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We determine those imaginary quadratic number fields whose 2-class groups have rank 3 and 4-rank ≤2 and such that the 2-class groups of their Hilbert 2-class fields have rank 2. This result, along with previous work, gives a complete determination of all complex quadratic number fields for which the 2-class groups of their 2-class fields have rank at most 2.

A family $\mathcal{F}$ of elliptic curves defined over number fields is said to be typically bounded in torsion if the torsion subgroups $E(F)[$tors$]$ of those elliptic curves $E_{/F}\in \mathcal{F}$ can be made uniformly bounded after removing from $\mathcal{F}$ those whose number field degrees lie in a subset of $\mathbb{Z}^+$ with arbitrarily small upper density. For every number field $F$, we prove unconditionally that the family $\mathcal{E}_F$ of elliptic curves defined over number fields and with $F$-rational $j$-invariant is typically bounded in torsion. For any integer $d\in\mathbb{Z}^+$, we also strengthen a result on typically bounding torsion for the family $\mathcal{E}_d$ of elliptic curves defined over number fields and with degree $d$ $j$-invariant., 17 pages, to appear in Journal of Number Theory

Let K / F be a finite Galois extension of number fields and let σ be an absolutely irreducible, self-dual, complex valued representation of Gal ( K / F ) . Let p be an odd prime and consider two elliptic curves E 1 , E 2 defined over Q with good, ordinary reduction at primes above p and equivalent mod-p Galois representations. In this article, we study the variation of the parity of the multiplicities of σ in the representation space associated to the p ∞ -Selmer groups of E 1 and E 2 over K. We also compare the root numbers for the twists of E 1 and E 2 over F by σ and show that the p-parity conjecture holds for the twist of E 1 / F by σ if and only if it holds for the twist of E 2 / F by σ. We also express Mazur-Rubin-Nekovař's arithmetic local constants in terms of certain local Iwasawa invariants.

We develop efficient algorithms for determining the number of global cubic fields of given discriminant. They are based on earlier work of Hasse for cubic number fields. Our new ingredients allow to enlarge the range of computations considerably. For explicit calculations we used Magma.

Let f n ( k ) be the number of subrings of index k in Z n . We show that results of Brakenhoff imply a lower bound for the asymptotic growth of subrings in Z n , improving upon lower bounds given by Kaplan, Marcinek, and Takloo-Bighash. Further, we prove two new lower bounds for f n ( p e ) when e ≥ n − 1 . Using these bounds, we study the divergence of the subring zeta function of Z n and its local factors. Lastly, we apply these results to the problem of counting orders in a number field.

The p-rationality of a totally real abelian number field can be checked from the values L ( 2 − p , χ ) of Dirichlet L-functions, for all non-principal even Dirichlet characters associated to the field. Using this criterion and the properties of the generalized Bernoulli numbers, we study the p-rationality of Q ( ζ 2 l + 1 ) + , the maximal real subfield of Q ( ζ 2 l + 1 ) , for Sophie Germain primes l and odd primes p that are primitive roots modulo l. We prove that Q ( ζ 2 l + 1 ) + is p-rational for such pairs if p 4 l . We also prove that the Siegel's heuristics on the equidistribution of the residues of Bernoulli numbers modulo p imply that Q ( ζ 2 l + 1 ) + is p-rational for all but finitely many p that are primitive roots modulo l.

Let f be a polynomial in n variables over some number field and Z a subscheme of affine n-space. The notion of motivic oscillation index of f at Z was initiated by Cluckers in [7] and Cluckers-Mustaţǎ-Nguyen in [12] . In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta function of f has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of l-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if f is a polynomial ‘of Thom-Sebastiani type’ with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of A − D − E type. In an appendix, we answer affirmatively a recent question of Cluckers-Mustaţǎ-Nguyen in [12] on poles of maximal order of twisted Igusa's local zeta functions.

A number field is called a Polya field if the module of integer-valued polynomials over its ring of integers has a regular basis. Let L be a field which is a compositum of two quadratic Polya fields. Some questions were raised on Polyaness of L in [7] . Part was solved in [3] and [8] . Here we develop a general strategy allowing us to treat the remaining cases but also to find all these previous results.

