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Denote by h = h (− p) the class number of the imaginary quadratic field ℚ (− p) with p prime. It is well known that h = 1 for p ∈ { 3 , 7 , 1 1 , 1 9 , 4 3 , 6 7 , 1 6 3 }. Recently, all the solutions of the Diophantine equation x 2 + p s = 4 y n with h = 1 were given by Chakraborty et al. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, Publ. Math. Debrecen 97(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation x 2 + p s = 2 r y n in unknown integers (x , y , s , r , n) , where s ≥ 0 , r ≥ 3 , n ≥ 3 , h ∈ { 1 , 2 , 3 } and gcd (x , y) = 1. To do this, we use the known results from the modularity of Galois representations associated with Frey–Hellegoaurch elliptic curves, the symplectic method and elementary methods of classical algebraic number theory. The aim of this paper is to extend the above results of Chakraborty et al. [ABSTRACT FROM AUTHOR]

In this paper, we generalize Ramanujan's sum to the ring of integers of an algebraic number field. We also obtain the orthogonality properties of Ramanujan's sum in the ring of integers. [ABSTRACT FROM AUTHOR]

For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov–Ribet method) of the fact that if G is a finitely generated and torsion-free multiplicative subgroup of a number field K having rank r , then the ratio between n r and the Kummer degree [ K (ζ n , G n) : K (ζ n) ] is bounded independently of n. We then apply this result to generalize to higher rank a theorem of Ziegler from 2006 about the multiplicative order of the reductions of algebraic integers (the multiplicative order must be in a given arithmetic progression, and an additional Frobenius condition may be considered). [ABSTRACT FROM AUTHOR]

For an algebraic number field K , let d K denote the discriminant of an algebraic number field K. It is well known that if K 1 , K 2 are algebraic number fields with coprime discriminants, then K 1 , K 2 are linearly disjoint over the field ℚ of rational numbers and d K 1 K 2 = d K 1 n 2 d K 2 n 1 , n i being the degree of K i over ℚ. In this paper, we prove that the converse of this result holds in relative extensions of algebraic number fields. We also give some more necessary and sufficient conditions for the analogue of the above equality to hold for algebraic number fields K 1 , K 2 linearly disjoint over K 1 ∩ K 2 . [ABSTRACT FROM AUTHOR]

We prove an equivariant version of Beilinson's conjecture on non-critical L -values of strongly modular abelian varieties over number fields. The proof builds on Beilinson's theorem on modular curves as well as a modularity result for endomorphism algebras. As an application, we prove a weak version of Zagier's conjecture on L (E , 2) and Deninger's conjecture on L (E , 3) for non-CM strongly modular Q -curves. [ABSTRACT FROM AUTHOR]

In this paper, we consider number fields containing quadratic subfields with minimal index that is large relative to the discriminant of the number field. We give new upper bounds on the minimal index, and construct families with the largest possible minimal index. [ABSTRACT FROM AUTHOR]

We study the Mahler measure of generators of a Galois extension with Galois group the full symmetric group. We prove that two classical constructions of generators give always algebraic numbers of big height. These results answer a question of Smyth and provide some evidence to a conjecture which asserts that the height of such a generator grows to infinity with the degree of the extension. [ABSTRACT FROM AUTHOR]

We establish new cases of quadratic number fields K unramified away from a prime p and ∞ whose absolute Galois group has no irreducible two-dimensional continuous Galois representations in GL2(F¯p). Our work builds on methods of Moon–Taguchi and Şengün and the usual analytic techniques of Odlyzko and Poitou where we note one of the new conditional cases arises via a correction of Poitou’s estimate. The results here seem optimal in that it seems these methods alone will yield no further cases either due to prohibitive computational issues or a failure of the analytic obstructions. [ABSTRACT FROM AUTHOR]

Let denote the ring of algebraic integers of an algebraic number field , where is a root of an irreducible trinomial belonging to . In this paper, we give necessary and sufficient conditions involving only for a given prime to divide the index of the subgroup in . In particular, we deduce necessary and sufficient conditions for to be equal to . [ABSTRACT FROM AUTHOR]

