We obtain nontrivial estimates of quadratic character sums of division polynomials $\Psi_n(P)$, $n=1,2, \dots$, evaluated at a given point $P$ on an elliptic curve over a finite field of $q$ elements. Our bounds are nontrivial if the order of $P$ is at least $q^{1/2 + \varepsilon}$ for some fixed $\varepsilon > 0$. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences that was recently brought up by K. Lauter and the second author.

We provide a criterion for the central norm to be any value in the simple continued fraction expansion of $\sqrt{D}$ for any non-square integer $D>1$. We also provide a simple criterion for the solvability of the Pell equation $x^2-Dy^2=-1$ in terms of congruence conditions modulo $D$.

We consider the diophantine equation $x^2 + y^6 = z^e$, $e \geq 4$. We show that, when $e$ is a multiple of $4$ or $6$, this equation has no solutions in positive integers with $x$ and $y$ relatively prime. As a corollary, we show that there exists no primitive Pythagorean triangle one of whose leglengths is a perfect cube, while the hypotenuse length is an integer square.

The purpose of this paper is to solve various differential equations having Eisenstein series as coefficients using various tools and techniques. The solutions are given in terms of modular forms, modular functions, and equivariant forms.

We describe an $E_8$ correspondence for the multiplicative eta-products of weight at least $4$.

We present another example of a $3$-variable polynomial defining a $K3$-hypersurface and having a logarithmic Mahler measure expressed in terms of a Dirichlet $L$-series.

Let $C$ be a hyperelliptic curve given by the equation $y^2=f(x)$ for $f\in\mathbb{Z}[x]$ without multiple roots. We say that points $P_{i}=(x_{i}, y_{i})\in C(\mathbb{Q})$ for $i=1,2,\dots, m$ are in arithmetic progression if the numbers $x_{i}$ for $i=1,2,\dots, m$ are in arithmetic progression. In this paper we show that there exists a polynomial $k\in\mathbb{Z}[t]$ with the property that on the elliptic curve $\mathcal{E}': y^2=x^3+k(t)$ (defined over the field $\mathbb{Q}(t)$) we can find four points in arithmetic progression that are independent in the group of all $\mathbb{Q}(t)$-rational points on the curve $\mathcal{E}'$. In particular this result generalizes earlier results of Lee and V\'{e}lez. We also show that if $n\in\mathbb{N}$ is odd, then there are infinitely many $k$'s with the property that on curves $y^2=x^n+k$ there are four rational points in arithmetic progressions. In the case when $n$ is even we can find infinitely many $k$'s such that on curves $y^2=x^n+k$ there are six rational points in arithmetic progression.

We obtain new results about the number of trinomials $t^n + at + b$ with integer coefficients in a box $(a,b) \in [C, C+A] \times [D, D+B]$ that are irreducible modulo a prime $p$. As a by-product we show that for any $p$ there are irreducible polynomials of height at most $p^{1/2+o(1)}$, improving on the previous estimate of $p^{2/3+o(1)}$ obtained by the author in 1989.

Zarhin proves that if $C$ is the curve $y^2=f(x)$ where $\textrm{Gal}_{\mathbb{Q}}(f(x))=S_n$ or $A_n$, then ${\textrm{End}}_{\overline{\mathbb{Q}}}(J)=\mathbb{Z}$. In seeking to examine his result in the genus $g=2$ case supposing other Galois groups, we calculate $\textrm{End}_{\overline{\mathbb{Q}}}(J)\otimes_{\mathbb{Z}} \mathbb{F}_2$ for a genus $2$ curve where $f(x)$ is irreducible. In particular, we show that unless the Galois group is $S_5$ or $A_5$, the Galois group does not determine ${\textrm{End}}_{\overline{\mathbb{Q}}}(J)$.

In this paper, we prove that a non--zero power series $F(z)\in\mathbb{C} [\mkern-3mu[ z]\mkern-3mu] $ satisfying $$F(z^d)=F(z)+\frac{A(z)}{B(z)},$$ where $d\geq 2$, $A(z),B(z)\in\mathbb{C}[z]$ with $A(z)\neq 0$ and $\deg A(z),\deg B(z)
Keywords:functional equations, transcendence, power series
Categories:11B37, 11J81

Let $p$ be a prime, and let $\zeta_p$ be a primitive $p$-th root of unity. The lattices in Craig's family are $(p-1)$-dimensional and are geometrical representations of the integral $\mathbb Z[\zeta_p]$-ideals $\langle 1-\zeta_p \rangle^i$, where $i$ is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions $p-1$ where $149 \leq p \leq 3001$, Craig's lattices are the densest packings known. Motivated by this, we construct $(p-1)(q-1)$-dimensional lattices from the integral $\mathbb Z[\zeta _{pq}]$-ideals $\langle 1-\zeta_p \rangle^i \langle 1-\zeta_q \rangle^j$, where $p$ and $q$ are distinct primes and $i$ and $j$ are positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties.

