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An alternative construction of equivariant Tamagawa numbersJun 11 2019We propose a new formulation of the equivariant Tamagawa number conjecture (ETNC) for non-commutative coefficients. We remove Picard groupoids, determinant functors, virtual objects and relative K-groups. Our Tamagawa numbers lie in an idele group instead ... More
A Classification of order $4$ class groups of $\mathbb{Q}{(\sqrt{n^2+1})}$Feb 14 2019We classify all order 4 class groups of the family of real quadratic fields $\mathbb{Q}{(\sqrt{n^2+1})}$. The main tools used are special values of Dedekind zeta functions attached to these fields and generalized Dedekind sum.
A note on certain real quadratic fields with class number upto threeDec 06 2018Dec 10 2018We obtain criteria for the class number of certain Richaud-Degert type real quadratic fields to be $3$. We also treat a couple of families of real quadratic fields of Richaud-Degert type that were not considered earlier, and obtain similar criteria for ... More
The smallest regulator for number fields of degree 7 with five real placesOct 08 2018In 2016 Astudillo, Diaz y Diaz and Friedman published sharp lower bounds for regulators of number fields of all signatures up to degree seven, except for fields of degree seven having five real places. We deal with this signature, proving that the field ... More
Large values of Dirichlet $L$- functions inside the critical stripMar 10 2018Apr 16 2018In the present paper, we study large values of Dirichlet $L$- functions inside the critical strip. For every $1/2<\sigma<1$, we show that for $q$ sufficiently large, there exists a non-principal character $\chi$ modulo $q$ and a constant $c(\sigma)>0$ ... More
On Iwasawa theory of Rubin-Stark units and narrow class groupsJan 29 2018Let $K$ be a totally real number field of degree $r$. Let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{2}$-extension of $K$ and let $L_{\infty}$ be a finite extension of $K_{\infty}$, abelian over $K$. The goal of this paper is to compare the characteristic ... More
Average values of L-functions in even characteristicJan 05 2017Mar 02 2017Let $k = \mathbb{F}_{q}(T)$ be the rational function field over a finite field $\mathbb{F}_{q}$, where $q$ is a power of $2$. In this paper we solve the problem of averaging the quadratic $L$-functions $L(s, \chi_{u})$ over fundamental discriminants. ... More
Iwasawa theory of Rubin-Stark units and class groupAug 10 2016Let $K$ be a totally real number field of degree $r=[K:\mathbb{Q}]$ and let $p$ be an odd rational prime. Let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$ and let $L_{\infty}$ be a finite extension of $K_{\infty}$, abelian over ... More
Reconstructing decomposition subgroups in arithmetic fundamental groups using regulatorsSep 17 2014Jul 23 2015Our main goal in the present article is to explain how one can reconstruct the decomposition subgroups and norms of points on an arithmetic curve inside its fundamental group if the following data are given: the fundamental group, a part of the cyclotomic ... More
A Study of Kummer's Proof of Fermat's Last Theorem for Regular PrimesJul 11 2013We study Kummer's approach towards proving the Fermat's last Theorem for regular primes. Some basic algebraic prerequisites are also discussed in this report, and also a brief history of the problem is mentioned. We review among other things the Class ... More
On the existence of large degree Galois representations for fields of small discriminantNov 06 2012Jul 28 2014Let $L/K$ be a Galois extension of number fields. We prove two lower bounds on the maximum of the degrees of the irreducible complex representations of ${\rm Gal}(L/K)$, the sharper of which is conditional on the Artin Conjecture and the Generalized Riemann ... More
Refined class number formulas and Kolyvagin systemsSep 22 2009We use the theory of Kolyvagin systems to prove (most of) a refined class number formula conjectured by Darmon. We show that for every odd prime $p$, each side of Darmon's conjectured formula (indexed by positive integers $n$) is "almost" a $p$-adic Kolyvagin ... More
Some New Maps and Ideals in Classical Iwasawa Theory with ApplicationsMay 27 2009Aug 03 2010We introduce a new ideal {\mathfrak D} of the p-adic Galois group-ring associated to a real abelian field and a related ideal {\mathfrak J} for imaginary abelian fields. Both result from an equivariant, Kummer-type pairing applied to Stark units in a ... More
A non-abelian Stickelberger theoremDec 19 2008Mar 11 2010Let L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring Z_(p)[G] that annihilates ... More
Stickelberger elements and Kolyvagin systemsAug 19 2008Mar 30 2011In this paper, we construct (many) Kolyvagin systems out of Stickelberger elements, utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. We show how to apply this construction to prove results on the odd parts ... More
$Λ$-adic Kolyvagin systemsJun 04 2007Mar 30 2011In this paper, we study the deformations of Kolyvagin systems that are known to exist in a wide variety of cases, by the work of B. Howard, B. Mazur, and K. Rubin for the residual Galois representations, along the cyclotomic Iwasawa algebra. We prove, ... More
Kolyvagin systems of Stark unitsMar 14 2007Mar 04 2008In this paper we construct, using Stark elements of Rubin [Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 33-62], Kolyvagin systems for certain modified Selmer structures (that are adjusted to have core rank one in the sense of [Mem. Amer. Math. Soc. ... More
Quadratic Function Fields with Exponent Two Ideal Class GroupApr 28 2005It has been shown by Madden that there are only finitely many quadratic extensions of k(x), k a finite field, in which the ideal class group has exponent two and the infinity place of k(x) ramifies. We give a characterization of such fields that allow ... More
First-hit analysis of algorithms for computing quadratic irregularityNov 08 2000Nov 10 2000The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta function at odd negative ... More
Comparison of algorithms to calculate quadratic irregularity of prime numbersOct 29 2000In previous work, the author has extended the concept of regular and irregular primes to the setting of arbitrary totally real number fields k_{0}, using the values of the zeta function \zeta_{k_{0}} at negative integers as our ``higher Bernoulli numbers''. ... More