In this paper, another version of Schmidt's subspace theorem is given and the corresponding Wirsing type result is also considered. [ABSTRACT FROM AUTHOR]

We compute all possible numbers that are the Frobenius number of a numerical semigroup when multiplicity and genus are fixed. Moreover, we construct explicitly numerical semigroups in each case. [ABSTRACT FROM AUTHOR]

Let be an algebraic number field with in the ring of algebraic integers of and be the minimal polynomial of over the field of rational numbers. In 1977, Uchida proved that if and only if does not belong to for any maximal ideal of the polynomial ring (see [ Osaka J. Math. 14 (1977) 155-157]). In this paper, we apply the above result to obtain some necessary and sufficient conditions involving the coefficients of for to equal when is a trinomial of the type . In the particular case when , it is deduced that is an integral basis of if and only if either (i) and or (ii) divides and . [ABSTRACT FROM AUTHOR]

In this paper, we characterize whether the pure sextic fields with square-free integers m ≢ ±1( 9) have power integral bases or do not; if m ≡ 2, 3 (Mod 4), then have power integral bases. We prove this by determining relative integral bases of such fields with respect to their cubic and quadratic subfields. Based on the works of Kovács and Pethő, several examples on application of monogenic fields to CNS (Canonical Number System) are shown. [ABSTRACT FROM AUTHOR]

We prove that the dimension of the Hecke module generated by a certain eta-quotient is equal to the class number of an imaginary quadratic field. To do this, we relate the eta-quotient to the Hecke theta series attached to a ray class character of the imaginary quadratic field. [ABSTRACT FROM AUTHOR]

In this short paper, we investigate exceptional sets in the Waring-Goldbach problem for fifth powers. For example, we show that all but O(N213/214) integers subject to the necessary local conditions can be represented as the sum of 11 fifth powers of primes. This improves the previous result. [ABSTRACT FROM AUTHOR]

We give formulas for the class numbers of bicyclic biquadratic number fields containing an imaginary quadratic field of class number one. The class number is expressed as a finite sum in terms of the basic Jacobi elliptic functions, playing a similar role as the trigonometric sine in Dirichlet classical class number formula. [ABSTRACT FROM AUTHOR]

Let A be a sequence of natural numbers, rA(n) be the number of ways to represent n as a sum of consecutive elements in A, and MA(x) ≔ ∑n ≤ x rA(n). We give a new short proof of LeVeque's formula regarding MA(x) when A is an arithmetic progression, and then extend the proof to give asymptotic formulas for the case when A behaves almost like an arithmetic progression, and also when A is the set of primes in an arithmetic progression. [ABSTRACT FROM AUTHOR]

Let G be a finite group and S be a subset of G\{0}. We call S an additive basis of G if every element of G can be expressed as a sum over a non-empty subset in some order. Let cr(G) be the smallest integer t such that every subset of G\{0} of cardinality t forms an additive basis of G. In this paper, we determine cr(G) for all groups G of order pq, where p, q are primes. [ABSTRACT FROM AUTHOR]

In a recent work, Gun, Murty and Rath formulated the Strong Chowla-Milnor conjecture and defined the Strong Chowla-Milnor space. In this paper, we proved a non-trivial lower bound for the dimension of these spaces. We also obtained a conditional improvement of this lower bound and noted that an unconditional improvement of this lower bound will lead to irrationality of both ζ(k) and ζ(k)/πk for all odd positive integers k. Following Gun, Murty and Rath, we define generalized Zagier spaces Vp(K) for multiple zeta values over a number field K. We prove that the dimension of V4d+2(K) for d ≥ 1, is at least 2, assuming a conjecture of Gun, Murty and Rath. [ABSTRACT FROM AUTHOR]