Let $A(n_1,n_2,\dots,n_{m-1})$ be the normalized Fourier coefficients of a Maass cusp form on $\textrm{GL}(m)$. In this paper, we study the cancellation of $A (n_1,n_2,\dots,n_{m-1})$ over Beatty sequences.

Dubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number $\alpha$ by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved, and we establish its analog for the ultrametric Mahler measure.

Let $T_{n}$ denote the $n$-th Chebyshev polynomial of the first kind, and let $U_{n}$ denote the $n$-th Chebyshev polynomial of the second kind. We give an explicit formula for the resultant $\operatorname{res}( T_{m}, T_{n} )$. Similarly, we give a formula for $\operatorname{res}( U_{m}, U_{n} )$.

In this paper, we apply the saddle-point method in conjunction with the theory of the Nörlund-Rice integrals to derive precise asymptotic formula for the generalized Li coefficients established by Omar and Mazhouda. Actually, for any function $F$ in the Selberg class $\mathcal{S}$ and under the Generalized Riemann Hypothesis, we have $$ \lambda_{F}(n)=\frac{d_{F}}{2}n\log n+c_{F}n+O(\sqrt{n}\log n), $$ with $$ c_{F}=\frac{d_{F}}{2}(\gamma-1)+\frac{1}{2}\log(\lambda Q_{F}^{2}),\ \lambda=\prod_{j=1}^{r}\lambda_{j}^{2\lambda_{j}}, $$ where $\gamma$ is the Euler's constant and the notation is as below.

Soient $p_1,p_2,p_3$ et $q$ des nombres premiers distincts tels que $p_1\equiv p_2\equiv p_3\equiv -q\equiv 1 \pmod{4}$, $k = \mathbf{Q} (\sqrt{p_1}, \sqrt{p_2}, \sqrt{p_3}, \sqrt{q})$ et $\operatorname{Cl}_2(k)$ le $2$-groupe de classes de $k$. A. Fröhlich a démontré que $\operatorname{Cl}_2(k)$ n'est jamais trivial. Dans cet article, nous donnons une extension de ce résultat, en démontrant que le rang de $\operatorname{Cl}_2(k)$ est toujours supérieur ou égal à $2$. Nous démontrons aussi, que la valeur $2$ est optimale pour une famille infinie de corps $k$.

Questions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions.

We give an infinite family of congruent number elliptic curves each with rank at least three.

It is shown that an old direct argument of Erdős and Heilbronn may be elaborated to yield a result of the current inverse type.

Let $f$ be a classical newform of weight $2$ on the upper half-plane $\mathcal H^{(2)}$, $E$ the corresponding strong Weil curve, $K$ a class number one imaginary quadratic field, and $F$ the base change of $f$ to $K$. Under a mild hypothesis on the pair $(f,K)$, we prove that the period ratio $\Omega_E/(\sqrt{|D|}\Omega_F)$ is in $\mathbb Q$. Here $\Omega_F$ is the unique minimal positive period of $F$, and $\Omega_E$ the area of $E(\mathbb C)$. The claim is a specialization to base change forms of a conjecture proposed and numerically verified by Cremona and Whitley.

Consider a finite morphism $f: X \rightarrow Y$ of smooth, projective varieties over a finite field $\mathbf{F}$. Suppose $X$ is the vanishing locus in $\mathbf{P}^N$ of $r$ forms of degree at most $d$. We show that there is a constant $C$ depending only on $(N,r,d)$ and $\deg(f)$ such that if $|{\mathbf{F}}|>C$, then $f(\mathbf{F}): X(\mathbf{F}) \rightarrow Y(\mathbf{F})$ is injective if and only if it is surjective.

No abstract.

In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series.

We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

Let $\phi$ be a Drinfeld module of generic characteristic, and let X be a sufficiently generic affine subvariety of $\mathbb{G_a^g}$. We show that the intersection of X with a finite rank $\phi$-submodule of $\mathbb{G_a^g}$ is finite.