Tame torsion and the tame inverse Galois problem

Fix a positive integer g and a squarefree integer m. We prove the existence of a genus g curve $$C/{\mathbb {Q}}$$ C / Q such that the mod m representation of its Jacobian is tame. The method is to analyse the period matrices of hyperelliptic Mumford curves, which could be of independent interest. As an application, we study the tame version of the inverse Galois problem for symplectic matrix groups over finite fields.

Squarefree Integers in Arithmetic Progressions to Smooth Moduli

Let $\varepsilon> 0$ be sufficiently small and let $0 < \eta < 1/522$ . We show that if X is large enough in terms of $\varepsilon $ , then for any squarefree integer $q \leq X^{196/261-\varepsilon }$ that is $X^{\eta }$ -smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression $a \pmod {q}$ , with $(a,q) = 1$ . In the case of squarefree, smooth moduli this improves upon previous work of Nunes, in which $196/261 = 0.75096\ldots $ was replaced by $25/36 = 0.69\overline {4}$ . This also establishes a level of distribution for a positive density set of moduli that improves upon a result of Hooley. We show more generally that one can break the $X^{3/4}$ -barrier for a density 1 set of $X^{\eta }$ -smooth moduli q (without the squarefree condition). Our proof appeals to the q-analogue of the van der Corput method of exponential sums, due to Heath-Brown, to reduce the task to estimating correlations of certain Kloosterman-type complete exponential sums modulo prime powers. In the prime case we obtain a power-saving bound via a cohomological treatment of these complete sums, while in the higher prime power case we establish savings of this kind using p-adic methods.

Skew braces of squarefree order

Let [Formula: see text] be a squarefree integer, and let [Formula: see text], [Formula: see text] be two groups of order [Formula: see text]. Using our previous results on the enumeration of Hopf–Galois structures on Galois extensions of fields of squarefree degree, we determine the number of skew braces (up to isomorphism) with multiplicative group [Formula: see text] and additive group [Formula: see text]. As an application, we enumerate skew braces whose order is the product of three distinct primes, in particular proving a conjecture of Bardakov, Neshchadim and Yadav on the number of skew braces of order [Formula: see text] for primes [Formula: see text].

Rational Points of Some Elliptic Curves Related to the Tilings of the Equilateral Triangle

Let n be a positive and squarefree integer. We show that the equilateral triangle can be dissected into $$n\cdot k^2$$ n · k 2 congruent triangles for some k if and only if $$n\le 3$$ n ≤ 3 , or at least one of the curves $$C_n :y^2 =x(x-n)(x+3n)$$ C n : y 2 = x ( x - n ) ( x + 3 n ) and $$C_{-n} : y^2 =x(x+n)(x-3n)$$ C - n : y 2 = x ( x + n ) ( x - 3 n ) has a rational point with $$y\ne 0$$ y ≠ 0 . We prove that if p is a positive prime such that $$p\equiv 7$$ p ≡ 7 (mod 24), then $$C_p$$ C p and $$C_{-p}$$ C - p do not have such points. Consequently, for these primes the equilateral triangle cannot be dissected into $$p\cdot k^2$$ p · k 2 congruent triangles for any k.

Converegence of a series leading to an analogue of Ramanujan's assertion on squarefree integers

Let d be a squarefree integer. We prove that(i) Pnμ(n)nd(n′) converges to zero, where n′ is the product of prime divisors of nwith ( dn ) = +1. We use the Prime Number Theorem.(ii) Q( dp )=+1(1 −1ps ) is not analytic at s=1, nor is Q( dp )=−1(1 −1ps ) .(iii) The convergence (i) leads to a proof that asymptotically half the squarefree ideals have an even number of prime ideal factors (analogue of Ramanujan’s assertion).

Spectral Multiplicity for Maaß Newforms of Non-Squarefree Level

We show that if a positive integer $q$ has $s(q)$ odd prime divisors $p$ for which $p^2$ divides $q$, then a positive proportion of the Laplacian eigenvalues of Maaß newforms of weight $0$, level $q$, and principal character occur with multiplicity at least $2^{s(q)}$. Consequently, the new part of the cuspidal spectrum of the Laplacian on $\Gamma_0(q) \backslash \mathbb{H}$ cannot be simple for any odd non-squarefree integer $q$. This generalises work of Strömberg who proved this for $q = 9$ by different methods.

A Bertrand postulate for a subclass of primes

Let d be a squarefree integer and consider the subclass of primes with Legendre symbol ( d/p ) = +1. It is shown that for x large enough (x; 2x] contain a prime of this type.

SOLUTIONS OF THE CUBIC FERMAT EQUATION IN QUADRATIC FIELDS

We give necessary and sufficient conditions on a squarefree integer d for there to be non-trivial solutions to x3 + y3 = z3 in [Formula: see text], conditional on the Birch and Swinnerton-Dyer conjecture. These conditions are similar to those obtained by J. Tunnell in his solution to the congruent number problem.

ON THE DIOPHANTINE EQUATION x2 + d2l + 1 = yn

Let d > 0 be a squarefree integer and denote by h = h(−d) the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. It is well known (see e.g. [25]) that for a given positive integer N there are only finitely many squarefree d's for which h(−d) = N. In [45], Saradha and Srinivasan and in [28] Le and Zhu considered the equation in the title and solved it completely under the assumption h(−d) = 1 apart from the case d ≡ 7 (mod 8) in which case y was supposed to be odd. We investigate the title equation in unknown integers (x, y, l, n) with x ≥ 1, y ≥ 1, n ≥ 3, l ≥ 0 and gcd(x, y) = 1. The purpose of this paper is to extend the above result of Saradha and Srinivasan to the case h(−d) ∈ {2, 3}.

On the 2-Rank of the Hilbert Kernel of Number Fields

Let $E/F$ be a quadratic extension of number fields. In this paper, we show that the genus formula for Hilbert kernels, proved by M. Kolster and A. Movahhedi, gives the 2-rank of the Hilbert kernel of $E$ provided that the 2-primary Hilbert kernel of $F$ is trivial. However, since the original genus formula is not explicit enough in a very particular case, we first develop a refinement of this formula in order to employ it in the calculation of the 2-rank of $E$ whenever $F$ is totally real with trivial 2-primary Hilbert kernel. Finally, we apply our results to quadratic, bi-quadratic, and tri-quadratic fields which include a complete 2-rank formula for the family of fields $\mathbb{Q}(\sqrt{2},\sqrt{\delta )}$ where $\delta $ is a squarefree integer.