Automorphism Research Articles

Fields of moduli and the arithmetic of tame quotient singularities

Given a perfect field $k$ with algebraic closure $\overline {k}$ and a variety $X$ over $\overline {k}$, the field of moduli of $X$ is the subfield of $\overline {k}$ of elements fixed by field automorphisms $\gamma \in \operatorname {Gal}(\overline {k}/k)$ such that the Galois conjugate $X_{\gamma }$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset \overline {k}$ such that $X$ descends to $k'$. In this paper, we extend the formalism and define the field of moduli when $k$ is not perfect. Furthermore, Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and prove that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher-dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As a consequence, we prove that a variety $X$ of dimension $d$ with a smooth marked point $p$ such that $\operatorname {Aut}(X,p)$ is finite, étale and of degree prime to $d!$ is defined over its field of moduli.

Countably many asymptotic tensor ranks

In connection with recent work on gaps in the asymptotic subranks of complex tensors the question arose whether the number of nonnegative real numbers that arise as the asymptotic subrank of some complex tensor is countable. In this short note we settle this question in the affirmative, for all tensor invariants that are algebraic in the sense that they are invariant under field automorphisms of the complex numbers.

Quasi-inner automorphisms of Drinfeld modular groups

Let A be the set of elements in an algebraic function field K over \mathbb{F}_{q} which are integral outside a fixed place \infty . Let G=\mathrm{GL}_{2}(A) be a Drinfeld modular group . The normalizer of G in \mathrm{GL}_{2}(K) , where K is the quotient field of A , gives rise to automorphisms of G , which we refer to as quasi-inner . Modulo the inner automorphisms of G , they form a group \mathrm{Quinn}(G) which is isomorphic to \mathrm{Cl}(A)_{2} , the 2 -torsion in the ideal class group \mathrm{Cl}(A) . The group \mathrm{Quinn}(G) acts on all kinds of objects associated with G . For example, it acts freely on the cusps and elliptic points of G . If \mathcal{T} is the associated Bruhat–Tits tree, the elements of \mathrm{Quinn}(G) induce non-trivial automorphisms of the quotient graph G \backslash \mathcal{T} , generalizing an earlier result of Serre. It is known that the ends of G \backslash \mathcal{T} are in one-to-one correspondence with the cusps of G . Consequently, \mathrm{Quinn}(G) acts freely on the ends. In addition, \mathrm{Quinn}(G) acts transitively on those ends which are in one-to-one correspondence with the vertices of G \backslash \mathcal{T} whose stabilizers are isomorphic to \mathrm{GL}_{2}(\mathbb{F}_{q}) .

Finding the Fixing Number of Johnson Graphs J(n, k) for k Є {2; 3}

The graph invariant, aptly named the fixing number, is the smallest number of vertices that, when fixed, eliminate all non-trivial automorphisms (or symmetries) of a graph. Although many graphs have established fixing numbers, Johnson graphs, a family of graphs related to the graph isomorphism problem, have only partially classified fixing numbers. By examining specific orbit sizes of the automorphism group of Johnson graphs and classifying the subsequent remaining subgroups of the automorphism group after iteratively fixing vertices, we provide exact minimal sequences of fixed vertices, in turn establishing the fixing number of infinitely many Johnson graphs. KEYWORDS: Graph Automorphism Groups; Symmetry Breaking; Fixing Number; Determining Number; Johnson Graphs; Kneser Graphs; Graph Invariants; Permutation Groups; Minimal Sized Bases.

Counting spanning trees of (1, [formula omitted])-periodic graphs

Let N≥2 be an integer, a (1, N)-periodic graph G is a periodic graph whose vertices can be partitioned into two sets V1={v∣σ(v)=v} and V2={v∣σi(v)≠vfor any1<i<N}, where σ is an automorphism with order N of G. The subgraph of G induced by V1 is called a fixed subgraph. Yan and Zhang (2011) studied the enumeration of spanning trees of a special type of (1, N)-periodic graphs with V1=0̸ for any non-trivial automorphism with order N. In this paper, we obtain a concise formula for the number of spanning trees of (1, N)-periodic graphs. Our result can reduce to Yan and Zhang’s when V1 is empty. As applications, we give a new closed formula for the spanning tree generating function of cobweb lattices, and obtain formulae for the number of spanning trees of circulant graphs Cn(s1,s2,…,sk), K1⋁Cn(s1,s2,…,sk) and K2⋁Cn(s1,s2,…,sk).