In all available papers, on power integral bases of any pure sextic number fields K generated by a complex root α of a monic irreducible polynomial f ( x ) = x 6 − m ∈ Z [ x ] , it was assumed that the rational integer m ≠ ∓ 1 is square free. In this paper, we investigate the monogeneity of any pure sextic number field, where the condition m is a square free rational integer is omitted. We start by calculating an integral basis of Z K ; the ring of integers of K. In particular, we characterize when Z K = Z [ α ] , that is when Z K is monogenic and generated by α. We give sufficient conditions on m, which warranty that K is not monogenic. We finish the paper by investigating the case, where m = e 5 and e ≠ ∓ 1 is a square free rational integer.

Suppose an elliptic curve E / Q has good supersingular reduction at a prime p, and satisfies 1 + p − # E ( Z / p Z ) = 0 , which is always true if p is supersingular and p > 3 by the Hasse inequality. Expanding Kobayashi's plus/minus-local points over the cyclotomic Z p -extension of Q p , where π is any element in C p with positive p-adic valuation, we construct doubly indexed plus/minus local points over Q p ( π p n − m , μ p n ) (which we will call local Iwasawa Kummer extensions) as n , m vary. Then, we explore various possible plus/minus Selmer groups over Z p -extensions of number fields over which p is ramified, or doubly indexed plus/minus Selmer groups over Iwasawa Kummer extensions, and study or speculate their properties using the above plus/minus local points. Finally, again using the above points, we study the following: Suppose k is a ramified quadratic extension of Q p . We find a bound of [ E ( m k ( μ p n ) ) : E + ( m k ( μ p n ) ) + E − ( m k ( μ p n ) ) ] as n varies.

By Dirichlet's Unit Theorem, under the log embedding the units in the ring of integers of a number field form a lattice, called the log-unit lattice. We investigate the geometry of these lattices when the number field is a biquadratic or cyclic cubic extension of $\mathbb{Q}$. In the biquadratic case, we determine when the log-unit lattice is orthogonal. In the cyclic cubic case, we show that the log-unit lattice is always equilateral triangular., Comment: 14 pages

Let O be the ring of integers of a number field, and let n ≥ 3 . This paper studies bi-interpretability of the ring of integers Z with the special linear group SL n ( O ) , the general linear group GL n ( O ) and the subgroup T n ( O ) of GL n ( O ) consisting of all the uppertriangular matrices. For each of these groups we provide a complete characterization of arbitrary models of their complete first-order theories.

A basic problem in transcendental number theory is to determine the arithmetic properties of analytic functions of the form f ( z ) = ∑ k = 0 ∞ a k z k where the coefficients a k ∈ K belong to an algebraic number field. In particular, one of the most basic problems is to determine if f ( α ) is algebraic or transcendental for non-zero algebraic arguments α. For example, if f ( z ) is a transcendental Mahler function, then under generic conditions f ( α ) is transcendental for all non-zero algebraic numbers with | α | 1 . Also, if f ( z ) is an E-function, then there exist algorithms which completely determine the arithmetic properties of f ( n ) ( α ) for non-zero algebraic numbers α. In contrast to these and other algorithmic results, we construct three functions f ( z ) , g ( z ) , and h ( z ) with computable rational coefficients for which no algorithms exist that determine if f ( n ) ∈ Q , g ( n ) ( 1 ) ∈ Q , or ∫ 0 1 h ( z ) z n d z ∈ Q for integral n ≥ 0 . Our results are an application of an undecidable variant of the Collatz Problem due to Lehtonen [9] .

It is an open problem whether the Iwasawa λ-invariants of the Z p -extensions of a fixed number field are bounded. Using the class-field theoretic tool of logarithmic class groups, we obtain bounds for the λ-invariants of Z p -extensions of suitable field extensions of imaginary quadratic number fields. We also prove the Gross-Kuz'min Conjecture for certain families of non-cyclotomic Z p -extensions.

The present paper is devoted to methods of the automatic verifying of the Kronecker-Vahlen theorem on the shortest length of the Euclidean algorithm in algebraic number fields with the infinite group of units. We provide explicit methods to prove that the Kronecker-Vahlen theorem fails in certain algebraic number field. In particular, we give a complete solution of the problem for quadratic norm-Euclidean number fields.

