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In this paper, it is proved that the group generated by Bass units contains a subgroup of finite index in the group of central units \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) of the integral group ring \mathbb {Z}G for a subgroup closed monomial group G with the property that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in G. If G is a generalized strongly monomial group, then it is also shown that the group generated by generalized Bass units contains a subgroup of finite index in \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)). Furthermore, for a generalized strongly monomial group G, the rank of \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) is determined. The formula so obtained is in terms of generalized strong Shoda pairs of G. [ABSTRACT FROM AUTHOR]

Let s(n) denote the sum of the proper divisors of the natural number n. We show that the number of n ≤ x such that s(n) is a sum of two squares has order of magnitude x/√log x, which agrees with the count of n ≤ x which are a sum of two squares. Our result confirms a special case of a conjecture of Erdős, Granville, Pomerance, and Spiro, who in a 1990 paper asserted that if A ⊂ N has asymptotic density zero (e.g., if A is the set of n ≤ x which are a sum of two squares), then s−1(A) also has asymptotic density zero. [ABSTRACT FROM AUTHOR]

We compute the classes of universal theta divisors of degrees zero and g-1 over the Deligne-Mumford compactification Mg,n of the moduli space of curves, with various integer weights on the points, in particular reproving a recent result of Müller. We also obtain a formula for the class in CHg(Mctg,n) (moduli of stable curves of compact type) of the double ramification cycle, given by the condition that a fixed linear combination of the marked points is a principal divisor, reproving a recent result of Hain. Our approach for computing the theta divisor is more direct, via test curves and the geometry of the theta divisor, and works easily over all of Mg,n. We used our extended result in another paper to study the partial compactification of the double ramification cycle. [ABSTRACT FROM AUTHOR]

Kollár gave a series of examples of rational surfaces of Picard number 1 with ample canonical divisor having cyclic singularities. In this paper, we construct several series of new examples in a geometric way, i.e., by blowing up several times inside a configuration of curves on the projective plane and then by contracting chains of rational curves. One series of our examples has the same singularities as Kollár's examples. [ABSTRACT FROM AUTHOR]

In this article we study the Titchmarsh divisor problem in the context of abelian varieties. Under the Generalized Riemann Hypothesis we obtain an asymptotic formula. [ABSTRACT FROM AUTHOR]

Let q be an odd prime power, denote by \mathbb {F}_q the finite field with q elements, and set A ≔\mathbb {F}_q[T], F ≔\mathbb {F}_q(T). Let \psi : A \to F\{\tau \} be a Drinfeld A-module over F, of rank r \geq 2, with End_{\overline {F}}(\psi) = A. For a non-zero ideal \mathfrak {n} of A, denote its unique monic generator by n, denote the degree of n as a polynomial in T by \deg n, and denote the \mathfrak {n}-division field of \psi by F(\psi [\mathfrak {n}]). A reciprocity law for \psi asserts that, if \gcd (charF, r) = 1 or if \mathfrak {n} is prime, then a non-zero prime ideal \mathfrak {p} \nmid \mathfrak {n} of A splits completely in F(\psi [\mathfrak {n}]) if and only if the Frobenius trace a_{1, \mathfrak {p}}(\psi) of \psi at \mathfrak {p} and the first component b_{1, \mathfrak {p}}(\psi) of the Frobenius index of \psi at \mathfrak {p} satisfy the congruences a_{1, \mathfrak {p}}(\psi) \equiv -r \pmod n and b_{1, \mathfrak {p}}(\psi) \equiv 0 \pmod n. We find the Dirichlet density of the set of non-zero prime ideals \mathfrak {p} for which the latter congruence never holds, that is, for which b_{1, \mathfrak {p}}(\psi) = 1. Using similar methods, we prove an asymptotic formula for the function of x defined by the average \frac {1}{\#\{\mathfrak {p}: \ \deg p = x \}} \sum _{\mathfrak {p}: \ \deg p = x} \tau _A(b_{1, \mathfrak {p}}(\psi)), where \mathfrak {p} = A p denotes an arbitrary non-zero prime ideal of A whose monic generator p \in A has degree x and where \tau _A(b_{1, \mathfrak {p}}(\psi)) denotes the number of monic divisors of b_{1, \mathfrak {p}}(\psi). [ABSTRACT FROM AUTHOR]

