We study the family of elliptic curves y² = x(x-a²)(x-b²) parametrized by Pythagorean triples (a,b,c). We prove that for a generic triple the lower bound of the rank of the Mordell-Weil group over ℚ is 1, and for some explicitly given infinite family the rank is 2. To each family we attach an elliptic surface fibered over the projective line. We show that the lower bounds for the rank are optimal, in the sense that for each generic fiber of such an elliptic surface its corresponding Mordell-Weil...

We generalize L. J. Mordell’s construction of cubic surfaces for which the Hasse principle fails.

This paper is a continuation of [19], where the divisibility criteria for initial prime numbers based on their representation in the decimal system were formalized. In the current paper we consider all primes up to 101 to demonstrate the method presented in [7].

For an irrational real number$\alpha$ and real number $\gamma$ we consider the inhomogeneous approximation constant$$M(\alpha ,\gamma ):=\underset{\left|n\right|\to \infty }{lim\; inf}\left|n\right|\left|\right|n\alpha -\gamma \left|\right|$$via the semi-regular negative continued fraction expansion of $\alpha$$$\alpha =\frac{1}{{}_{}}$$

Nous donnons une caractérisation complète de tous les morphismes binaires qui préservent les mots sturmiens et montrons que les mots infinis engendrés par ces morphismes sont rigides.

Special values of certain $L$ functions of the type $L(M,s)$ are studied where $M$ is a motive over a totally real field $F$ with coefficients in another field $T$, and$$L(M,s)=\prod _{𝔭}{L}_{𝔭}(M,𝒩{𝔭}^{-s})$$is an Euler product $𝔭$ running through maximal ideals of the maximal order ${𝒪}_F$ of $F$ and$$\begin{array}{cc}\hfill L_{𝔭}(M,X{)}^{-1}& =(1-{\alpha }^{\left(1\right)}(𝔭\left)X\right)·(1-{\alpha }^{\left(2\right)}(𝔭\left)X\right)·...·(1-\alpha (d\left)\right(𝔭\left)X\right)\hfill \\ & =1+A_1\left(𝔭\right)X+...+A_d\left(𝔭\right)X^d\hfill \end{array}$$being a polynomial with coefficients in $T$. Using the Newton and the Hodge polygons of $M$ one formulate a conjectural criterium for the existence of a $p$-adic analytic continuation of the special values. This conjecture is verified in a number of cases related to...

This is the last of a series of three papers where we compute the unramified cohomology of quadrics in degree up to 4. Complete results were obtained in the two previous papers for quadrics of dimension $\le 4$ and $\ge 11$. Here we deal with the remaining dimensions between 5 and 10. We also prove that the unramified cohomology of Pfister quadrics with divisible coefficients always comes from the ground field, and that the same holds for their unramified Witt rings. We apply these results to real quadrics....