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Acta Arithmetica

### A new formulation of the law of octic reciprocity for primes =+-3 (mod 8) and its consequences.

International Journal of Mathematics and Mathematical Sciences

### A Note on prime k-th power nonresidues.

Manuscripta mathematica

Acta Arithmetica

### Abelian fields and the Brumer-Stark conjecture

Compositio Mathematica

Acta Arithmetica

### Automorphic forms

Banach Center Publications

### Barnes' identities and representations of GL (2). I. Finite field case.

Journal für die reine und angewandte Mathematik

Integers

### Character sums in finite fields

Compositio Mathematica

### Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

Acta Arithmetica

Consider the families of curves ${C}^{n,A}:y²=xⁿ+Ax$ and ${C}_{n,A}:y²=xⁿ+A$ where A is a nonzero rational. Let ${J}^{n,A}$ and ${J}_{n,A}$ denote their respective Jacobian varieties. The torsion points of ${C}^{3,A}\left(ℚ\right)$ and ${C}_{3,A}\left(ℚ\right)$ are well known. We show that for any nonzero rational A the torsion subgroup of ${J}^{7,A}\left(ℚ\right)$ is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to ${J}^{7,A}\left(ℚ\right)\left[2\right]$ (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for ${J}^{3,A}$ (A ≠ 4) and ${J}^{5,A}$. We also almost...

### Criterion for 3 to be eleventh power

Acta Mathematica et Informatica Universitatis Ostraviensis

Acta Arithmetica

### Diophantine approximation by square-free numbers

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Acta Arithmetica

### Hereditary orders, Gauss sums and supercuspidal representations of GLN.

Journal für die reine und angewandte Mathematik

### Jacobi sums and explicit reciprocity laws

Compositio Mathematica

### Jacobi-sum Hecke characters and Gauss-sum identities

Compositio Mathematica

### Kummer's calculation of Lp (1,...).

Journal für die reine und angewandte Mathematik

### Legendre polynomials and supercongruences

Acta Arithmetica

Let p > 3 be a prime, and let Rₚ be the set of rational numbers whose denominator is not divisible by p. Let Pₙ(x) be the Legendre polynomials. In this paper we mainly show that for m,n,t ∈ Rₚ with m≢ 0 (mod p), ${P}_{\left[p/6\right]}\left(t\right)\equiv -\left(3/p\right){\sum }_{x=0}^{p-1}\left(\left(x³-3x+2t\right)/p\right)\left(modp\right)$ and $\left({\sum }_{x=0}^{p-1}\left(\left(x³+mx+n\right)/p\right)\right)²\equiv \left(\left(-3m\right)/p\right){\sum }_{k=0}^{\left[p/6\right]}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\left(\genfrac{}{}{0pt}{}{3k}{k}\right)\left(\genfrac{}{}{0pt}{}{6k}{3k}\right){\left(\left(4m³+27n²\right)/\left(12³·4m³\right)\right)}^{k}\left(modp\right)$, where (a/p) is the Legendre symbol and [x] is the greatest integer function. As an application we solve some conjectures of Z. W. Sun and the author concerning ${\sum }_{k=0}^{p-1}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\left(\genfrac{}{}{0pt}{}{3k}{k}\right)\left(\genfrac{}{}{0pt}{}{6k}{3k}\right)/{m}^{k}\left(modp²\right)$, where m is an integer not divisible by p.

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