We prove a conjectural formula for the Brumer–Stark units. Dasgupta and Kakde have shown the formula is correct up to a bounded root of unity. In this paper, we resolve the ambiguity in their result. We also remove an assumption from Dasgupta–Kakde’s result on the formula.

The lifting problem for universal quadratic forms over a totally real number field K consists of determining the existence or otherwise of a quadratic form with integer coefficients (or $\mathbb {Z}$-form) that is universal over K. We prove the nonexistence of universal $\mathbb {Z}$-forms over simplest cubic fields for which the integer parameter is big enough. The monogenic case is already known. We prove the nonexistence in the nonmonogenic case by using the existence of a totally positive nonunit algebraic integer in K with minimal (codifferent) trace equal to one.

By analogy with the trace of an algebraic integer $\alpha $ with conjugates $\alpha _1=\alpha , \ldots , \alpha _d$ , we define the G-measure $ {\mathrm {G}} (\alpha )= \sum _{i=1}^d ( |\alpha _i| + 1/ | \alpha _i | )$ and the absolute ${\mathrm G}$ -measure ${\mathrm {g}}(\alpha )={\mathrm {G}}(\alpha )/d$ . We establish an analogue of the Schur–Siegel–Smyth trace problem for totally positive algebraic integers. Then we consider the case where $\alpha $ has all its conjugates in a sector $| \arg z | \leq \theta $ , $0 < \theta < 90^{\circ }$ . We compute the greatest lower bound $c(\theta )$ of the absolute G-measure of $\alpha $ , for $\alpha $ belonging to $11$ consecutive subintervals of $]0, 90 [$ . This phenomenon appears here for the first time, conforming to a conjecture of Rhin and Smyth on the nature of the function $c(\theta )$ . All computations are done by the method of explicit auxiliary functions.

We prove that in each degree divisible by 2 or 3, there are infinitely many totally real number fields that require universal quadratic forms to have arbitrarily large rank.

We consider a concrete family of $2$ -towers $(\mathbb {Q}(x_n))_n$ of totally real algebraic numbers for which we prove that, for each $n$ , $\mathbb {Z}[x_n]$ is the ring of integers of $\mathbb {Q}(x_n)$ if and only if the constant term of the minimal polynomial of $x_n$ is square-free. We apply our characterization to produce new examples of monogenic number fields, which can be of arbitrary large degree under the ABC-Conjecture.

We study totally positive definite quadratic forms over the ring of integers $\mathcal {O}_K$ of a totally real biquadratic field $K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$. We restrict our attention to classic forms (i.e. those with all non-diagonal coefficients in $2\mathcal {O}_K$) and prove that no such forms in three variables are universal (i.e. represent all totally positive elements of $\mathcal {O}_K$). Moreover, we show the same result for totally real number fields containing at least one non-square totally positive unit and satisfying some other mild conditions. These results provide further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of $\mathcal {O}_K$; we prove several new results about their properties.

We prove that a nonreal algebraic number $\unicode[STIX]{x1D703}$ with modulus greater than $1$ is a complex Pisot number if and only if there is a nonzero complex number $\unicode[STIX]{x1D706}$ such that the sequence of fractional parts $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ has a finite number of limit points. Also, we characterise those complex Pisot numbers $\unicode[STIX]{x1D703}$ for which there is a convergent sequence of the form $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ for some $\unicode[STIX]{x1D706}\in \mathbb{C}^{\ast }$. These results are generalisations of the corresponding real ones, due to Pisot, Vijayaraghavan and Dubickas.

Until recently, the ‘plus part’ of the class numbers of cyclotomic fields had only been determined for fields of root discriminant small enough to be treated by Odlyzko’s discriminant bounds.

However, by finding lower bounds for sums over prime ideals of the Hilbert class field, we can now establish upper bounds for class numbers of fields of larger discriminant. This new analytic upper bound, together with algebraic arguments concerning the divisibility properties of class numbers, allows us to unconditionally determine the class numbers of many cyclotomic fields that had previously been untreatable by any known method.

In this paper, we study in particular the cyclotomic fields of composite conductor.

In this paper we generalize some results, obtained by Shimura, on the special values of L-functions of l-adic representations attached to quadratic CM-base change of Hilbert modular forms twisted by finite order characters. The generalization is to the case of the special values of L-functions of arbitrary base change to CM-number fields of l-adic representations attached to Hilbert modular forms twisted by some finite-dimensional representations.

Let θ be a real number greater than 1, and let (()) be the fractional part function. Then, θ is said to be a Z-number if there is a non-zero real number λ such that ((λθn)) < for all n ∈ ℕ. Dubickas (A. Dubickas, Even and odd integral parts of powers of a real number, Glasg. Math. J., 48 (2006), 331–336) showed that strong Pisot numbers are Z-numbers. Here it is proved that θ is a strong Pisot number if and only if there exists λ ≠ 0 such that ((λα)) < for all. Also, the following characterisation of Pisot numbers among real numbers greater than 1 is shown: θ is a Pisot number ⇔ ∃ λ ≠ 0 such thatfor allan ∈ {0,1}, N ∈ ℕ}, where ‖λα‖ = min{((λα)), 1 − ((λα))}.

In this article we prove Tate conjecture for a large class of compact quaternionic Shimura fourfolds.

In this paper we compute and continue meromorphically to the entire complex plane the zeta functions of twisted quaternionic Shimura varieties of dimension 3. The twist of the quaternionic Shimura varieties is done by a mod ℘ representation of the absolute Galois group.