Automorphisms of the supersingular isogeny graph

We provide a condition for which the supersingular l-isogeny graph in characteristic p has only one nontrivial automorphism, given by the action of Frobenius. For a fixed p, our condition is known to hold for a density 1 set of primes l.

On the Codes over a Family of Rings and Their Applications to DNA Codes

In this paper, the structures of the linear codes over a family of the rings $A_{t}=Z_{4}\left[ u_{1},\ldots ,u_{t}\right] \left/ \left\langle u_{i}^{2}-u_{i},u_{i}u_{j}-u_{j}u_{i}\right\rangle \right. $ are given, where $i,j=1,2,\ldots ,t$, $i\neq j$, $Z_{4}=\{0,1,2,3\}$. A map between the elements of the $A_{t}$ and the alphabet $\left\{ A,T,C,G\right\} ^{2^{t}}$ is constructed. The DNA codes are obtained with three different methods, by using the cyclic, skew cyclic codes over a family of the rings $A_{t}$ and $\theta _{i}$-set, where $\theta _{i}$ is a non trivial automorphism on $A_{i}$, for $i=1,2,\ldots ,t$.

Twisted Goppa Codes With an Efficient Decoding Algorithm and Quasi-Cyclic Properties

In this paper, we introduce twisted Goppa codes, which generalize classical Goppa codes by adding a twisted term. Then we provide an efficient decoding algorithm for twisted Goppa codes. The Niederreiter cryptosystem is bassed on linear error-correcting codes in which the public key is a parity check matrix. When twisted Goppa codes are applied to the Niederreiter cryptosystem, the public key size is overlarge. To reduce the public key size, we construct quasi-cyclic twisted Goppa codes via a non-trivial automorphism group carefully selecting the defining set and the matched polynomial. Moreover, we obtain a family of cyclic twisted Goppa codes.

Optimal Universal Quantum Circuits for Unitary Complex Conjugation

Let $U_d$ be a unitary operator representing an arbitrary $d$-dimensional unitary quantum operation. This work presents optimal quantum circuits for transforming a number $k$ of calls of $U_d$ into its complex conjugate $\bar{U_d}$. Our circuits admit a parallel implementation and are proven to be optimal for any $k$ and $d$ with an average fidelity of $\left\langle{F}\right\rangle =\frac{k+1}{d(d-k)}$. Optimality is shown for average fidelity, robustness to noise, and other standard figures of merit. This extends previous works which considered the scenario of a single call ($k=1$) of the operation $U_d$, and the special case of $k=d-1$ calls. We then show that our results encompass optimal transformations from $k$ calls of $U_d$ to $f(U_d)$ for any arbitrary homomorphism $f$ from the group of $d$-dimensional unitary operators to itself, since complex conjugation is the only non-trivial automorphisms on the group of unitary operators. Finally, we apply our optimal complex conjugation implementation to design a probabilistic circuit for reversing arbitrary quantum evolutions.

The distinguishing index of graphs with infinite minimum degree

AbstractThe distinguishing index of a graph is the least number of colors necessary to obtain an edge coloring of that is preserved only by the trivial automorphism. We show that if is a connected ‐regular graph for some infinite cardinal then , proving a conjecture of Lehner, Pilśniak, and Stawiski. We also show that if is a graph with infinite minimum degree and at most vertices of degree for every infinite cardinal , then . In particular, if has infinite minimum degree and order at most .

Towards a classification of charge-3 monopoles with symmetry

We classify all possible charge-3 monopole spectral curves with non-trivial automorphism group and within these identify those with elliptic quotients. By focussing on elliptic quotients the transcendental constraints for a monopole spectral curve become ones regarding periods of elliptic functions. We construct the Nahm data and new monopole spectral curves with D_6 and V_4 symmetry, the latter based on an integrable complexification of Euler’s equations, and for which energy density isosurfaces are plotted. Extensions of our approach to higher charge and hyperbolic monopoles are discussed.

