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The Ohno-Nakagawa (O-N) reflection theorem is an unexpectedly simple identity relating the number of \mathrm{GL}_2 \mathbb{Z}classes of binary cubic forms (equivalently, cubic rings) of two different discriminants D -27D it generalizes cubic reciprocity and the Scholz reflection theorem. In this paper, we present a new approach to this theorem using Fourier analysis on the adelic cohomology H^1(\mathbb{A}_K, M)of a finite Galois module, modeled after the celebrated Fourier analysis o...

We set up a general framework to study Tate cohomology groups of Galois modules along \mathbb{Z}_pextensions of number fields. Under suitable assumptions on the Galois modules, we establish the existence of a five-term exact sequence in a certain quotient category whose objects are simultaneously direct and inverse systems, subject to some compatibility. The exact sequence allows one, in particular, to control the behaviour of the Tate cohomology groups of the units along \mathbb{Z}_pexten...

Let \pibe a SL(3,\mathbb{Z})Hecke Maass-cusp form, fbe a SL(2,\mathbb{Z})holomorphic cusp form or Maass-cusp form with normalized Fourier coefficients \lambda_{\pi}(r,n) \text{ and }\lambda_{f}(n)respectively and \chibe any non-trivial character mod pwhere pis a prime. Then we have $S_{\pi ,f,\chi }(N)\ll_{\pi , f, \epsilon} N^{3/4}p^{11/16 +{\eta /4}}(Np)^\epsilon .

On some new families of k-Mersenne and generalized k-Gaussian Mersenne numbers and their polynomials.

In this paper, we define new generalized k-Mersenne numbers and give a formula of generalized Mersenne polynomials and further we study their properties. Moreover, we define Gaussian Mersenne numbers and obtain some identities like Binet Formula, Cassini's identity, D'Ocagne's Identity, and generating functions. The generalized Gaussian Mersenne numbers are described and the relation with classical Mersenne numbers are explained. We also introduce a generalization of Gaussian Mersenne polynomial...

Let 1 < c < 24/19 We show that the number of integers n \le Nthat cannot be written as [p_1^c] + [p_2^c](p_1 p_2primes) is O(N^{1-\sigma+\varepsilon}) Here \sigmais a positive function of c(given explicitly) and \varepsilonis an arbitrary positive number.

A geometric approach to elliptic curves with torsion groups \mathbb{Z}/10\mathbb{Z} \mathbb{Z}/12\mathbb{Z} \mathbb{Z}/14\mathbb{Z} and \mathbb{Z}/16\mathbb{Z}

We give new parametrisations of elliptic curves in Weierstrass normal form y^2=x^3+ax^2+bxwith torsion groups \mathbb{Z}/10\mathbb{Z}and \mathbb{Z}/12\mathbb{Z}over \mathbb{Q} and with \mathbb{Z}/14\mathbb{Z}and \mathbb{Z}/16\mathbb{Z}over quadratic fields. Even though the parametrisations are equivalent to those given by Kubert and Rabarison, respectively, with the new parametrisations we found three infinite families of elliptic curves with torsion group $\mathbb{Z}/12\mathbb...

We show that the split family of cubic Thue Equations \[ X(X-A_nY)(X-B_nY)-Y^3 = \pm 1, \] with linear recurrence sequences A_n, B_nsatisfying a dominant root condition each, has but the trivial solutions when the parameter nis smaller than an effectively computable constant.

Let \mathbb{Q}(\alpha)and \mathbb{Q}(\beta)be linearly disjoint number fields and let \mathbb{Q}(\theta)be their compositum. We prove that the first-degree prime ideals of \mathbb{Z}[\theta]may almost always be constructed in terms of the first-degree prime ideals of \mathbb{Z}[\alpha]and \mathbb{Z}[\beta] and vice-versa. We also classify the cases in which this correspondence does not hold, by providing explicit counterexamples. We show that for every pair of coprime integers ...

We prove that for a given odd number m\geq3 for all but finitely many primes `p', class number of \mathbb{Q}(\sqrt{1-2m^p})is divisible by `p' and this collection of fields is infinite for a fixed `m'. We also prove that the class number of \mathbb{Q}(\sqrt{1-2m^p})is divisible by `p' whenever 2m^p-1is a square-free integer. For the above two results we conclude some corollaries by replacing `p' by an square-free odd integer t\geq3 We prove that for any pair of twin primes $p_1,p_...

We study the dynamics of stochastic families of rational maps on the projective line. As such families can be infinite and may not typically be defined over a single number field, we introduce the concept of generalized adelic measures, generalizing previous notions introduced by Favre and Rivera-Letelier and Mavraki and Ye. Generalized adelic measures are defined over the measure space of places of an algebraic closure of the rationals, using a framework established by Allcock and Vaaler. This ...

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