Let $ K $ be a number field, $ S $ a finite set of places of $ K $, and $ \mathcal{O}_S $ be the ring of $ S $-integers. Moreover, let $$ G_n^{(0)} Z^d + \cdots + G_n^{(d-1)} Z + G_n^{(d)} $$ be a polynomial in $ Z $ having simple linear recurrences of integers evaluated at $ n $ as coefficients. Assuming some technical conditions we give a description of the zeros $ (n,z) \in \mathbb{N} \times \mathcal{O}_S $ of the above polynomial. We also give a result in the spirit of Hilbert irreducibility for such polynomials., Comment: 13 pages

Ellenberg proved that the abc conjecture would follow if this conjecture were known for sums $a+b=c$ such that $D\mid abc$ for some integer~$D$. Mochizuki proved a theorem with an opposite restriction, that the full abc conjecture would follow if it were known for abc sums that are not highly divisible. We prove both theorems for general number fields., 22 pages

We consider a G-function F ( z ) = ∑ k = 0 ∞ A k z k ∈ K 〚 z 〛 , where K is a number field, of radius of convergence R and annihilated by the G-operator L ∈ K ( z ) [ d / d z ] , and a parameter β ∈ Q ∖ Z ⩽ 0 . We define a family of G-functions F β , n [ s ] ( z ) = ∑ k = 0 ∞ A k ( k + β + n ) s z k + n indexed by the integers s and n. Fix α ∈ K ⁎ ∩ D ( 0 , R ) . Let Φ α , β , S be the K -vector space generated by the values F β , n [ s ] ( α ) , n ∈ N , 0 ⩽ s ⩽ S . We show that there exist some positive constants u K , F , β and v F , β such that u K , F , β log ⁡ ( S ) ⩽ dim K ⁡ Φ α , β , S ⩽ v F , β S . This generalizes a previous theorem of Fischler and Rivoal (2017), which is the case β = 0 . Our proof is an adaptation of their article [6] , making use of the Andre-Chudnovsky-Katz Theorem on the structure of the G-operators and of the saddle point method. Moreover, we rely on Dwork and Andre's quantitative results on the size of G-operators to obtain an explicit formula for the constant u K , F , β , which was not given in [6] in the case β = 0 .

Let F be a number field and let i ≥ 2 be an integer. In this paper, we study the positive etale wild kernel WK 2 i − 2 et , + F , which is the twisted analogue of the 2-primary part of the narrow class group. If E / F is a Galois extension of number fields with Galois group G, we prove a genus formula relating the order of the groups ( WK 2 i − 2 et , + E ) G and WK 2 i − 2 et , + F .

Let E 1 and E 2 be elliptic curves defined over a number field K. Suppose that for all but finitely many primes l, and for all finite extension fields L / K , dim F l ⁡ Sel l ( L , E 1 ) = dim F l ⁡ Sel l ( L , E 2 ) . We prove that E 1 and E 2 are isogenous over K.

We present the basics of the local theory, which arises from global Rankin-Selberg integrals, attached to pairs of irreducible globally generic cuspidal automorphic representations of the quasi-split unitary group U 2 n + 1 and Res E / F GL m , for a quadratic extension of number fields E / F , when m > n .

Let l be a rational prime number. Assuming the Gross-Kuz'min conjecture along a Z l -extension K ∞ of a number field K, we show that there exist integers μ ˜ , λ ˜ and ν ˜ such that the exponent e ˜ n of the order l e ˜ n of the logarithmic class group C l ˜ n for the n-th layer K n of K ∞ is given by e ˜ n = μ ˜ l n + λ ˜ n + ν ˜ , for n big enough. We show some relations between the classical invariants μ and λ, and their logarithmic counterparts μ ˜ and λ ˜ for some class of Z l -extensions. Additionally, we provide numerical examples for the cyclotomic and the non-cyclotomic case.

Let K be a number field of degree k ≥ 8 . In this article, we investigate a certain density theorem for zeros of the Dedekind zeta function attached to K. In this context, our theorem strengthens a result of Heath-Brown. Further, we investigate an upper-bound of the error term for an ideal counting function attached to K. When K is a cyclotomic field, we prove a stronger upper-bound for this error term.

We study the arithmetic (geometric) progressions in the $x$-coordinates of quadratic points on smooth projective planar curves defined over a number field $k$. Unless the curve is hyperelliptic, we prove that these progressions must be finite. We, moreover, show that the arithmetic gonality of the curve determines the infinitude of these progressions in the set of $\overline{k}$-points with field of definition of degree at most $n$, $n\ge 3$.

