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Mahesh Kakde

Publications and source records attributed to Mahesh Kakde.

18 recordsLinked to original sources
arXiv2025

On a refinement of the Birch and Swinnerton-Dyer Conjecture in positive characteristic

We formulate a refined version of the Birch and Swinnerton-Dyer conjecture for abelian varieties over global function fields. This refinement incorporates both families of congruences between the leading terms of Artin-Hasse-Weil $L$-series and also strong restrictions on the Galois structure of natural Selmer complexes and constitutes a precise analogue for abelian varieties over function fields of the equivariant Tamagawa number conjecture for abelian varieties over number fields. We then provide strong supporting evidence for this conjecture including giving a full proof, modulo only the assumed finiteness of Tate-Shafarevich groups, in an important class of examples.

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arXiv2023

On The Equivariant Tamagawa Number Conjecture

Let $F$ be a totally real field and $K$ a finite abelian CM extension of $F$. Using class field theory, we show that our previous result giving a strong form of the Brumer-Stark conjecture implies the minus part of the equivariant Tamagawa number conjecture for the Tate motive associated to $K/F$. We work integrally over $\mathbf{Z}$, in particular the prime 2 is not inverted. This note can be viewed as our perspective on the recent work of Bullach, Burns, Daoud, and Seo. Following their philosophy, we show that the functorial properties (i.e. norm compatibilities) connecting the strong Brumer-Stark conjecture for varying number fields actually implies the minus part of the equivariant Tamagawa number conjecture.

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arXiv2023

The Residually Indistinguishable Case of Ribet's Method for GL2

Ribet's method provides a strategy for constructing a nontrivial extension of a $p$-adic Galois representation $ρ_1$ by another such representation $ρ_2$. Suppose we are working over a local ring. An important assumption that occurs throughout literature is that the representations $ρ_i$ are residually distinguishable i.e. are residually non-isomorphic. The main theorem of this paper is a general version of Ribet's Lemma for $\rm{GL}_2$ where we do not impose the assumption that the associated characters are residually distinguished.

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arXiv2023

Brumer-Stark Units and Explicit Class Field Theory

Let $F$ be a totally real field of degree $n$ and $p$ an odd prime. We prove the $p$-part of the integral Gross--Stark conjecture for the Brumer--Stark $p$-units living in CM abelian extensions of $F$. In previous work, the first author showed that such a result implies an exact $p$-adic analytic formula for these Brumer--Stark units up to a bounded root of unity error, including a ``real multiplication'' analogue of Shimura's celebrated reciprocity law from the theory of Complex Multiplication. In this paper we show that the Brumer--Stark units, along with $n-1$ other easily described elements (these are simply square roots of certain elements of $F$) generate the maximal abelian extension of $F$. We therefore obtain an unconditional construction of the maximal abelian extension of any totally real field, albeit one that involves $p$-adic integration for infinitely many primes $p$. Our method of proof of the integral Gross--Stark conjecture is a generalization of our previous work on the Brumer--Stark conjecture. We apply Ribet's method in the context of group ring valued Hilbert modular forms. A key new construction here is the definition of a Galois module $\nabla_{\!\sL}$ that incorporates an integral version of the Greenberg--Stevens $\sL$-invariant into the theory of Ritter--Weiss modules. This allows for the reinterpretation of Gross's conjecture as the vanishing of the Fitting ideal of $\nabla_{\!\sL}$. This vanishing is obtained by constructing a quotient of $\nabla_{\!\sL}$ whose Fitting ideal vanishes using the Galois representations associated to cuspidal Hilbert modular forms..

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arXiv2022

On the Brumer-Stark Conjecture

Let $H/F$ be a finite abelian extension of number fields with $F$ totally real and $H$ a CM field. Let $S$ and $T$ be disjoint finite sets of places of $F$ satisfying the standard conditions. The Brumer-Stark conjecture states that the Stickelberger element $Θ^{H/F}_{S, T}$ annihilates the $T$-smoothed class group $\text{Cl}^T(H)$. We prove this conjecture away from $p=2$, that is, after tensoring with $\mathbf{Z}[1/2]$. We prove a stronger version of this result conjectured by Kurihara that gives a formula for the 0th Fitting ideal of the minus part of the Pontryagin dual of $\text{Cl}^T(H) \otimes \mathbf{Z}[1/2]$ in terms of Stickelberger elements. We also show that this stronger result implies Rubin's higher rank version of the Brumer-Stark conjecture, again away from 2. Our technique is a generalization of Ribet's method, building upon on our earlier work on the Gross-Stark conjecture. Here we work with group ring valued Hilbert modular forms as introduced by Wiles. A key aspect of our approach is the construction of congruences between cusp forms and Eisenstein series that are stronger than usually expected, arising as shadows of the trivial zeroes of $p$-adic $L$-functions. These stronger congruences are essential to proving that the cohomology classes we construct are unramified at $p$.

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arXiv2022

On the Brumer-Stark Conjecture and Refinements

We state the Brumer-Stark conjecture and motivate it from two perspectives. Stark's perspective arose in his attempts to generalize the classical Dirichlet class number formula for the leading term of the Dedekind zeta function at $s=1$ (equivalently, $s=0$). Brumer's perspective arose by generalizing Stickelberger's work regarding the factorization of Gauss sums and the annihilation of class groups of cyclotomic fields. These viewpoints were synthesized by Tate, who stated the Brumer-Stark conjecture in its current form. The conjecture considers a totally real field $F$ and a finite abelian CM extension $H/F$. It states the existence of $p$-units in $H$ whose valuations at places above $p$ are related to the special values of the $L$-functions of the extension $H/F$ at $s=0$. Essentially equivalently, the conjecture states that a Stickelberger element associated to $H/F$ annihilates the (appropriately smoothed) class group of $H$. This conjecture has been refined by many authors in multiple directions. We conclude by stating our results toward these various conjectures and summarizing the proofs. In particular, we prove the Brumer-Stark conjecture, Rubin's higher rank version, and Kurihara's conjecture, all "away from 2." We also prove strong partial results toward Gross's conjecture and the exact $p$-adic analytic formula for Brumer-Stark units. The key technique involved in the proofs is Ribet's method. We demonstrate congruences between Hilbert modular Eisenstein series and cusp forms, and use the associated Galois representations to construct Galois cohomology classes. These cohomology classes are interpreted in terms of Ritter-Weiss modules, from which results on class groups may be deduced.