Fast hashing to [formula omitted] on pairing-friendly curves with the lack of twists

Pairing-friendly curves with the lack of twists, such as BW13-P310 and BW19-P286, have been receiving attention in pairing-based cryptographic protocols as they provide fast operation in the first pairing subgroup G1 at the 128-bit security level. However, they also incur a performance penalty for hashing to G2 simultaneously since G2 is totally defined over a full extension field. Furthermore, the previous methods for hashing to G2 focus on pairing-friendly curves admitting a twist, which can not be employed for our selected curves.In this paper, we propose a general method for hashing to G2 on curves with the lack of twists. More importantly, we further optimize the general algorithm on curves with non-trivial automorphisms, which is certainly suitable for BW13-P310 and BW19-P286. Theoretical estimations show that the latter would be more efficient than the former. For comparing the performance of the two proposed algorithms in detail, high speed software implementation over BW13-P310 is also provided on a 64-bit processor. Experimental results show that the general algorithm can be sped up by up to 88% if the computational cost of cofactor multiplication for G2 is only considered, while the improved method is up to 71% faster than the general one for the whole process.

On the degrees of irreducible characters fixed by some field automorphism in 𝑝-solvable groups

It is known that, if all the real-valued irreducible characters of a finite group have odd degree, then the group has normal Sylow 2 2 -subgroup. This result is generalized for Sylow p p -subgroups, for any prime number p p , while assuming the group to be p p -solvable. In particular, it is proved that a p p -solvable group has a normal Sylow p p -subgroup if p p does not divide the degree of any irreducible character of the group fixed by a field automorphism of order p p .

Recovering affine linearity of functions from their restrictions to affine lines

Motivated by recent results of Tao–Ziegler [Discrete Anal. 2016] and Greenfeld–Tao (2022 preprint) on concatenating affine-linear functions along subgroups of an abelian group, we show three results on recovering affine linearity of functions $$f: V \rightarrow W$$ from their restrictions to affine lines, where V, W are $${\mathbb {F}}$$ -vector spaces and $$\dim V \geqslant 2$$ . First, if $$\dim V < |{\mathbb {F}}|$$ and $$f: V \rightarrow {\mathbb {F}}$$ is affine-linear when restricted to affine lines parallel to a basis and to certain “generic” lines through 0, then f is affine-linear on V. (This extends to all modules M over unital commutative rings R with large enough characteristic.) Second, we explain how a classical result attributed to von Staudt (1850 s) extends beyond bijections: If $$f: V \rightarrow W$$ preserves affine lines $$\ell $$ , and if $$f(v) \not \in f(\ell )$$ whenever $$v \not \in \ell $$ , then this also suffices to recover affine linearity on V, but up to a field automorphism. In particular, if $${\mathbb {F}}$$ is a prime field $${\mathbb {Z}}/p{\mathbb {Z}}$$ ( $$p>2$$ ) or $${\mathbb {Q}}$$ , or a completion $${\mathbb {Q}}_p$$ or $${\mathbb {R}}$$ , then f is affine-linear on V. We then quantitatively refine our first result above, via a weak multiplicative variant of the additive $$B_h$$ -sets initially explored by Singer [Trans. Amer. Math. Soc. 1938], Erdös–Turán [J. London Math. Soc. 1941], and Bose–Chowla [Comment. Math. Helv. 1962]. Weak multiplicative $$B_h$$ -sets occur inside all rings with large enough characteristic, and in all infinite or large enough finite integral domains/fields. We show that if R is among any of these classes of rings, and $$M = R^n$$ for some $$n \geqslant 3$$ , then one requires affine linearity on at least $$\left( {\begin{array}{c}n\\ \lceil n/2 \rceil \end{array}}\right) $$ -many generic lines to deduce the global affine linearity of f on $$R^n$$ . Moreover, this bound is sharp.

