Class groups of real cyclotomic fields
We prove that every finite abelian group G occurs as a subgroup of the class group of infinitely many real cyclotomic fields.
math.NT↗
arXiv subjects
Mohit Mishra
Publications and source records attributed to Mohit Mishra.
Class group of real cyclotomic fields
In this article, we prove that every finite abelian group $G$ of odd order occurs as a subgroup of the class group of infinitely many real cyclotomic fields.
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On the structure of order 4 class groups of $\mathbb{Q}(\sqrt{n^2+1})$
Groups of order $4$ are isomorphic to either $\mathbb{Z}/4\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. We give certain sufficient conditions permitting to specify the structure of class groups of order $4$ in the family of real quadratic fields $\mathbb{Q}{(\sqrt{n^2+1})}$ as $n$ varies over positive integers. Further, we compute the values of Dedekind zeta function attached to these quadratic fields at the point $-1$. As a side result, we show that the size of the class group of this family could be made as large as possible by increasing the size of the number of distinct odd prime factors of $n$.
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Primary rank of the class group of real cyclotomic fields
Let $H= \mathbb{Q}(ζ_{n} + {ζ_{n}}^{-1})$ and $\ell$ be an odd prime such that $q \equiv 1 \pmod \ell$ for some prime factor $q$ of $n$. We get a bound on the $\ell$-rank of the class group of $H$(under some conditions) in terms of the $\ell$-rank of the class group of real quadratic subfield contained in $H$. This is an extension of a recent work of E. Agathocleous (with alternate hypothesis) where she handles $\ell=3$ case. As an application of our main result we relate the $\ell$-rank of real quadratic subfields of $H$.
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Lower bound for class number of certain real quadratic fields
Let $d$ be a square-free positive integer and $h(d)$ the class number of the real quadratic field $\mathbb{Q}{(\sqrt{d})}.$ In this paper we give an explicit lower bound for $h(n^2+r)$, where $r=1,4$, and also establish an equivalent criteria to attain this lower bound in terms of special value of Dedekind zeta function. Our bounds enable us to reduce the real quadratic families considered in Chowla and Yokoi's conjecture to comparatively small subfamily. We also give an equivalent criteria for having an alternate proof of both the conjectures. Also applying our results, we obtain some criteria for class group of prime power order to be cyclic.
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A note on certain real quadratic fields with class number upto three
We 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 the class number of such fields to be 2 and 3.
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ΔBreakpad: Diversified Binary Crash Reporting
This paper introduces ΔBreakpad. It extends the Breakpad crash reporting system to handle software diversity effectively and efficiently by replicating and patching the debug information of diversified software versions. Simple adaptations to existing open source compiler tools are presented that on the one hand introduce significant amounts of diversification in the code and stack layout of ARMv7 binaries to mitigate the widespread deployment of code injection and code reuse attacks, while on the other hand still supporting accurate crash reporting. An evaluation on SPEC2006 benchmarks demonstrates that the corresponding computational, storage, and communication overheads are small.
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