In this paper, we study the notion of norm-Euclidean ideal class, which was defined by Lenstra [Euclidean ideal classes, Astérisque 61 (1979) 121-131]. Using a slight modification of an algorithm determining among other properties the Euclidean minimum described in [P. Lezowski, Computation of the Euclidean minimum of algebraic number fields, preprint (2011); http://hal.archives-ouvertes.fr/hal-00632997/en/], we give new examples of number fields with norm-Euclidean ideal classes. Extending the results of Cioffari [The Euclidean condition in pure cubic and complex quartic fields, Math. Comput. 33 (1979) 389-398], we also establish the complete list of pure cubic number fields which admit a norm-Euclidean ideal class. Finally, we show that ${\mathbf{Q}}(\sqrt{2}, \sqrt{35})$, which is known to admit a non-principal Euclidean ideal class, has no norm-Euclidean ideal class. [ABSTRACT FROM AUTHOR]

Let L/K be a Galois extension of number fields. The problem of counting the number of prime ideals 픭 of K with fixed Frobenius class in Gal(L/K) and norm satisfying a congruence condition is considered. We show that the square of the error term arising from the Chebotarëv Density Theorem for this problem is small "on average". The result may be viewed as a variation on the classical Barban-Davenport-Halberstam Theorem. [ABSTRACT FROM AUTHOR]

The Spiegelungssatz is an inequality between the 4-ranks of the narrow ideal class groups of the quadratic fields ${\mathbb{Q}}(\sqrt{D})$ and ${\mathbb{Q}}(\sqrt{-D})$. We provide a combinatorial proof of this inequality. Our interpretation gives an affine system of equations that allows to describe precisely some equality cases. [ABSTRACT FROM AUTHOR]

Given a fixed quadratic extension K of ℚ, we consider the distribution of elements in K of norm one (denoted $\mathscr{N}$). When K is an imaginary quadratic extension, $\mathscr{N}$ is naturally embedded in the unit circle in ℂ and we show that it is equidistributed with respect to inclusion as ordered by the absolute Weil height. By Hilbert's Theorem 90, an element in $\mathscr{N}$ can be written as $\alpha/\overline{\alpha}$ for some $\alpha \in \mathscr{O}_K$, which yields another ordering of $\mathscr{N}$ given by the minimal norm of the associated algebraic integers. When K is imaginary we also show that $\mathscr{N}$ is equidistributed in the unit circle under this norm ordering. When K is a real quadratic extension, we show that $\mathscr{N}$ is equidistributed with respect to norm, under the map β ↦ log|β|(mod log|ϵ2|) where ϵ is a fundamental unit of $\mathscr{O}_K$. [ABSTRACT FROM AUTHOR]

Let K be an algebraic number field with non-trivial class group G and let $\mathcal{O}_K$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let Fk (x) denote the number of non-zero principal ideals $a\mathcal{O}_K$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that Fk (x) behaves, for x → ∞, asymptotically like x(log x)-1+1/|G|(log log x)Nk(G). We study Nk (G) with new methods from Combinatorial Number Theory. [ABSTRACT FROM AUTHOR]

For a given conic bundle X over a curve C defined over 픽q, we count irreducible branch covers of C in X of degree d and height e ≫ 1. As a special case, we get the number of algebraic numbers of degree d and height e over the function field 픽q(C). [ABSTRACT FROM AUTHOR]

What is the probability for a number field of composite degree d to have a nontrivial subfield? As the reader might expect the answer heavily depends on the interpretation of probability. We show that if the fields are enumerated by the smallest height of their generators the probability is zero, at least if d > 6. This is in contrast to what one expects when the fields are enumerated by the discriminant. The main result of this paper is an estimate for the number of algebraic numbers of degree d = en and bounded height which generate a field that contains an unspecified subfield of degree e. If n > max{e2 + e, 10}, we get the correct asymptotics as the height tends to infinity. [ABSTRACT FROM AUTHOR]

