inside Pic(X) by proving that it contains a translate of reg(S) and is contained in a translate of Nef(X), with each bound translating by a fixed vector as n increases. Along the way we will also see some surprising behavior for multigraded regularity of modules.

The symbolic powers of an ideal is an algebraic construction that encodes important information about its underlying algebraic variety. Symbolic powers are related to several important theorems and theories from different areas of mathematics, ranging from Krull’s principal ideal theorem in algebraic geometry, to the theory of evolutions in relation with Fermat’s last theorem.

In the first talk, I will introduce the notion of symbolic powers and discuss some of its basic properties. In the second talk I will present some results originating from my research. In particular, I will show connections with algebraic geometry, combinatorics, and linear optimization.

The symbolic powers of an ideal is an algebraic construction that encodes important information about its underlying algebraic variety. Symbolic powers are related to several important theorems and theories from different areas of mathematics, ranging from Krull’s principal ideal theorem in algebraic geometry, to the theory of evolutions in relation with Fermat’s last theorem.

In the first talk, I will introduce the notion of symbolic powers and discuss some of its basic properties. In the second talk I will present some results originating from my research. In particular, I will show connections with algebraic geometry, combinatorics, and linear optimization.

In this talk I'll introduce exceptional primes for a

natural number m, and talk about some of their properties with a focus

on m = 3 and 4. Then I'll talk about how these primes are connected

to cyclotomic lambda invariants for imaginary quadratic fields.

I will report on joint work with R. Sujatha and V. Vatsal. Two

$p$-ordinary Hecke-eigenforms are are congruent at a prime $\varpi|p$ if

all but finitely many of their Fourier coefficients are congruent modulo

$\varpi$. R. Greenberg and V. Vatsal showed in 2000 that the

Iwasawa-invariants of congruent modular forms are related. As a result, if

$\mu$-invariant vanishes and the main conjecture holds for a given

Hecke-eigenform, then the same is true for a congruent Hecke-eigenform.

This involves studying the behavior of Selmer groups and p-adic L-functions

with respect to congruences. We generalize these results to symmetric

square representations.

The main task at hand is that the p-adic L-functions for the symmetric

square exhibit congruences. In this setting, the normalized L-values for

$sym^2(f)$ can be expressed in terms of the Petersson inner product of $f$

with a nearly holomorphic function. This function is expressed as the

product of a theta function and an Eisentein series. The ordinary

holomorphic projection of this function is shown to have nice properties.

The Petersson inner product is modified and related to an abstractly

defined algebraic pairing due to Hida, and the two pairing are related up

to a "canonical period". Under further hypotheses, it is shown that this

canonical period is suitably well behaved. For this, we assume a certain

version of Ihara's lemma, which is known in certain cases.

With these preparations, we are able to show that normalized L-values for

the symmetric square behave well with respect to congruence, and hence, the

p-adic L-functions too. It follows that the analytic Iwasawa invariants for

congruent Hecke-eigencuspforms are related. Such results for the algebraic

Iwasawa invariants follow from work of R. Greenberg and V. Vatsal. Just as

in the classicial case, the results have implications to the main

conjecture. If time permits, we will introduce the role of the fine-Selmer

group and discuss a condition for the vanishing on the $\mu$-invariant that

can be stated purely in terms of the residual representation.

In 2005, the speaker and John Jones of Arizona State

University constructed an online database of low degree p-adic fields.

A newer version is integrated into the LMFDB at https://urldefense.com/v3/__http://www.lmfdb.org/padicField/__;!!IKRxdwAv5BmarQ!NM-i0RLwDPqjIaB6NNp7H-BAmVZeFeJPMeTxrqO1Ogpw7eSquETloANyTxhIeFXxIw$ . The talk will discuss possible improvements based on the fact that totally ramified finite degree

extensions of p-adic fields have near-canonical defining polynomials.

the classification of Λ modules is a well known topic and its applications are apparent in Iwasawa Theory. Related to the classification idea is the adjoint problem, introduced by Koike 1999 [4]. The adjoint of a module is a mysterious object in Iwasawa Theory, hence the application of knowing their classification is not fully annotated in contemporary research. There is however, material from Iwasawa’s 1973 letter to Andre Weil that can reveal situations where understanding the adjoint of a module in the set Mf(T) can be advantageous [2]. We explore this material and show the results obtained for lamba =3 and 4

The p-adic variation is an important tool in arithmetic problems. I will explain one approach to p-variation based on the study of the cohomology of arithmetic manifolds, giving some ideas of recents works about the construction of p-adic L-functions and the study of the geometry of eigenvarieties for GL(2n).

