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Katharina Müller

Publications and source records attributed to Katharina Müller.

At least 19 recordsLinked to original sources
arXiv2026

Hilbert's tenth problem for families of $ \mathbb{Z}_p $-extensions of imaginary quadratic fields

Via a novel application of Iwasawa theory, we study Hilbert's tenth problem for number fields occurring in $\mathbb{Z}_p$-towers of imaginary quadratic fields $K$. For a odd prime $p$, the lines $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ are identified with $\mathbb{Z}_p$-extensions $ K_{a,b}/K $. Under certain conditions on $ K $ that involve explicit elliptic curves, we identify a line $(a_0,b_0) \in \mathbb{P}^1(\mathbb{Z}/p\mathbb{Z})$ such that for all $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ with $(a, b)\not\equiv (a_0, b_0)\pmod{p}$, Hilbert's tenth problem has a negative answer in all finite layers of $ K_{a,b} $. Using results of Bhargava et al., we prove unconditionally that a positive proportion of imaginary quadratic fields meet our criterion when $p=3$. For $p=11,13,31,37$, the analogous conclusions obtained from the rank-zero twist families of Kriz--Li are conditional on the vanishing of the $p$-primary Tate--Shafarevich groups for a positive relative proportion of those twists.

math.NT↗
arXiv2026

Iwasawa theory of (directed) Cayley graphs

In this article, we prove the Defect Conjecture for Cayley graphs of abelian groups, dihedral groups, and groups of the form $\mathbb{Z}/p\mathbb{Z} \rtimes \mathbb{Z}/(p-1)\mathbb{Z}$ where $p\geq 3$ is a prime. We also compute the Iwasawa invariants of the Bowen--Franks groups associated with these graphs in several cases.

math.NT↗
arXiv2026

Modularity theorems for Eisenstein congruences in prime-power level

Let $p, N \geq 5$ be primes such that $N \equiv 1 \bmod p$. We prove modularity theorems at levels $N$ and $N^2$, showing that suitable Eisenstein localizations of the weight-$2$ $p$-adic Hecke algebra at these levels are isomorphic to certain natural quotients of a universal pseudodeformation ring. This universal ring parametrizes pseudorepresentations that are residually Eisenstein, unramified outside $Np$ and finite-flat at $p$, and satisfy appropriate conditions at $N$ depending on the level.

math.NT↗
arXiv2026

Second-Harmonic Imaging of Magnetic Domains in Thin Film Hematite

Hematite is an antiferromagnetic oxide and candidate altermagnetic insulator whose Néel order reorients from an easy-axis to an easy-plane phase at the Morin transition. Interpreting altermagnetic transport and symmetry sensitive optical responses requires knowledge of the Néel vector orientation relative to the crystal axes and of the domain structure within the probed device. Here, we show that polarization-resolved second-harmonic generation (SHG) microscopy resolves magnetic symmetry and domains in epitaxial (0001)-oriented hematite films. Across the Morin temperature, the SHG polarization anisotropy evolves from an approximately sixfold pattern consistent with the easy-axis orientation to pronounced twofold patterns consistent with the easy-plane orientation. A symmetry analysis based on magnetic-dipole and electric-quadrupole contributions reproduces this evolution. Importantly, the interference between SHG amplitudes that are odd and even under reversal of the magnetic order renders opposite Néel-vector orientations optically distinguishable. Consistently, opposite directions of an applied in-plane magnetic field produce distinct SHG responses in our experiments. Using this magnetic contrast, we image micrometer-scale domains, their reorganization across the Morin transition, and their reconfiguration under magnetic and thermal cycling, including a remanent change after cycling through the spin-flop transition. These results establish SHG microscopy as a local probe of magnetic symmetry, Néel-vector orientation, and domain evolution in hematite films.