Let K be a field of degree n over Q, the field of rational numbers, with ring of integers O. Fix an integer m > 1, say with $m=p_{1}^{r_{1}} \cdots p_{t}^{r_{t}}$ as a product of distinct prime powers, and let χ be a numerical character modulo m of conductor f(χ). Set ζm = exp(2πi/m) and let M be any ideal of O satisfying Tr M ⊆ mZ and N(1 + M) ⊆ 1 + f(χ)Z, where Tr and N are the trace and norm maps for K/Q. Then the Gauss sum \[ G_{{\it M}}(\chi)= \sum_{\alpha \in O{\!/\!}{\it M}^{*}} \chi(N\alpha) \zeta_{m}^{Tr \; \alpha} \] is well-defined. If in addition N(1 + M) ⊆ 1 + mZ, then the Kloosterman sums \[ R_{{\it M}}(\eta,z)= \sum_{\alpha\in O{\!/\!}{\it M}^{*} } \eta(N\alpha) \zeta_{m}^{Tr\,\alpha + z/N\alpha}, \quad (z \in {\bf Z}/m{\bf Z}^{*}) \] are well-defined for any numerical character η (mod m). The computation of GM(χ) and RM(η, z) is shown to reduce to their determination for m = pr, a power of a prime p, where M is comprised solely of ideals of K lying above p. In this setting we first explicitly determine GM(χ) for m = pr (r > 1) generalizing Mauclaire's classical result for K = Q. Relying on the recent evaluation of Kloosterman sums for prime powers in p-adic fields, we then proceed to compute the Kloosterman sums RM(η, z) here for m = pr (r > 1) when o(η) | p -1. This determination generalizes Salie's result in the classical case K = Q with o(η) = 1 or 2. [ABSTRACT FROM AUTHOR]

Recent work of Fili and the author examines an ultrametric version of the Mahler measure, denoted M∞(α) for an algebraic number α. We show that the computation of M∞(α) can be reduced to a certain search through a finite set. Although it is an open problem to record the points of this set in general, we provide some examples where it is reasonable to compute and our result can be used to determine M∞(α). [ABSTRACT FROM AUTHOR]

By examining the transformation formulae between unilateral series and bilateral ones derived from the bilateral Bailey lemma, we establish numerous identities of false theta functions, including most of the known ones discovered mainly by Rogers [40] and Ramanujan in his Lost Notebook [39]. [ABSTRACT FROM AUTHOR]

In this paper, we study the p-adic behavior of Weil numbers in the cyclotomic ℤp-extension of the pth cyclotomic field. We determine the characteristic ideal of the quotient of semi-local units by Weil numbers in terms of the characteristic ideals of some classical modules that appear in the Iwasawa theory. In a recent preprint [9] by Nguyen Quang Do and Nicolas, a generalization of this result to a semi-simple case was obtained. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve of conductor p. Given a cyclic subgroup C of order p in E[p], we construct a modular point PC on E, called self-point, as the image of (E,C) on X0(p) under the modular parametrization X0(p) → E. We prove that the point is of infinite order in the Mordell–Weil group of E over the field of definition of C. One can deduce a lower bound on the growth of the rank of the Mordell–Weil group in its PGL2(ℤp)-tower inside ℚ(E[p∞]). [ABSTRACT FROM AUTHOR]

The aim of this paper is to study certain family of elliptic curves $\{\mathscr X_H\}_H$ defined over a number field F arising from hyperplane sections of some cubic surface $\mathscr X/F$ associated to a cyclic cubic extension K/F. We show that each $\mathscr X_H$ admits a 3-isogeny ϕ over F and the dual Selmer group $S^{(\widehat{\phi})}(\widehat{\mathscr X_H}/F)$ is bounded by a kind of unit/class groups attached to K/F. This is proven via certain rational function on the elliptic curve $\mathscr X_H$ with nice property. We also prove that the Shafarevich–Tate group coincides with a class group of K as a special case. [ABSTRACT FROM AUTHOR]