Text We construct, for specific imaginary quadratic number fields with class number 1, an arithmetic site of Connes-Consani type. The main difficulty here is that their constructions and part of their results strongly rely on the natural order existing on real numbers which is compatible with basic arithmetic operations. We first define what we call an arithmetic site for such number fields, we then calculate the points of those arithmetic sites and we express them in terms of the adele class space considered by Connes to give a spectral interpretation of zeroes of Hecke L functions of number fields. We get that the points of our arithmetic site are related not only to the zeroes of the Dedekind zeta function of the number field considered but also to other Hecke L functions involving non-trivial characters at the archimedean place. We then study the relation between the spectrum of the ring of integers of the number field and the arithmetic site. An appendix by Alain Connes shows that the compatibility with the Frobenius yields the correct notion of morphisms of semirings in this context. Video For a video summary of this paper, please visit https://www.youtube.com/watch?v=lWF9gbfvcYY&feature=youtu.be .

In 2016, Jang and Kim stated a conjecture about the norms of indecomposable integers in real quadratic number fields Q ( D ) where D > 1 is a squarefree integer. Their conjecture was later disproved by Kala for D ≡ 2 mod 4 . We investigate such indecomposable integers in greater detail. In particular, we find the minimal D in each congruence class D ≡ 1 , 2 , 3 mod 4 that provides a counterexample to the Jang-Kim Conjecture; provide infinite families of such counterexamples; and state a refined version of the Jang-Kim Conjecture. Lastly, we prove a slightly weaker version of our refined conjecture that is of the correct order of magnitude, showing the Jang-Kim Conjecture is only wrong by at most O ( D ) .

We define X-base Fibonacci-Wieferich primes, which generalize Wieferich primes, where X is a finite set of algebraic numbers. We show that there are infinitely many non-X-base Fibonacci-Wieferich primes, assuming the abc-conjecture of Masser-Oesterle-Szpiro for number fields. We also provide a new conjecture concerning the rank of the free part of the abelian group generated by all elements in X and give some heuristics that support the conjecture.

A post-critically finite rational map ϕ of prime degree p and a base point β yield a tower of finitely ramified iterated extensions of number fields, and sometimes provide an arboreal Galois representation with a p-adic Lie image. In this paper, we take ϕ to be the monic Chebyshev polynomial x 2 − 2 , and we examine the size of the 2-part of the ideal class group of extensions in the resulting tower. In some cases, we prove an analogue of Greenberg's conjecture from Iwasawa theory. A key tool is a general theorem on p-indivisibility of class numbers of relative cyclic extensions of degree p 2 .

Let { n P + Q } n ≥ 0 be a sequence of points on an elliptic curve defined over a number field K. In this paper, we study the denominators of the x-coordinates of this sequence. We prove that, if Q is a torsion point of prime order, then for n large enough there always exists a primitive divisor. Later on, we show the link between the study of the primitive divisors and a Lang-Trotter conjecture. Indeed, given two points P and Q on the elliptic curve, we prove a lower bound for the number of primes p such that P is in the orbit of Q modulo p.

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K × . We consider Kummer extensions of G of the form K ( ζ 2 m , G 2 n ) / K ( ζ 2 m ) , where n ⩽ m . In the paper by Debry and Perucca (2016) [1] , the degrees of those extensions have been evaluated in terms of divisibility parameters over K ( ζ 4 ) . We prove how properties of G over K explicitly determine the divisibility parameters over K ( ζ 4 ) . This result yields a clear computational advantage, since no field extension is required.

A number field with trivial Polya group [2] is called a Polya field. We define “relative Polya group Po ( L / K ) ” for L / K a finite extension of number fields, generalizing the Polya group. Using cohomological tools in [1] , we compute some relative Polya groups. As a consequence, we generalize Leriche's results in [17] and prove the triviality of relative Polya group for the Hilbert class field of K. Then we generalize our previous results [19] on Polya S 3 -extensions of Q to dihedral extensions of Q of order 2l, for l an odd prime. We also improve Leriche's upper bound in [16] on the number of ramified primes in Polya D l -extensions of Q and prove that for a real (resp. imaginary) Polya D l -extension of Q at most 4 (resp. 2) primes ramify.

H. W. Lenstra \cite{lenstra} introduced the notion of an Euclidean ideal class, which is a generalization of norm-Euclidean ideals in number fields. Later, families of number fields of small degree were obtained with an Euclidean ideal class (for instance, in \cite{hester1} and \cite{cathy}). In this paper, we construct certain new families of biquadratic number fields having a non-principal Euclidean ideal class and this extends the previously known families given by H. Graves \cite{hester1} and C. Hsu \cite{cathy}., Comment: It is a preliminary version. Comments and suggestions are welcome

We present a new algorithm for computing the endomorphism ring of an ordinary abelian surface over a finite field which is subexponential and generalizes an algorithm of Bisson and Sutherland for elliptic curves. The correctness of this algorithm only requires the heuristic assumptions required by the algorithm of Biasse and Fieker [2] which computes the class group of an order in a number field in subexponential time. Thus we avoid the multiple heuristic assumptions on isogeny graphs and polarized class groups which were previously required. The output of the algorithm is an ideal in the maximal totally real subfield of the endomorphism algebra, generalizing the elliptic curve case.