Constructing CM Fields for NFS to Accelerate DL Computation in Non-Prime Finite Fields

The hardness of discrete logarithm problem (DLP) over finite fields is the security foundation of many cryptographic protocols. When the characteristic is not small, the state-of-the-art algorithms for solving DLP are the number field sieve (NFS) and its variants. In the relation collection step, to translate the relations between prime ideals to those of elements one needs to use the Schirokauer map. Besides, if the number field has non-trivial automorphisms, one can use them to accelerate the factor-base logarithms computation. However, the Schirokauer map is not compatible with automorphisms. To exploit automorphism efficiently, we focus on the method to construct fields in NFS such that the fields on both sides have non-trivial automorphisms with the logarithms of units being zero. 1. We construct two families of CM polynomials of arbitrary even degree with small coefficients, corresponding to the automorphisms being <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ↦ – <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ↦ 1/ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> . 2. We show how to combine these polynomials with the JLSV <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and Conjugation polynomial selection methods on both sides. 3. We also generalize our method to the multiple number field sieve and the extended tower number field sieve.

A Note on Eigenvalues and Asymmetric Graphs

This note is intended as a contribution to the study of quantitative measures of graph complexity that use entropy measures based on symmetry. Determining orbit sizes of graph automorphism groups is a key part of such studies. Here we focus on an extreme case where every orbit contains just a single vertex. A permutation of the vertices of a graph G is an automorphism if, and only if, the corresponding permutation matrix commutes with the adjacency matrix of G. This fact establishes a direct connection between the adjacency matrix and the automorphism group. In particular, it is known that if the eigenvalues of the adjacency matrix of G are all distinct, every non-trivial automorphism has order 2. In this note, we add a condition to the case of distinct eigenvalues that makes the graph asymmetric, i.e., reduces the automorphism group to the identity alone. In addition, we prove analogous results for the Google and Laplacian matrices. The condition is used to build an O(n3) algorithm for detecting identity graphs, and examples are given to demonstrate that it is sufficient, but not necessary.

Sets of real numbers closed under Turing equivalence: applications to fields, orders and automorphisms

In the first half of this paper, we study the way that sets of real numbers closed under Turing equivalence sit inside the real line from the perspective of algebra, measure and orders. Afterwards, we combine the results from our study of these sets as orders with a classical construction from Avraham to obtain a restriction about how non trivial automorphism of the Turing degrees (if they exist) interact with 1-generic degrees.

On Some Regular Two-Graphs up to 50 Vertices

Regular two-graphs on up to 36 vertices are classified, and recently, the classification of regular two-graphs on 38 and 42 vertices having at least one descendant with a nontrivial automorphism group has been performed. The first unclassified cases are those on 46 and 50 vertices. It is known that there are at least 97 regular two-graphs on 46 vertices leading to 2104 descendants and 54 regular two-graphs on 50 vertices leading to 785 descendants. In this paper, we classified all strongly regular graphs with parameters (45,22,10,11), (49,24,11,12), and (50,21,8,9) that have Z6 as the automorphism group and constructed regular two-graphs from SRGs (45,22,10,11), SRGs (49,24,11,12), and SRGs (50,21,8,9) that have automorphisms of order six. In this way, we enumerated all regular two-graphs on up to 50 vertices that have at least one descendant with an automorphism group of order six or at least one strongly regular graph associated with an automorphism group of order six. We found 236 new regular two-graphs on 46 vertices leading to 3172 new SRG (45,22,10,11) and 51 new regular two-graphs on 50 vertices leading to 398 new SRG (49,24,11,12).

A note on Quebbemann’s extremal lattices of rank 64

By constructing explicit examples, we show that the method of Quebbemann yields many isomorphism classes of extremal lattices of rank 64. Many of these examples have no non-trivial automorphisms.

Rigid birational involutions of ℙ 3 and cubic threefolds

We construct families of birational involutions on ℙ 3 or on a smooth cubic threefold which do not fit into a non-trivial elementary relation of Sarkisov links. As a consequence, we construct new homomorphisms from their group of birational transformations, effectively re-proving their non-simplicity. We also prove that these groups admit a free product structure. Finally, we produce automorphisms of these groups that are not generated by inner and field automorphisms.