相关论文: $\mathbb{Q}$-curves and the Lebesgue-Nagell equati…

We prove a new inequality for the Hodge number h^1,1 of irregular complex smooth projective surfaces of general type without irregular pencils of genus >1. More specifically we show that if the irregularity q satisfies q=2^k+1 then…

We study the family of algebraic curves of genus $\geq 1$ defined by the affine equations $y^s=ax^r+b$ over a number field $k$, where $r \geq 2$ and $s\geq 2$ are fixed integers. Assuming the strong version of Lang's conjecture on varieties…

We consider all genus 2 curves over Q given by an equation y^2 = f(x) with f a squarefree polynomial of degree 5 or 6, with integral coefficients of absolute value at most 3. For each of these roughly 200000 isomorphism classes of curves,…

Consider a field $k$ of characteristic $0$, not necessarily algebraically closed, and a fixed algebraic curve $f=0$ defined by a tame polynomial $f\in k[x,y]$ with only quasi-homogeneous singularities. We prove that the space of holomorphic…

For a square-free integer $N$, we present a procedure to compute $\mathbb{Q}$-curves parametrized by rational points of the modular curve $X_0^*(N)$ when this is hyperelliptic.

We obtain additional Diophantine applications of the methods surrounding Darmon's program for the generalized Fermat equation developed in the first part of this series of papers. As a first application, we use a multi-Frey approach…

Everybody knows from school how to solve a quadratic equation of the form $x^2-px+q=0$ graphically. But this method can become tedious if several equations ought to be solved, as for each pair $(p,q)$ a new parabola has to be drawn.…

We show that the kernel $K(x,y)={1\over 2}+\lfloor {1\over xy}\rfloor -{1\over xy}$ ($0<x,y\leq 1$) has infinitely many positive eigenvalues and infinitely many negative eigenvalues. Our interest in this kernel is motivated by the…

For an even integer $k$, let $r_{2k}(n)$ be the number of representations of $n$ as a sum of $2k$ squares. The quantity $r_{2k}(n)$ is appoximated by the classical singular series $\rho_{2k}(n) \asymp n^{k-1}$. Deligne's bound on the…

We present a relation between the Mathieu equation and a particular elliptic curve. We find that the Floquet exponent of the Mathieu equation, for both $q<<1$ and $q>>1$, can be obtained from the integral of a differential one form along…

By considering mirror symmetry applied to conformal field theories corresponding to strings propagating in quintic hypersurfaces in projective 4-space, Candelas, de la Ossa, Green and Parkes calculated the ``number of rational curves on the…

Note that the family of closed curves C_N={(x,y)\in R^2;x^(2N)+y^(2N)=1} for N=1,2,3,... approaches the boundary of [-1,1]^2 as N \to \infty. In this paper we exhibit a natural parameterization of these curves and generalize to a larger…

We study the following two-parameter variant of the Erd\H os-Falconer distance problem. Given $E,F \subset {\Bbb F}_q^{k+l}$, $l \geq k \ge 2$, the $k+l$-dimensional vector space over the finite field with $q$ elements, let $B_{k,l}(E,F)$…

For a non-degenerate irreducible curve $C$ of degree $d$ in $\mathbb{P}^3$ over $\mathbb{F}_q$, we prove that the number $N_q(C)$ of $\mathbb{F}_q$-rational points of $C$ satisfies the inequality $N_q(C) \leq (d-2)q+1$. Our result improves…

Let $\mathbb F_q$ denote the finite field with $q$ elements. In this paper we use the relationship between suitable polynomials and number of rational points on algebraic curves to give the exact number of elements $a\in \mathbb F_q$ for…

Suppose q is a complex number of modulus one and different from 1,-1. Let O(R^2_q) be the *-algebra with two hermitean generators x and y satisfying the relation xy=qyx. Using operator representations of the *-algebra O(R^2_q) on Hilbert…

For a large class (heuristically most) of irreducible binary cubic forms $F(x,y) \in \mathbb Z[x,y]$, Bennett and Dahmen proved that the generalized superelliptic equation $F(x,y)=z^l$ has at most finitely many solutions in $x,y \in \mathbb…

We develop a systematic procedure for constructing quantum many-body problems whose spectrum can be partially or totally computed by purely algebraic means. The exactly-solvable models include rational and hyperbolic potentials related to…

Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds…

By finding all integral points on certain elliptic and hyperelliptic curves we completely solve the Diophantine equation $\binom{n}{k}=\binom{m}{l}+d$ for $-3\leq d\leq 3$ and $(k,l)\in\{(2,3),\; (2,4),\;(2,5),\; (2,6),\; (2,8),\; (3,4),\;…