Let s(n)=\sum _{d\mid n,~d

Given a collection of boundary divisors in the moduli space \overline {\mathcal {M}}_{0,n} of stable genus-zero n-pointed curves, Giansiracusa proved that their intersection is nonempty if and only if all pairwise intersections are nonempty. We give a complete classification of the pairs (g,n) for which the analogous statement holds in \overline {\mathcal {M}}_{g,n}. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the next extremal case for a Castelnuovo-type bound $ \mathrm{reg} V \le \lceil (\deg V - 1)/ {\mbox{\rm codim} } V \rceil + 1$. A Buchsbaum variety with the maximal regularity is known to be a divisor on a variety of minimal degree if the degree of the variety is large enough. We show that a Buchsbaum variety satisfying $ \mathrm{reg} V = \lceil (\deg V - 1)/ {\mbox{\rm codim} } V \rceil$ $ \deg V \gg 0$ [ABSTRACT FROM AUTHOR]

We have shown in an earlier paper that the divisor class group of the centre of a smooth PI algebra with trivial $ K_0$ [ABSTRACT FROM AUTHOR]

In this paper we use the divisorial Zariski decomposition to give a more precise version of the algebraic Morse inequalities. [ABSTRACT FROM AUTHOR]

First we prove some elementary but useful identities in the group ring of $mathbb {Q}/mathbb {Z}$. Our identities have potential applications to several unsolved problems which involve sums of Farey fractions. In this paper we use these identities, together with some analytic number theory and results about divisors in short intervals, to estimate the cardinality of a class of sets of fundamental interest. [ABSTRACT FROM AUTHOR]

In this article, we study the Piltz divisor problem, which is sometimes called the generalized Dirichlet divisor problem, over number fields. We establish an identity akin to Voronoï's formula concerning the error term in the Dirichlet divisor problem. [ABSTRACT FROM AUTHOR]

Using the irreducibility of a natural irreducible representation of the theta group of an ample line bundle on an abelian variety, we derive a bound for the number of n-torsion points that lie on a given theta divisor. We present also two alternate approaches to attacking the case n = 2. [ABSTRACT FROM AUTHOR]

Given a positive integer n the k-fold divisor function dk(n) equals the number of ordered k-tuples of positive integers whose product equals n. In this article we study the variance of sums of dk(n) in short intervals and establish asymptotic formulas for the variance of sums of dk(n) in short intervals of certain lengths for k = 3 and for k ≥ 4 under the assumption of the Lindelöf hypothesis. [ABSTRACT FROM AUTHOR]

We establish a uniform asymptotic formula with a power saving error term for the average of the divisor functionτ(n) := _∑k|n 1 over values of the quadratic polynomial x²+|D| where D < 0 is a fundamental discriminant. [ABSTRACT FROM AUTHOR]

We run Mori's program for the moduli space of stable pointed rational curves with divisor K +Σaiψi. We prove that the birational model for the pair is either the Hassett space of weighted pointed stable rational curves for the same weights or the GIT quotient of the product of projective lines with the linearization given by the same weights. [ABSTRACT FROM AUTHOR]

We give a new proof of Bogomolov's instability theorem. Further-more, we prove that it is equivalent to a statement which characterizes when the first cohomology group of a suitable divisor does not vanish. [ABSTRACT FROM AUTHOR]

The elementary divisors of the incidence matrix of lines in PG(3, q) are computed, where two lines are incident if and only if they are skew. [ABSTRACT FROM AUTHOR]