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We show that for any integer $N$, there are only finitely many cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$, with $n$ varying, whose conductor is $N$ and whose weights are in the interval $\{0,1,...,23\}$. More generally, we define a simple sequence $(r(w))_{w \geq 0}$ such that for any integer $w$, any number field $E$ whose root-discriminant is less than $r(w)$, and any ideal $N$ in the ring of integers of $E$, there are only finitely many cuspidal algebraic automorphic representations of general linear groups over $E$ whose conductor is $N$ and whose weights are in the interval $\{0,1,...,w\}$. Assuming a version of GRH, we also show that we may replace $r(w)$ with $8 \pi e^{\gamma-H_w}$ in this statement, where $\gamma$ is Euler's constant and $H_w$ the $w$-th harmonic number. The proofs are based on some new positivity properties of certain real quadratic forms which occur in the study of the Weil explicit formula for Rankin-Selberg $L$-functions. Both the effectiveness and the optimality of the methods are discussed., Comment: 30 pages, 1 table

Quadratic form reduction and lattice reduction are fundamental tools in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra-Lenstra-Lov\'asz reduction algorithm (so-called LLL) has been improved in many ways through the past decades and remains one of the central methods used for reducing integral lattice basis. In particular, its floating-point variants-where the rational arithmetic required by Gram-Schmidt orthogonalization is replaced by floating-point arithmetic-are now the fastest known. However, the systematic study of the reduction theory of real quadratic forms or, more generally, of real lattices is not widely represented in the literature. When the problem arises, the lattice is usually replaced by an integral approximation of (a multiple of) the original lattice, which is then reduced. While practically useful and proven in some special cases, this method doesn't offer any guarantee of success in general. In this work, we present an adaptive-precision version of a generalized LLL algorithm that covers this case in all generality. In particular, we replace floating-point arithmetic by Interval Arithmetic to certify the behavior of the algorithm. We conclude by giving a typical application of the result in algebraic number theory for the reduction of ideal lattices in number fields., Comment: 23 pages

This article deals with universal deformations of dihedral representations with a particular focus on the question when the universal deformation is dihedral. Results are obtained in three settings: (1) representation theory, (2) algebraic number theory, (3) modularity. As to (1), we prove that the universal deformation is dihedral if all infinitesimal deformations are dihedral. Concerning (2) in the setting of Galois representations of number fields, we give sufficient conditions to ensure that the universal deformation relatively unramified outside a finite set of primes is dihedral, and discuss in how far these conditions are necessary. As side-results, we obtain cases of the unramified Fontaine-Mazur conjecture, and in many cases positively answer a question of Greenberg and Coleman on the splitting behaviour at p of p-adic Galois representations attached to newforms. As to (3), we prove a modularity theorem of the form `R=T' for parallel weight one Hilbert modular forms for cases when the minimal universal deformation is dihedral., 43 pages; minor corrections and improvements following referee's comments

Let O be the ring of S-integers in a number field k. We prove that if the group of units O^* is infinite then every matrix in $\Gamma$ = SL_2(O) is a product of at most 9 elementary matrices. This completes a long line of research in this direction. As a consequence, we obtain that $\Gamma$ is boundedly generated as an abstract group., Comment: Final version - to appear in `Algebra and Number Theory'

In the recent PhD thesis of Bouw, an algorithm is examined that computes the group structure of the principal units of a p-adic number field completion. In the same thesis, this algorithm is used to compute Hilbert norm residue symbols. In the present paper, we will demonstrate two other applications. The first application is the computation of an important invariant of number field completions, called ibeta. The algorithm that computes ibeta is deterministic and runs in polynomial time. The second application uses Hilbert norm residue symbols and yields a probabilistic algorithm that computes the m-th power residue symbol (a/b)m in arbitrary number fields K. This probabilistic algorithm relies on LLL-reduction and sampling of nearprimes. Using heuristics, we analyse its complexity to be polynomial expected time in n = [K: ℚ], log |ΔK| and the input bit size - a tentative conclusion corroborated by timing experiments. An implementation of the algorithm in Magma will be available at https://github.com/kodebro/powerresiduesymbol.

Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs and their minimal defining algebraic number field. Testing or sharpening these conjectures requires that the SICs are expressed exactly, rather than as numerical approximations. While many exact solutions of SICs have been constructed previously using Gr\"obner bases, this method has probably been taken as far as is possible with current computer technology (except in special cases where there are additional symmetries). Here we describe a method for converting high-precision numerical solutions into exact ones using an integer relation algorithm in conjunction with the Galois symmetries of a SIC. Using this method we have calculated 69 new exact solutions, including 9 new dimensions where previously only numerical solutions were known, which more than triples the number of known exact solutions. In some cases the solutions require number fields with degrees as high as 12,288. We use these solutions to confirm that they obey the number-theoretic conjectures and we address two questions suggested by the previous work., Comment: 22 pages + 19 page appendix with many data tables. v2: published version

For a finite abelian p-group A of rank d = dim A/pA, let A := be its (logarithmic) mean exponent. We study the behavior of themean exponent of p-class groups in pro-p towers L/K of number fields. Via a combination of results from analytic and algebraic number theory, we construct infinite tamely ramified pro-p towers in which the mean exponent of p-class groups remains bounded. Several explicit examples are given with p = 2. Turning to group theory, we introduce an invariant attached to a finitely generated pro-p group G; when G = Gal(L/K), where L is the Hilbert p-class field tower of a number field K, measures the asymptotic behavior of the mean exponent of p-class groups inside L/K. We compare and contrast the behavior of this invariant in analytic versus non-analytic groups. We exploit the interplay of group-theoretical and number-theoretical perspectives on this invariant and explore some open questions that arise as a result, which may be of independent interest in group theory.