Let p be a prime number. In [15] , we studied the class semigroup of the ring of integers of the cyclotomic Z p -extension of the rationals. In this paper, we generalize the result to some Z S -extensions of number fields. Moreover, we investigate the relation between the class semigroup and Iwasawa invariants.

Let E / F be a quadratic extension of number fields and D a quaternion algebra over F which E embeds. Flicker and Rallis conjectured that a cuspidal automorphic representation π of GL ( 2 n , E ) is the unstable base change lift of a generic cuspidal automorphic representation σ of the quasi-split unitary group U ( 2 n ) if and only if it is distinguished by GL ( 2 n , F ) . We conjecture that π is distinguished by GL ( n , D ) if and only if σ is generic with respect to certain non-degenerate character attached to D. We use the relative trace formula to prove the n = 1 case of our conjecture.

We determine all degree 7 number fields where the absolute value of the discriminant is ≤ 2 ⋅ 10 8 . To accomplish this large-scale computation, we distributed the computation over many servers on the internet through the BOINC network.

Motivated by analogies with basic density theorems in analytic number theory, we introduce a notion (and variations) of the homological density of one space in another. We use Weil's number field/function field analogy to predict coincidences for limiting homological densities of various sequences Z n ( d 1 , … , d m ) ( X ) of spaces of 0-cycles on manifolds X. The main theorem in this paper is that these topological predictions, which seem strange from a purely topological viewpoint, are indeed true. One obstacle to proving such a theorem is the combinatorial complexity of all possible “collisions” of points. This problem does not arise in the simplest (and classical) case ( m , n ) = ( 1 , 2 ) of configuration spaces. To overcome this obstacle we apply the Bjorner–Wachs theory of lexicographic shellability from algebraic combinatorics.

Tate first proposed a method to determine $K_2\mathcal{O}_F,$ the tame kernel of $F,$ and gave the concrete computations for some special quadratic fields with small discriminant. After that, many examples for quadratic fields with larger discriminants are given, and similar works also have been done for cubic fields and for some special quartic fields with discriminants not large. In the present paper, we investigate the case of more general imaginary cyclic quartic field $F=\mathbb{Q}\Big(\sqrt{-(D+B\sqrt{D})}\Big)$ with class number one and large discriminants. The key problem is how to decrease the huge theoretical bound appearing in the computation to a manageable one and the main difficulty is how to deal with the large-scale data emerged in the process of computation. To solve this problem we have established a general architecture for the computation, in particular we have done the works: (1) the PARI's functions are invoked in C++ codes; (2) the parallel programming approach is used in C++ codes; (3) in the design of algorithms and codes, the object-oriented viewpoint is used, so an extensible program is obtained. As an application of our program, we prove that $K_2\mathcal{O}_F$ is trivial in the following three cases: $B=1,D=2$ or $B=2, D=13$ or $B=2, D=29.$ In the last case, the discriminant of $F$ is 24389, hence, we can claim that our architecture also works for the computation of the tame kernel of a number field with discriminant less than 25000.

For a finite separable field extension K/k, all subfields can be obtained by intersecting so-called principal subfields of K/k. In this work we present a way to quickly compute these intersections. If the number of subfields is high, then this leads to faster run times and an improved complexity., Comment: Slides available at: http://www.math.fsu.edu/~hoeij/2017/Presentation.pdf

For a nice algebraic variety $X$ over a number field $F$, one of the central problems of Diophantine Geometry is to locate precisely the set $X(F)$ inside $X(\A_F)$, where $\A_F$ denotes the ring of ad\`eles of $F$. One approach to this problem is provided by the finite descent obstruction, which is defined to be the set of adelic points which can be lifted to twists of torsors for (certain) finite \'etale group schemes over $F$ on $X$. More recently, Kim proposed an iterative construction of another subset of $X(\A_F)$ which contains the set of rational points. In this paper, we compare the two constructions. Our main result shows that the two approaches are essentially equivalent., Comment: New version; improved exposition; corrected a few typographical errors. New Appendix. 20 pages

Let E be an elliptic curve over the rational field Q. Let K be a quadratic extension of Q. We show that (under mild conditions on E) for every r > 0 , there are infinitely many quadratic twists E d / Q of E / Q such that .

