Products of Vector Valued Eisenstein Series

We prove that products of at most two vector valued Eisenstein series that originate in level 1 span all spaces of cusp forms for congruence subgroups. This can be viewed as an analogue in the level aspect to a result that goes back to Rankin, and Kohnen and Zagier, which focuses on the weight aspect. The main feature of the proof are vector valued Hecke operators. We recover several classical constructions from them, including classical Hecke operators, Atkin-Lehner involutions, and oldforms. As a corollary to our main theorem, we obtain a vanishing condition for modular forms reminiscent of period relations deduced by Kohnen and Zagier in the context of their previously mentioned result.Comment: accepted for publication in Forum Mathematicu

Arakelov class groups of random number fields

This is the final version. Available from Springer via the DOI in this record. Data availability statement: Data sharing is not applicable to this article, as no datasets were generated or analysed during the present work.The main purpose of the paper is to formulate a probabilistic model for Arakelov class groups in families of number fields, offering a correction to the Cohen–Lenstra–Martinet heuristic on ideal class groups. To that end, we show that Chinburg’s Ω(3) conjecture implies tight restrictions on the Galois module structure of oriented Arakelov class groups. As a consequence, we construct a new infinite series of counterexamples to the Cohen–Lenstra–Martinet heuristic, which have the novel feature that their Galois groups are non-abelian.Engineering and Physical Sciences Research Council (EPSRC)Engineering and Physical Sciences Research Council (EPSRC

Towards a Generic Model Theory: Automatic Bisimulations for Atomic, Molecular and First-order Logics

After observing that the truth conditions of connectives of non-classical logics are generally defined in terms of formulas of first-order logic (FOL), we introduce protologics, a class of logics whose connectives are defined by arbitrary first-order formulas. Then, we identify two subclasses of protologics which are particularly well-behaved. We call them atomic and molecular logics. Notions of invariance for atomic and molecular logics can be automatically defined from the truth conditions of their connectives, bisimulations do not need to be defined by hand on a case by case basis for each logic. Moreover, molecular logics behave as 'paradigmatic logics': every first-order logic and every protologic is as expressive as a molecular logic. Then, we prove a series of model-theoretical results for molecular logics which characterize them as fragments of FOL and which provide criteria for axiomatizability and definability of a class of models in these logics. In particular, we rediscover van Benthem's theorem for modal logic as a specific instance of our generic theorems and other results for modal intuitionistic logic and temporal logic. We also discover a wide range of novel results, such as for the Lambek calculus. Then, we apply our method and generic results to FOL and find out novel invariance notions for FOL, that we call predicate bisimulation and first-order bisimulation. They refine the usual notions of isomorphism and partial isomorphism. We prove generalizations as well as new versions of the Keisler theorems for countable languages in which isomorphisms are replaced by predicate bisimulations and first-order bisimulations

Genetic regulation of the tissue specific expression of Pyruvate kinase

SIGLEAvailable from British Library Document Supply Centre- DSC:D82962 / BLDSC - British Library Document Supply CentreGBUnited Kingdo