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arXiv2020

On Constant Terms of Eisenstein Series

We calculate the constant terms of certain Hilbert modular Eisenstein series at all cusps. Our formula relates these constant terms to special values of Hecke $L$-series. This builds on previous work of Ozawa, in which a restricted class of Eisenstein series were studied. Our results have direct arithmetic applications---in separate work we apply these formulas to prove the Brumer-Stark conjecture away from $p=2$ and to give an exact analytic formula for Brumer-Stark units.

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arXiv2018

A Note on the Main Conjecture over Q

In this note we show how the main conjecture of the Iwasawa theory over Q has a natural place in the context of the Galois representation of the Galois group $Gal(\bar Q/Q)$ on the etale pro-p fundamental group of the projective line minus three points. However we still need to assume the Vandiver conjecture to get a proof of the main conjecture in this context.

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arXiv2018

Cup products in the etale cohomology of number fields

This paper concerns cup product pairings in étale cohomology related to work of M. Kim and of W. McCallum and R. Sharifi. We will show that by considering Ext groups rather than cohomology groups, one arrives at a pairing which combines invariants defined by Kim with a pairing defined by McCallum and Sharifi. We also prove a formula for Kim's invariant in terms of Artin maps in the case of cyclic unramified Kummer extensions. One consequence is that for all $n > 1$, there are infinitely many number fields $F$ over which there are both trivial and non-trivial Kim invariants associated to cyclic groups of order $n$.

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arXiv2016

On the Gross-Stark Conjecture

In 1980, Gross conjectured a formula for the expected leading term at $s=0$ of the Deligne--Ribet $p$-adic $L$-function associated to a totally even character $ψ$ of a totally real field $F$. The conjecture states that after scaling by $L(ψω^{-1}, 0)$, this value is equal to a $p$-adic regulator of units in the abelian extension of $F$ cut out by $ψω^{-1}$. In this paper, we prove Gross's conjecture.

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arXiv2014

Non-commutative q-expansions

In this short note we partially answer a question of Fukaya and Kato by constructing a $q$-expansion with coefficients in a non-commutative Iwasawa algebra whose constant term is a non-commutative p-adic zeta function.

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arXiv2013

Kato's Local epsilon conjecture: $l \neq p$ case

Kato and Fukaya conjectured existence of certain elements in certain ($p$-adic) Iwasawa algebras which are related to Deligne-Langland's ($l$-adic) local epsilon constants. We prove the existence of these elements in the $l \neq p$ case.

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arXiv2011

The main conjecture of Iwasawa theory for totally real fields

The purpose of this paper is to prove the main conjecture of non-commutative Iwasawa theory for p-adic Lie extensions, for an odd prime p, of totally real number fields assuming that the Iwasawa mu invariant of a certain totally real number field vanishes.

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arXiv2010

K_1 of some Iwasawa algebras

Noncommutative Iwasawa theory has created a lot of interest in Whitehead groups of Iwasawa algebras of compact p-adic Lie groups with a quotient isomorphic to the additive group of p-adic integers. In this paper we compute Whitehead groups of Iwasawa algebra of a pro-p compact p-adic Lie group of dimension one. We also give results on Whitehead groups of the localisation of such Iwasawa algebras at the canonical Ore set defined by Coates, Fukaya, Kato, Sujatha and Venjakob.

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arXiv2010

K_1 of some noncommutative group rings

In this article I generalise previous computations (by K. Kato, T. Hara and myself) of K_1 (only up to p-power torsion) of p-adic group rings of finite non-abelian p-groups in terms of p-adic group rings of abelian subquotients of the group. Such computation have applications in non-commutative Iwasawa theory due to a strategy proposed by D. Burns, K. Kato (and a modification by T. Hara) for deducing non-commutative main conjectures from commutative main conjectures and certain congruences between special L-values. However, I do not say anything about Iwasawa theory in this article. The details of applications in Iwasawa theory will be presented in a separate paper.

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arXiv2008

Proof of the Main Conjecture of Noncommutative Iwasawa Theory for Totally Real Number Fields in Certain Cases

Fix an odd prime $p$. Let $G$ be a compact $p$-adic Lie group containing a closed, normal, pro-$p$ subgroup $H$ which is abelian and such that $G/H$ is isomorphic to the additive group of $p$-adic integers $\mathbbZ_p$ . First we assume that $H$ is finite and compute the Whitehead group of the Iwasawa algebra, $Λ(G)$, of $G$. We also prove some results about certain localisation of $Λ(G)$ needed in Iwasawa theory. Let $F$ be a totally real number field and let $F_{\infty}$ be an admissible $p$-adic Lie extension of $F$ with Galois group $G$. The computation of the Whitehead groups are used to show that the Main Conjecture for the extension $F_{\infty}/F$ can be deduced from certain congruences between abelian $p$-adic zeta functions of Delige and Ribet. We prove these congruences with certain assumptions on $G$. This gives a proof of the Main Conjecture in many interesting cases such as $\mathbb{Z}_p\rtimes

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