arXiv subjects

Heiko Knospe

Publications and source records attributed to Heiko Knospe.

7 recordsLinked to original sources

arXiv2025

On a multiplicative non-Hecke twist of motivic L-functions

We investigate the twisting of motivic $L$-functions by a family of multiplicative characters $\psi$, defined on prime ideals $\mathfrak{p}$ via $\psi(\mathfrak{p})=\alpha^{N(\mathfrak{p})}$ for a fixed $\alpha \in \mathbb{C}$. One can extend $\psi$ to a continuous non-Hecke character on the idele group of a number field. For $|\alpha|<1$, the resulting $\psi$-twisted $L$-function has interesting analytic properties: an enhanced half-plane of absolute convergence, preservation of the Euler product structure, and meromorphic continuation to the complex plane. We give applications to Dirichlet $L$-functions and $L$-functions associated to modular forms. Furthermore, we show that $\psi$-twisting allows the construction of convergent $p$-adic Dirichlet series and $p$-adic Euler products which have some similarities with their complex counterparts.

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arXiv2024

Special values of $p$-adic $L$-functions and Iwasawa $\lambda$-invariants of Dirichlet characters

We investigate the Iwasawa $\lambda$-invariant $\lambda_p(\chi)$ of the $p$-adic $L$-function associated with a Dirichlet character $\chi$ for odd primes $p$. Using special values of the $p$-adic $L$-function and its derivative, we derive novel and computationally efficient criteria to distinguish between the cases $\lambda = 0$, $\lambda = 1$, $\lambda = 2$, and $\lambda \geq 3$. In particular, we study the case when the $p$-adic $L$-function vanishes at $s=0$. Building on the results of Ferrero-Greenberg and of Gross-Koblitz, we establish explicit conditions for $\lambda_p(\chi) >1 $ and $\lambda_p(\chi) > 2$. Furthermore, we extend the methods of Ernvall-Mets\"ankyl\"a and of Dummit et al. to calculate the $\lambda$-invariant. This is achieved by twisting $\chi$ by characters $\psi$ of $p$-power order and using the values of the $p$-adic $L$-function at $s=2-p, \dots, 0$. Additionally, we utilize the value at $s=1$ to compute $\lambda_p(\chi)$. Finally, these results are applied to generate numerical data on the distribution of $\lambda$-invariants, when either the prime $p$ or the Dirichlet character $\chi$ is fixed.

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arXiv2022

On Iwasawa $λ$-invariants for abelian number fields and random matrix heuristics

Following both Ernvall-Metsänkylä and Ellenberg-Jain-Venkatesh, we study the density of the number of zeroes (i.e. the cyclotomic $λ$-invariant) for the $p$-adic zeta-function twisted by a Dirichlet character $χ$ of any order. We are interested in two cases: (i) the character $χ$ is fixed and the prime $p$ varies, and (ii) $\text{ord}(χ)$ and the prime $p$ are both fixed but $χ$ is allowed to vary. We predict distributions for these $λ$-invariants using $p$-adic random matrix theory and provide numerical evidence for these predictions. We also study the proportion of $χ$-regular primes, which depends on how $p$ splits inside $\mathbb{Q}(χ)$. Finally in an extensive Appendix, we tabulate the values of the $λ$-invariant for every character $χ$ of conductor $\leq 1000$ and for odd primes $p$ of small size.

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arXiv2021

Dirichlet Series Expansions of p-adic L-Functions

We study $p$-adic $L$-functions $L_p(s,χ)$ for Dirichlet characters $χ$. We show that $L_p(s,χ)$ has a Dirichlet series expansion for each regularization parameter $c$ that is prime to $p$ and the conductor of $χ$. The expansion is proved by transforming a known formula for $p$-adic $L$-functions and by controlling the limiting behavior. A finite number of Euler factors can be factored off in a natural manner from the $p$-adic Dirichlet series. We also provide an alternative proof of the expansion using $p$-adic measures and give an explicit formula for the values of the regularized Bernoulli distribution. The result is particularly simple for $c=2$, where we obtain a Dirichlet series expansion that is similar to the complex case.

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arXiv2020

A note on regularized Bernoulli distributions and p-adic Dirichlet expansions

We consider Bernoulli distributions and their regularizations, which are measures on the $p$-adic integers $\mathbb{Z}_p$. It is well known that their Mellin transform can be used to define $p$-adic $L$-functions. We show that for $p>2$ one of the regularized Bernoulli distributions is particularly simple and equal to a measure on $\mathbb{Z}_p$ that takes the values $\pm \frac{1}{2}$ on clopen balls. We apply this to $p$-adic $L$-functions for Dirichlet characters of $p$-power conductor and obtain Dirichlet series expansions similar to the complex case. Such expansions were studied by D. Delbourgo, and this contribution provides an approach via $p$-adic measures.

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arXiv2018

On the Spectrum of Nonstandard Dedekind Rings

The methods of nonstandard analysis are applied to algebra and number theory. We study nonstandard Dedekind rings, for example an ultraproduct of the ring of integers of a number field. Such rings possess a rich structure and have interesting relations to standard Dedekind rings and their completions. We use lattice theory to classify the ideals of nonstandard Dedekind rings. The nonzero prime ideals are contained in exactly one maximal ideal and can be described using valuation theory. The localisation at a prime ideal gives a valuation ring and we determine the value group and the residue field. The spectrum of a nonstandard Dedekind ring is described using lattices and value groups. Furthermore, we investigate the Riemann-Zariski space and the valuation spectrum of nonstandard Dedekind rings and their quotient fields.

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arXiv2016

Nonstandard Measure Spaces with Values in non-Archimedean Fields

The aim of this contribution is to bring together the areas of $p$-adic analysis and nonstandard analysis. We develop a nonstandard measure theory with values in a complete non-Archimedean valued field $K$, e.g. the $p-$adic numbers $\mathbb{Q}_p$. The corresponding theory for real-valued measures is well known by the work of P. A. Loeb, R. M. Anderson and others. We first review some of the standard facts on non-Archimedean measures and briefly sketch the prerequisites from nonstandard analysis. Then internal measures on rings and algebras with values in a nonstandard field ${^*K}$ are introduced. We explain how an internal measure induces a $K$-valued Loeb measure. The standard-part map between a Loeb space and the underlying standard measure space is measurable almost everywhere. We establish liftings from measurable functions to internal simple functions. Furthermore, we prove that standard measure spaces can be described as push-downs of hyperfinite internal measure spaces. This result is an analogue of a well-known Theorem on hyperfinite representations of Radon spaces. Then standard integrable functions are related to internal $S$-integrable functions and integrals are represented by hyperfinite sums. Finally, the results are applied to measures and integrals on $\mathbb{Z}_p$ and $\mathbb{Z}_p^{\times}$. We obtain explicit series expansions for the $p$-adic zeta function and the $p$-adic Euler-Mascheroni constant which we use for computations.

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