We obtain a Voronoi–Oppenheim summation formula for divisor functions of totally real number fields. This generalizes a formula proved by Oppenheim in 1927. We use a similar method to the one developed by Beineke and Bump in order to prove the classical Oppenheim summation using a certain Eisenstein series and representation theory. Our formula has a simple formulation for real quadratic number fields.

In this paper, a simple and elementary method is given for deriving estimates of sums of arithmetic functions in F q [ t ] . The method is the function field analogue of a result first proved by Stefan A. Burr in 1973 in the number field case. A novelty of this paper is that we are able to extend Burr's result, in the function field context, and obtain secondary main terms for the appropriate sums involving the divisor functions d r ( f ) with an error term that improves the one given by Burr.

Let $X$ be a smooth projective variety with a fibration into varieties that either satisfy a condition on representability of zero-cycles or that are torsors under an abelian variety. We study the classes in the Brauer group that never obstruct the Hasse principle for $X$. We prove that if the generic fiber has a zero-cycle of degree $d$ over the generic point, then the Brauer classes whose orders are prime to $d$ do not play a role in the Brauer--Manin obstruction. As a result we show that the odd torsion Brauer classes never obstruct the Hasse principle for del Pezzo surfaces of degree 2, certain K3 surfaces, and Kummer varieties., 9 pages, minor changes in text and presentation

We introduce mod $l$ Milnor invariants of a Galois element associated to Ihara's Galois representation on the pro-$l$ fundamental group of a punctured projective line ($l$ being a prime number), as arithmetic analogues of Milnor invariants of a pure braid. We then show that triple quadratic (resp. cubic) residue symbols of primes in the rational (resp. Eisenstein) number field are expressed by mod $2$ (resp. mod $3$) triple Milnor invariants of Frobenius elements. For this, we introduce dilogarithmic mod $l$ Heisenberg ramified covering ${\cal D}^{(l)}$ of $\mathbb{P}^1$, which may be regarded as a higher analog of the dilogarithmic function, for the gerbe associated to the mod $l$ Heisenberg group, and we study the monodromy transformations of certain functions on ${\cal D}^{(l)}$ along the pro-$l$ longitudes of Frobenius elements for $l=2,3$., Comment: 33 pages, 1 figure

In 2016 Astudillo, Diaz y Diaz and Friedman published sharp lower bounds for regulators of number fields of all signatures up to degree seven, except for fields of degree seven having five real places. We deal with this signature, proving that the field with the first discriminant has minimal regulator. The new element in the proof is an extension of Pohst's geometric method from the totally real case to fields having one complex place.

We describe the powers of irreducible polynomials occurring as characteristic polynomials of automorphisms of even unimodular lattices over number fields. This generalizes results of Gross & McMullen and Bayer-Fluckiger & Taelman.

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$. Let $F(s)= \sum_{n=1}^\infty f(n)n^{-s}$ be the associated Dirichlet series and $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ be the truncated Dirichlet series. In this setting, we obtain new Hal\'asz-type results for the logarithmic mean value of $f$. More precisely, we prove estimates for the sum $\sum_{n=1}^x f(n)/n$ in terms of the size of $|F(1+1/\log x)|$ and show that these estimates are sharp. As a consequence of our mean value estimates, we establish non-trivial zero-free regions for these partial sums $F_N(s)$. In particular, we study the zero distribution of partial sums of the Dedekind zeta function of a number field $K$. More precisely, we give some improved results for the number of zeros up to height $T$ as well as new zero density results for the number of zeros up to height $T$, lying to the right of $\Re(s) =\sigma$, where $\sigma > 1/2$., Comment: 27 pages

Let p be a rational prime. We study Galois (co-)invariants of Iwasawa modules attached to Z p -extensions of number fields, by encoding the corresponding orders in so-called T-ranks. We show that the growth of T-ranks at small layers of a Z p -extension bounds the over-all growth. This allows for effective algorithms for checking whether the T-ranks of the layers in a Z p -extension remain bounded. We apply this theoretical tool for checking the boundedness of T-ranks in order to prove a conjecture of Gross for all number fields containing exactly two primes above p, and we verify this conjecture by computations for many cubic non-normal number fields with three such primes. Moreover, our method can also be used to verify numerically Leopoldt's Conjecture for K and p, provided that p is totally split in K. Finally, we study the problem of semi-simplicity.