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1. SERRE WEIGHTS AND BREUIL’S LATTICE CONJECTURE IN DIMENSION THREE

   https://doi.org/10.1017/fmp.2020.1  

   LE, DANIEL; LE HUNG, BAO V.; LEVIN, BRANDON; MORRA, STEFANO (January 2020, Forum of Mathematics, Pi)

   We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$ -arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $$p$$ . This is a generalization to $$\text{GL}_{3}$$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights $(2,1,0)$ as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame $$n$$ -dimensional Galois representations’, Duke Math. J.   149 (1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math.   212 (1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group $$\text{GL}_{3}(\mathbb{F}_{q})$$ . 

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2. Extremal weights and a tameness criterion for mod $$p$$ Galois representations

   https://doi.org/10.4171/JEMS/1513  

   Le, Daniel; Le_Hung, Bao Viet; Levin, Brandon; Morra, Stefano (September 2024, Journal of the European Mathematical Society)

   We study the weight part of Serre’s conjecture for genericn-dimensional modpGalois representations. We first generalize Herzig’s conjecture to the case where the field is ramified atpand prove the weight elimination direction of the conjecture. We then introduce a new class of weights associated ton-dimensional local modprepresentations which we callextremal weights. Using a “Levi reduction” property of certain potentially crystalline Galois deformation spaces, we prove the modularity of these weights. As a consequence, we deduce the weight part of Serre’s conjecture for unit groups of some division algebras in generic situations. 

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3. Nutrient management offsets the effect of deoxygenation and warming on nitrous oxide emissions in a large US estuary

   https://doi.org/10.1126/sciadv.adq5014  

   Tang, Weiyi; Da, Fei; Tracey, John C; Intrator, Naomi; Kunes, Moriah A; Lee, Jenna A; Wan, Xianhui Sean; Jayakumar, Amal; Friedrichs, Marjorie_A M; Ward, Bess B (December 2024, Science Advances)

   Many estuaries experience eutrophication, deoxygenation and warming, with potential impacts on greenhouse gas emissions. However, the response of N₂O production to these changes is poorly constrained. Here we applied nitrogen isotope tracer incubations to measure N₂O production under experimentally manipulated changes in oxygen and temperature in the Chesapeake Bay—the largest estuary in the United States. N₂O production more than doubled from nitrification and increased exponentially from denitrification when O₂was decreased from >20 to <5 micromolar. Raising temperature from 15° to 35°C increased N₂O production 2- to 10-fold. Developing a biogeochemical model by incorporating these responses, N₂O emissions from the Chesapeake Bay were estimated to decrease from 157 to 140 Mg N year⁻¹from 1986 to 2016 and further to 124 Mg N year⁻¹in 2050. Although deoxygenation and warming stimulate N₂O production, the modeled decrease in N₂O emissions, attributed to decreased nutrient inputs, indicates the importance of nutrient management in curbing greenhouse gas emissions, potentially mitigating climate change. 

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4. The classifying element for quotients of Fermat curves

   https://doi.org/10.1080/00927872.2024.2346299  

   Duque-Rosero, Juanita; Pries, Rachel (May 2024, Communications in Algebra)

   Suppose C is a cyclic Galois cover of the projective line branched at the three points 0, 1, and ∞. Under a mild condition on the ramification, we determine the structure of the graded Lie algebra of the lower central series of the fundamental group of C in terms of a basis which is well-suited to studying the action of the absolute Galois group of Q. 

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5. Galois Symmetry Induced by Hecke Relations in Rational Conformal Field Theory and Associated Modular Tensor Categories

   https://doi.org/10.1088/1751-8121/ab8e03  

   Harvey, J.A.; Hu, Y.; Wu, Y. (January 2020, Journal of physics)

   null (Ed.)

   Hecke operators relate characters of rational conformal field theories (RCFTs) with different central charges, and extend the previously studied Galois symmetry of modular representations and fusion algebras. We show that the conductor N of an RCFT and the quadratic residues modulo N play an important role in the computation and classification of Galois permutations. We establish a field correspondence in different theories through the picture of effective central charge, which combines Galois inner automorphisms and the structure of simple currents. We then make a first attempt to extend Hecke operators to the full data of modular tensor categories. The Galois symmetry encountered in the modular data transforms the fusion and the braiding matrices as well, and yields isomorphic structures in theories related by Hecke operators. 

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