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

Развитие науки в контексте глобализации

Let G denote a linear algebraic group over Q and K and L two number fields. Assumethat there is a group isomorphism G(AK,f ) ∼= G(AL,f ) of points on G over the finite adeles ofK and L, respectively. We establish conditions on the group G, related to the structure of its Borelgroups, under which K and L have isomorphic adele rings. Under these conditions, if K or L is aGalois extension of Q and G(AK,f ) ∼= G(AL,f ), 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 Q, thefinite Hecke algebras for GL(n) (for fixed n ≥ 2) are isomorphic by an isometry for the L1-norm, then the fields K and L are isomorphic. This can be viewed as an analogue in the theory ofautomorphic representations of the theorem of Neukirch that the absolute Galois group of a numberfield determines the field, if it is Galois over Q

Galois representations and Galois groups over Q

In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve and let J(C) be the associated Jacobian variety. Assume that there exists a prime p such that J(C) has semistable reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes l (if they exist) such that the Galois representation attached to the l-torsion of J(C) is surjective onto the group GSp(2n, l). In particular we realize GSp(6, l) as a Galois group over Q for all primes l in [11, 500000].Comment: Minor changes. 13 pages. This paper contains results of the collaboration started at the conference Women in numbers - Europe, (October 2013), by the working group "Galois representations and Galois groups over Q

When is a polarised abelian variety determined by its p\boldsymbol{p}-divisible group?

We study the Siegel modular variety $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ of genus $g$ and its supersingular locus $\mathcal{S}_g$. As our main result we determine precisely when $\mathcal{S}_g$ is irreducible, and we list all $x$ in $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ for which the corresponding central leaf $\mathcal{C}(x)$ consists of one point, that is, for which $x$ corresponds to a polarised abelian variety which is uniquely determined by its associated polarised $p$-divisible group. The first problem translates to a class number one problem for quaternion Hermitian lattices. The second problem also translates to a class number one problem, whose solution involves mass formulae, automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus $g=4$.Comment: New title, 39 page

Isomorphism classes of Drinfeld modules over finite fields

We study isogeny classes of Drinfeld $A$-modules over finite fields $k$ with commutative endomorphism algebra $D$, in order to describe the isomorphism classes in a fixed isogeny class. We study when the minimal order $A[\pi]$ of $D$ occurs as an endomorphism ring by proving when it is locally maximal at $\pi$, and show that this happens if and only if the isogeny class is ordinary or $k$ is the prime field. We then describe how the monoid of fractional ideals of the endomorphism ring $\mathcal{E}$ of a Drinfeld module $\phi$ up to $D$-linear equivalence acts on the isomorphism classes in the isogeny class of $\phi$, in the spirit of Hayes. We show that the action is free when restricted to kernel ideals, of which we give three equivalent definitions, and determine when the action is transitive. In particular, the action is free and transitive on the isomorphism classes in an isogeny class which is either ordinary or defined over the prime field, yielding a complete and explicit description in these cases.Comment: 21 page