Consider the congruence class Rm(a) = {a + im : i ∈ Z} and the infinite arithmetic progression Pm(a) = {a + im : i ∈ N0}. For positive integers a,b,c,d,m the sum of products set Rm(a)Rm(b) + Rm(c)Rm(d) consists of all integers of the form (a+im) · (b+jm)+(c+km)(d+ℓm) for some i,j,k,ℓ ∈ Z. It is proved that if gcd(a,b,c,d,m) = 1, then Rm(a)Rm(b) + Rm(c)Rm(d) is equal to the congruence class Rm(ab+cd), and that the sum of products set Pm(a)Pm(b)+Pm(c)Pm eventually coincides with the infinite arithmetic progression Pm(ab+cd). [ABSTRACT FROM AUTHOR]

If f is a non-constant polynomial with integer coefficients and q is an integer, we may regard f as a map from Z/qZ to Z/qZ. We show that the distribution of the (normalized) spacings between consecutive elements in the image of these maps becomes Poissonian as q tends to infinity along any sequence of square free integers such that the mean spacing modulo q tends to infinity. [ABSTRACT FROM AUTHOR]

Let n be a positive integer and φ(n) denotes the Euler phi function. It is well known that the power sum of n can be evaluated in closed form in terms of n. Also, the sum of all those φ(n) positive integers that are coprime to n and not exceeding n, is expressible in terms of n and φ(n). Although such results already exist in literature, but here we have presented some new analytical results in these connections. Some functional and integral relations are derived for the general power sums. [ABSTRACT FROM AUTHOR]

Some theta function identities are proved and used to give formulae for the number of representations of a positive integer by certain quaternary forms x2 + ey2 + fz2 + gt2 with e, f, g ∈ {1, 2, 4, 8}. [ABSTRACT FROM AUTHOR]

We give an elementary proof of the number of representations of an integer by the quaternary quadratic form x2 + xy + y2 + z2 + zt + t2. [ABSTRACT FROM AUTHOR]

Let ℤtr be the ring of totally real integers, Gal(ℚ) the absolute Galois group of ℚ, and e a positive integer. For each σ = (σ1,...,σe) ∈ Gal(ℚ)e let ℤtr(σ) be the fixed ring in ℤtr of σ1,...,σe. Then, the theory of all first order sentences θ that are true in ℤtr(σ) for almost all σ ∈ Gal(ℚ)e (in the sense of the Haar measure) is undecidable. [ABSTRACT FROM AUTHOR]

We introduce a method to find upper and lower bounds for the genus of numerical semigroups. Using it we prove some old and new bounds for it and for the Frobenius number of the semigroup. [ABSTRACT FROM AUTHOR]

The triple Euler sum defined by \[ \begin{array}{rcl} \zeta(p, q, r) & = & \sum\limits_{\ell=3}^{\infty} \frac{1}{\ell^{p}} \sum\limits_{k=2}^{\ell-1} \frac{1}{k^{q}} \sum\limits_{j=1}^{k-1} \frac{1}{j^{r}} \\ [10pt] & = & \sum\limits_{l=1}^{\infty} \sum\limits_{k=1}^{\infty} \sum\limits_{j=1}^{\infty} \frac{1}{(j+k+\ell)^{p}(j+k)^{q}j^{r}} \end{array} \] with positive integers p, q, r, p ≥ 2, has not been fully evaluated yet except for the case of q = r = 1 and some particular cases such as p + q + r ≤ 10 or p = q = r. With the general theory developed for double Euler sums, we are able to produce identities among triple Euler sums with a variable or two when q = r =. Performing differentiations with respect to the specified variable gives the explicit evaluations of ζ(p, q, r) for general p, q, r. In particular, the values of ζ(n, 1, 1), ζ(2n + 1, 1, 2), ζ(2n + 1, 2, 1) and ζ(2n, 2, 2) are given explicitly in terms of classical double Euler sums. Also ζ(2n, 1, 2m + 1) and ζ(2n + 1, 1, 2m) are obtained for all positive integers m and n. [ABSTRACT FROM AUTHOR]

We prove that for almost all real primitive characters χd of modulus |d|, the least positive integer nχd at which χd takes a value not equal to 0 and 1 satisfies nχd ≪ log|d|, and give a quite precise estimate on the size of the exceptional set. [ABSTRACT FROM AUTHOR]

