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We introduce a notion of Hecke-monicity for functions on certain moduli spaces associated to torsors of finite groups over elliptic curves, and show that it implies strong invariance properties under linear fractional transformations. Specifically, if a weakly Hecke-monic function has algebraic integer coefficients and a pole at infinity, then it is either a holomorphic genus-zero function invariant under a congruence group or of a certain degenerate type. As a special case, we prove the same conclusion for replicable functions of finite order, which were introduced by Conway and Norton in the context of monstrous moonshine. As an application, we introduce a class of Lie algebras with group actions, and show that the characters derived from them are weakly Hecke-monic. When the Lie algebras come from chiral conformal field theory in a certain sense, then the characters form holomorphic genus-zero functions invariant under a congruence group.
Given a nonzero integer , we know by Hermite’s Theorem that there exist only finitely many cubic number fields of discriminant . However, it can happen that two nonisomorphic cubic fields have the same discriminant. It is thus natural to ask whether there are natural refinements of the discriminant which completely determine the isomorphism class of the cubic field. Here we consider the trace form as such a refinement. For a cubic field of fundamental discriminant we show the existence of an element in Bhargava’s class group such that is completely determined by . By using one of Bhargava’s composition laws, we show that is a complete invariant whenever is totally real and of fundamental discriminant.
We classify the simple supersingular modules for the pro--Iwahori Hecke algebra of -adic by proving a conjecture by Vignéras about a modnumericalLanglands correspondence on the side of the Hecke modules. We define a process of induction for -modules in characteristic that reflects the parabolic induction for representations of the -adic general linear group and explore the semisimplification of the standard nonsupersingular -modules in light of this process.
We develop a patching machinery over the field of formal power series in two variables over an infinite field . We apply this machinery to prove that if is separably closed and is a finite group of order not divisible by , then there exists a -crossed product algebra with center if and only if the Sylow subgroups of are abelian of rank at most .
We consider models for genus-one curves of degree for , and , which arise in explicit -descent on elliptic curves. We prove theorems on the existence of minimal models with the same invariants as the minimal model of the Jacobian elliptic curve and provide simple algorithms for minimising a given model, valid over general number fields. Finally, for genus-one models defined over , we develop a theory of reduction and again give explicit algorithms for , and .