cond-mat.mtrl-sci↗
arXiv2026

A Heuristic approach to the Iwasawa theory of elliptic curves

Let $E_{/\mathbb{Q}}$ be an elliptic curve and $p$ an odd prime such that $E$ has good ordinary reduction at $p$ and the Galois representation on $E[p]$ is irreducible. Then Greenberg's $μ=0$ conjecture predicts that the Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ is cofinitely generated as a $\mathbb{Z}_p$-module. In this article we study this conjecture from a statistical perspective. We extend the heuristics of Poonen and Rains to obtain further evidence for Greenberg's conjecture. The key idea is that the vanishing of the $μ$-invariant can be detected by the intersection $M_1\cap M_2$ of two Iwasawa modules $M_1, M_2$ with additional properties in a given inner product space. The heuristic is based on showing that there is a probability measure on the space of pairs $(M_1, M_2)$ respect to which the event that $M_1\cap M_2$ is finite happens with probability $1$.

math.NT↗
arXiv2026

A new perspective on the rank of Mazur's Eisenstein Hecke algebra

Let $N, p \geq 5$ be primes such that $N \equiv 1 \bmod p$. We study the rank $r$ of the Hecke algebra that parametrizes modular forms of weight 2 and level $N$ that are Eisenstein modulo $p$. When $r$ is $2$ or $3$, we prove that $r-1$ equals the order of vanishing of the mod-$p$ reduction of a zeta element that interpolates Dirichlet $L$-values at $-1$, thereby recovering results of Merel and Lecouturier. This equality can fail in some cases when $r \geq 4$, and we provide a heuristic explanation of this failure. Our approach handles all of these cases uniformly by studying the analogous Hecke algebra in level $N^2$. When exactly one of $r-1$ or the order of vanishing equals $3$, we provide precise information about Galois orbits of cuspidal newforms in level $N^2$ that are Eisenstein modulo $p$.

math.NT↗
arXiv2026

Bowen--Franks groups and minus class groups of cyclotomic number fields with prime conductor

Let $p$ be an odd rational prime and consider the cyclotomic number field $K = \mathbb{Q}(ζ_{p})$ of conductor $p$. We construct a directed graph $Y$ on $p-1$ vertices for which the torsion part of the corresponding Bowen--Franks group is closely related to the minus part of the class group of $K$. In particular, both groups have the same cardinality up to an explicit power of $p$. Furthermore, they are both $\mathrm{Gal}(K/\mathbb{Q})$-modules, and we prove the equality of the cardinalities of their isotypic components after tensoring them with the valuation ring of an appropriate $\ell$-adic field for $\ell \nmid p-1$.

math.NT↗
arXiv2026

Iwasawa theory for abelian towers of digraphs

Let $p$ and $\ell$ be prime numbers, and $d\ge1$ an integer. We formulate and prove Iwasawa main conjectures of the Picard groups and Bowen--Franks groups in $\mathbb{Z}_p^d$-towers of digraphs. In particular, we relate the $\ell$ parts of these groups to certain $p$-adic $L$-functions defined using a voltage assignment. In the case where $\ell$ is not equal to $p$, we make use of the recent work of Bandini--Longhi to define the appropriate characteristic ideals. We also prove the growth of the $\ell$-part of these groups, generalizing classical results of Sinnott and Washington on ideal class groups of number fields. Finally, we introduce the concept of defect, which compare certain algebraic and analytic ranks related to Bowen--Franks groups and study their asymptotic behaviour in a $\mathbb{Z}_p^d$-tower.

math.NT↗
arXiv2025

Iwasawa Theory of Elliptic Curves in Quadratic Twist Families

In this article, we use two different approaches -- one algebraic and the other analytic -- to study the variation of Iwasawa invariants of rational elliptic curves in some quadratic twist families. The analytic approach involves a thorough investigation of half-integral weight modular forms. On the other hand, the algebraic proof requires studying the BDP-Selmer groups and the fine Selmer groups.

math.NT↗
arXiv2025

IHearYou: Linking Acoustic Features to DSM-5 Depressive Behavior Indicators

Depression affects over millions people worldwide, yet diagnosis still relies on subjective self-reports and interviews that may not capture authentic behavior. We present IHearYou, an approach to automated depression detection focused on speech acoustics. Using passive sensing in household environments, IHearYou extracts voice features and links them to DSM-5 (Diagnostic and Statistical Manual of Mental Disorders) indicators through a structured Linkage Framework instantiated for Major Depressive Disorder. The system runs locally to preserve privacy and includes a persistence schema and dashboard, presenting real-time throughput on a commodity laptop. To ensure reproducibility, we define a configuration-driven protocol with False Discovery Rate (FDR) correction and gender-stratified testing. Applied to the DAIC-WOZ dataset, this protocol reveals directionally consistent feature-indicator associations, while a TESS-based audio streaming experiment validates end-to-end feasibility. Our results show how passive voice sensing can be turned into explainable DSM-5 indicator scores, bridging the gap between black-box detection and clinically interpretable, on-device analysis.

