New Points on Curves

Let K be a field and let L/K be a finite extension. Let X/K be a scheme of finite type. A point of X(L) is said to be new if it does not belong to the union of X(F), when F runs over all proper subextensions of L. Fix now an integer g>0 and a finite separable extension L/K of degree d. We investigate in this article whether there exists a smooth proper geometrically connected curve of genus g with a new point in X(L). We show for instance that if K is infinite of characteristic different from 2 and g is bigger or equal to [d/4], then there exist infinitely many hyperelliptic curves X/K of genus g, pairwise non-isomorphic over the algebraic closure of K, and with a new point in X(L). When d is between 1 and 10, we show that there exist infinitely many elliptic curves X/K with pairwise distinct j-invariants and with a new point in X(L).Comment: To appear in Acta Arithmetic

Hecke algebra isomorphisms and adelic points on algebraic groups

Let $G$ denote a linear algebraic group over $\mathbf{Q}$ and $K$ and $L$ two number fields. Assume that there is a group isomorphism of points on $G$ over the finite adeles of $K$ and $L$, respectively. We establish conditions on the group $G$, related to the structure of its Borel groups, under which $K$ and $L$ have isomorphic adele rings. Under these conditions, if $K$ or $L$ is a Galois extension of $\mathbf{Q}$ and $G(\mathbf{A}_{K,f})$ and $G(\mathbf{A}_{L,f})$ are isomorphic, then $K$ and $L$ are isomorphic as fields. We use this result to show that if for two number fields $K$ and $L$ that are Galois over $\mathbf{Q}$, the finite Hecke algebras for $\mathrm{GL}(n)$ (for fixed $n > 1$) are isomorphic by an isometry for the $L^1$-norm, then the fields $K$ and $L$ are isomorphic. This can be viewed as an analogue in the theory of automorphic representations of the theorem of Neukirch that the absolute Galois group of a number field determines the field if it is Galois over $\mathbf{Q}$.Comment: 19 pages - completely rewritte