The Iwasawa Main Conjecture for modular motives is a conjecture of Barry Mazur, Ralph Greenberg and Kazuya Kato describing the variation of special values of L-functions of eigencuspforms under twists by cyclotomic characters. In this talk, I will explain its statement and meaning as well as outline its proof (under mild hypothesis on the residual Galois representation), and especially how to deduce the conjecture in general from the case of good reduction. This is joint work with Xin Wan.

This is a survey talk on zeta functions built from the spectrum of Laplace-type operators on Riemannian manifolds. Because the spectrum is unknown in general, standard number theoretic techniques involving functional equations, product formulas, and locations of zeros can't be handled. However, in parallel to number theory, special values of zeta functions have geometric significance. Geometric zeta functions have the advantage of fitting into infinite dimensional families parametrized by Riemannian metrics, and the use of supersymmetric techniques leads to a zeta function approach to the Atiyah-Singer index theorem. In the case of arithmetic quotients of algebraic groups, there is a relation between a special value of an L-function and the special value of a geometric variant of a zeta function, but such direct connections seem rare.

In 1980's, Mazur and Tate studied ``Iwasawa theory for elliptic curves over finite abelian extensions" and formulated various related conjectures. One of their conjectures says that the analytically defined Mazur-Tate element lies in the Fitting ideal of the dual Selmer group of an elliptic curve. We discuss some cases of the conjecture for modular forms of higher weight.

In 1996, Coates and Greenberg computed explicitly the module of universal norms associated with an abelian variety in a perfectoid field extension. The computation of this module is essential to Iwasawa theory, notably to prove "control theorems" for Selmer groups generalising Mazur's foundational work on the Iwasawa theory of abelian varieties over $\mathbb{Z}_p$-extensions. Coates and Greenberg then raised the natural question on possible generalisations of their result to general motives. In this talk, I will present a new approach to this question relying on the classification of vector bundles over the Fargues-Fontaine curve, which enables to answer Coates and Greenberg's question affirmatively in new cases.

I will discuss a paper by Michael Chou, Harris Daniels, Ivan Krijan,

and Filip Najman which, for an elliptic curve E/Q, computes all possible torsion

groups E(Q∞,p)tors, with Q∞,p representing the Zp-extension of Q. The paper

demonstrates that E(Q∞,p)tors = E(Q)tors for all primes p ≥ 5; however, in

the cases p = 2 or 3, there are infinitely many elliptic curves (with distinct

j-invariants) where torsion grows from Q to Q∞,p.

Given two p-ordinary elliptic curves with the same mod p Galois representation, Greenberg and Vatsal showed that their Iwasawa mu- and lambda-invariants ( over the cyclotomic Z_p-extension of Q) were related by an explicit formula. As an immediate consequence, this made it possible to propogate information about Iwasawa Main Conjecture in p-adic families of elliptic curves, while other applications have been found in recent years. The methods of Greenberg-Vatsal were later generalized to more general modular forms in both the p-ordinary and p-nonordinary settings, but most of these results assumed that the relevant Selmer groups were cotorsion Iwasawa modules. This talk will discuss the history of this theory before highlighting some recent work, joint with Antonio Lei, that extends the Greenberg-Vatsal methods to Selmer groups of positive corank.

In 1973 Kenkichi Iwasawa sent a paper to Andre Weil. The paper details much of the algebraic properties associated to Zp extensions of number fields, including the structure of the class groups in relation to the unramified extension of our Zp extension. This talk we detail the topics covered in this paper in relation to self adjoint Lambda modules. The purpose is to explain a connection between the self adjoint modules and understanding the galois group of the maximal unramified extension of a Zp extension. This is a critical area in using the main conjecture of Iwasawa theory.

I'll briefly review p-adic measure theory and define the Iwasawa invariants for a basic Z_p-extension over an abelian number field. Then I'll present a paper that finds bounds for the lambda-invariant, as well as an alternative proof that the mu-invariant is zero.

I will discuss recent work calculating the top weight cohomology of the moduli space A_g of principally polarized abelian varieties of dimension g for small values of g. The key idea is that this piece of cohomology is encoded combinatorially via the relationship between the boundary complex of a compactification of A_g and the moduli space of tropical abelian varieties. This is joint work with Madeline Brandt, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.