cs.SD↗
arXiv2025

Distributed Pulse-Wave Simulator for DDoS Dataset Generation

Pulse-wave Distributed Denial-of-Service (DDoS) attacks generate short, synchronized bursts of traffic that circumvent pattern-based detection and quickly exhaust traditional defense systems. This transient and spatially distributed behavior makes analysis extremely challenging, as no public datasets capture how such attacks evolve across multiple network domains. Since each domain observes only a partial viewpoint of the attack, a correlated, multi-vantage view is essential for comprehensive analysis, early detection, and attribution. This paper presents DPWS, an open-source simulator for generating distributed pulse-wave DDoS datasets. DPWS models multi-AS topologies and produces synchronized packet captures at multiple autonomous systems, showing the distributed structure of coordinated bursts. It enables fine-grained control of traffic parameters through a lightweight YAML interface. DPWS reproduces pulse-wave dynamics across multiple vantage points, exhibits natural fingerprint variability at equal aggregate rates, and scales with MPI in ns-3, providing a reproducible basis for studying pulse-wave behaviour and benchmarking distributed DDoS defenses, while sharing practical insights on ns-3 scalability and synchronization gained during development.

cs.NI↗
arXiv2025

Iwasawa Theory of graphs and their duals

In this article, we study questions pertaining to ramified $\mathbb{Z}_p^d$-extensions of a finite connected graph $X$. We also study the Iwasawa theory of dual graphs.

math.NT↗
arXiv2025

On $\mathbb{Z}_p$-towers of graph coverings arising from a constant voltage assignment

We investigate properties of $\mathbb{Z}_p$-towers of graph coverings that arise from a constant voltage assignment. We prove the existence and uniqueness (up to isomorphisms) of such towers. Furthermore, we study the Iwasawa invariants of these towers, and apply our results to towers of isogeny graphs enhanced with level structures, as well as towers arising from volcano graphs.

math.NT↗
arXiv2025

Isogeny graphs with level structures arrising from the Verschiebung map

We enhance an isogeny graph of elliptic curves by incorporating level structures defined by bases of the kernels of iterates of the Verschiebung map. We extend several previous results on isogeny graphs with level structures defined by geometric points to these graphs. Firstly, we prove that these graphs form $\mathbb{Z}_p$-towers of graph coverings as the power of the Verschiebung map varies. Secondly, we prove that the connected components of these graphs display a volcanic structure.

math.NT↗
arXiv2024

On towers of Isogeny graphs with full level structure

Let $p,q,l$ be three distinct prime numbers and let $N$ be a positive integer coprime to $pql$. For an integer $n\ge 0$, we define the directed graph $X_l^q(p^nN)$ whose vertices are given by isomorphism classes of elliptic curves over a finite field of characteristic $q$ equipped with a level $p^nN$ structure. The edges of $X_l^q(p^nN)$ are given by $l$-isogenies. We are interested in when the connected components of $X_l^q(p^nN)$ give rise to a tower of Galois covers as $n$ varies. We show that only in the supersingular case we do get a tower of Galois covers. We also study similar towers of isogeny graphs given by oriented supersingular curves, as introduced by Colò-Kohel, enhanced with a level structure.

math.NT↗
arXiv2024

On ordinary isogeny graphs with level structure

Let $l$ and $p$ be two distinct prime numbers. We study $l$-isogeny graphs of ordinary elliptic curves defined over a finite field of characteristic $p$, together with a level structure. Firstly, we show that as the level varies over all $p$-powers, the graphs form an Iwasawa-theoretic abelian $p$-tower, which can be regarded as a graph-theoretical analogue of the Igusa tower of modular curves. Secondly, we study the structure of the crater of these graphs, generalizing previous results on volcano graphs. Finally, we solve an inverse problem of graphs arising from the crater of $l$-isogeny graphs with level structures, partially generalizing a recent result of Bambury, Campagna and Pazuki.

math.NT↗