In this paper, we give necessary and sufficient conditions for 8-ranks of narrow class groups of $F = \mathbb{Q}(\sqrt{2pq})$, distinct primes p ≡ q ≡ 1 mod 4. The results are useful for numerical computations. [ABSTRACT FROM AUTHOR]

For positive integers 1 ≤ i ≤ k, we consider the arithmetic properties of quotients of Wronskians in certain normalizations of the Andrews–Gordon q-series \[ \prod_{1\leq n\not \equiv 0,\pm i\ ({\rm mod}\ 2k+1)} \frac{1}{1-q^n}. \] This study is motivated by their appearance in conformal field theory, where these series are essentially the irreducible characters of $\mathcal{M}(2,2k+1)$ Virasoro minimal models. We determine the vanishing of such Wronskians, a result whose proof reveals many partition identities. For example, if Pb(a;n) denotes the number of partitions of n into parts which are not congruent to 0, ±a (mod b), then for every positive integer n, we have \[ P_{27}(12; n) = P_{27}(6; n-1) + P_{27}(3; n-2). \] We also show that these quotients classify supersingular elliptic curves in characteristic p. More precisely, if 2k + 1 = p, where p ≥ 5 is prime, and the quotient is non-zero, then it is essentially the locus of supersingular j-invariants in characteristic p. [ABSTRACT FROM AUTHOR]

Qualified residue difference sets of power n are known to exist for n = 2,4,6, as do similar sets that include the zero element. Both classes of sets are proved non-existent for n = 8. [ABSTRACT FROM AUTHOR]

In this paper we consider the family of systems (2c + 1)U2 - 2cV2 = μ and (c - 2)U2 - cZ2 = -2μ of relative Pellian equations, where the parameter c and the root of unity μ are integers in the same imaginary quadratic number field $K=\mathbb{Q} (\sqrt{-D})$. We show that for |c| ≥ 3 only certain values of μ yield solutions of this system, and solve the system completely for |c| ≥ 1544686. Furthermore we will consider the related relative Thue equation \[ X^{4}-4cX^{3}Y+(6c+2)X^{2}Y^{2}+4cXY^{3}+Y^{4}=\mu \] and solve it by the method of Tzanakis under the same assumptions. [ABSTRACT FROM AUTHOR]

We state a form of the modular relation in which the functional equation appears in the form of an expression of one Dirichlet series in terms of the other multiplied by the quotient of gamma functions and illustrate it by some concrete examples including the results of Koshlyakov, Berndt and Wigert and Bellman. [ABSTRACT FROM AUTHOR]

Kohnen showed that the zeros of the Eisenstein series Ek in the standard fundamental domain other than i and ρ are transcendental. In this paper, we obtain similar results for a more general class of modular forms, using the earlier works of Kanou, Kohnen and the recent work of Getz. [ABSTRACT FROM AUTHOR]

We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p + 1. An extension of Artin's conjecture is that there are infinitely many such inert primes for which this order is maximal. This is known at present only under the Generalized Riemann Hypothesis. Unconditionally, we show that for any choice of 7 units in different real quadratic fields satisfying a certain simple restriction, there is at least one of the units which satisfies the above version of Artin's conjecture. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. For any CM number field K containing a primitive pth root of unity, class field theory and Kummer theory put together yield the well known reflection inequality λ+ ≤ λ- between the "plus" and "minus" parts of the λ-invariant of K. Greenberg's classical conjecture predicts the vanishing of λ+. We propose a weak form of this conjecture: λ+ = λ- if and only if λ+ = λ- = 0, and we prove it when K+ is abelian, p is totally split in K+, and certain conditions on the cohomology of circular units are satisfied (e.g. in the semi-simple case). [ABSTRACT FROM AUTHOR]

The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present a new type of discrepancy bound for sequences of s-tuples of successive nonlinear congruential pseudorandom numbers over a ring of integers ℤM. [ABSTRACT FROM AUTHOR]