Recently the authors showed that the algebraic integers of the form $-m+\zeta_k$ are bases of a canonical number system of $\mathbb{Z}[\zeta_k]$ provided $m\geq \phi(k)+1$, where $\zeta_k$ denotes a $k$-th primitive root of unity and $\phi$ is Euler's totient function. In this paper we are interested in the questions whether two bases $-m+\zeta_k$ and $-n+\zeta_k$ are multiplicatively independent. We show the multiplicative independence in case that $0 1$., Comment: 9 pages

It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric familiy of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $\xi$ of the polynomial $P_t(x)=x^4-tx^3-6x^2+tx+1$, assuming that $t>0$, $t\neq 3$ and $t^2+16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal index and all elements of minimal index in all fields in this family.

Using class field theory I give an example of a function field of genus 4 with class number one over the finite field F 2 . In a previous paper (see [2, Section 2] ) a proof of the nonexistence of such a function field is given. This counterexample shows that the proof in [2] is wrong and so the list of algebraic function fields with class number one given in [2] should admit one more example.

By Northcott's Theorem there are only finitely many algebraic points in affine $n$-space of fixed degree over a given number field and of height at most $X$. For large $X$ the asymptotics of these cardinalities have been investigated by Schanuel, Schmidt, Gao, Masser and Vaaler, and the author. In this paper we study the case where the coordinates of the points are restricted to algebraic integers, and we derive the analogues of Schanuel's, Schmidt's, Gao's and the author's results., to appear in Int. Math. Res. Notices

We study the distribution of algebraic integers with prescribed factorization properties in short intervals and prove that for a large class of such numbers from a fixed algebraic number field $K$ with a non-trivial class group, every interval of the form $(x, x+x^{\theta})$ with a fixed $\theta >1/2$ contains absolute value of the norm of such algebraic integer provided $x\geq x_0$. The constant $x_0$ effectively depends on $K$ and $\theta$.

Genus theory belongs to algebraic number theory and, in very broad terms, deals with the part of the ideal class group of a number field that is ‘easy to compute’. Historically, the importance of genus theory stems from the fact that it was the essential algebraic ingredient in the derivation of the classical reciprocity laws – from Gaus’s second proof over Kummer’s contributions up to Takagi’s ‘general’ reciprocity law for p-th power residues. The central theorem in genus theory is the principal genus theorem, which is hard to describe in just one sentence – readers not familiar with genus theory might want to glance into Section 2 before reading on. In modern terms, the principal genus theorem for abelian extensions k/Q describes the splitting of prime ideals of k in the genus field kgen of k, which by definition is the maximal unramified extension of k that is abelian over Q. In this note we outline the development of the principal genus theorem from its conception in the context of binary quadratic forms by Gaus (with hindsight, traces of genus theory can be found in the work of Euler on quadratic forms and idoneal numbers) to its modern formulation within the framework of class field theory. It is somewhat remarkable that, although the theorem itself is classical, the name ‘principal ideal theorem’ (‘Hauptgeschlechtssatz’ in German) was not used in the 19th century, and it seems that it was coined by Hasse in his Bericht [28] and adopted immediately by the abstract algebra group around Noether. It is even more remarkable that Gaus doesn’t bother to formulate the principal genus theorem except in passing: after observing in [25, §247] that duplicated classes (classes of forms composed with themselves) lie in the principal genus, the converse (namely the principal genus theorem) is stated for the first time in §261: si itaque omnes classes generis principalis ex duplicatione alicuius classis provenire possunt (quod revera semper locum habere in sequentibus demonstrabitur), . . . 1 The actual statement of the principal genus theorem is somewhat hidden in [25, §286], where Gaus formulates the following Problem. Given a binary form F = (A,B,C) of determinant D belonging to a principal genus: to find a binary form f from whose duplication we get the form F . It strikes us as odd that Gaus didn’t formulate this central result properly; yet he knew exactly what he was doing [25, §287]

For some algebraic number fields $F$ with only one fundamental unit, we give a lower bound of the extension degree of the ray class field of conductor a rational prime $p$ over the Hilbert class field of $F$.

Ono's number $p_D$ and the class number $h_D$, associated to an imaginary quadratic field with discriminant $-D$, are closely connected. For example, Frobenius-Rabinowitsch Theorem asserts that $p_D = 1$ if and only if $h_D = 1$. In 1986, T. Ono raised a problem whether the inequality $h_D \leq 2^{p_D}$ holds. However, in our previous paper [8], we saw that there are infinitely many $D$ such that the inequality does not hold. In this paper we give a modification to the inequality $h_D \leq 2^{p_D}$. We also discuss lower and upper bounds for Ono's number $p_D$.

We are interested in the arithmetic of ring spectra. To make sense of this we must work with structured ring spectra, such as S-algebras [EKMM], symmetric ring spectra [HSS] or Γ-rings [Ly]. We will refer to these as Salgebras. The commutative objects are then commutative S-algebras. The category of rings is embedded in the category of S-algebras by the Eilenberg– MacLane functor R →HR. We may therefore view an S-algebra as a generalization of a ring in the algebraic sense. The added flexibility of S-algebras provides room for new examples and constructions, which may eventually also shed light upon the category of rings itself. In algebraic number theory the arithmetic of the ring of integers in a number field is largely captured by its Picard group, its unit group and its Brauer group. These are