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Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.

9 votes
0 answers
322 views

Frey-Mazur for abelian varieties

Let $K$ be a number field. The Frey-Mazur conjecture asserts the existence of a constant $N_K$ such that for all primes $p>N_K$, and all pairs of elliptic curves $E_1$, $E_2/K$, if $\overline{\rho}_{E …
10 votes
Accepted

Trivial Weil-Châtelet group

Weil-Chatelet groups are huge. A theorem of Shafarevich states that if $n \ge 2$ and if $E$ is an elliptic curve (or an abelian variety) over a number field $k$ then $H^1(G_k,E)$ has infinitely many e …
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1 vote
0 answers
100 views

Points on the intersection of an affine quadric and cubic over a finite field

Are there absolute constants $N$ and $B$ such that the following is true? Let $p>B$ be a prime. Let $q(x_0,\dotsc,x_n)$ and $c(x_0,\dotsc,x_n)$ be homogeneous of degree $2$ and $3$ with coefficients …