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1. The Lambda Invariants at CM Points

   https://doi.org/10.1093/imrn/rnz230  

   Yang, Tonghai; Yin, Hongbo; Yu, Peng (November 2019, International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper, we show that $$\lambda (z_1) -\lambda (z_2)$$, $$\lambda (z_1)$$, and $$1-\lambda (z_1)$$ are all Borcherds products on $$X(2) \times X(2)$$. We then use the big CM value formula of Bruinier, Kudla, and Yang to give explicit factorization formulas for the norms of $$\lambda (\frac{d+\sqrt d}2)$$, $$1-\lambda (\frac{d+\sqrt d}2)$$, and $$\lambda (\frac{d_1+\sqrt{d_1}}2) -\lambda (\frac{d_2+\sqrt{d_2}}2)$$, with the latter under the condition $$(d_1, d_2)=1$$. Finally, we use these results to show that $$\lambda (\frac{d+\sqrt d}2)$$ is always an algebraic integer and can be easily used to construct units in the ray class field of $${\mathbb{Q}}(\sqrt{d})$$ of modulus $$2$$. In the process, we also give explicit formulas for a whole family of local Whittaker functions, which are of independent interest. 

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2. On the Convolution Inequality f ≥ f ⋆ f

   https://doi.org/10.1093/imrn/rnaa350  

   Carlen, Eric A; Jauslin, Ian; Lieb, Elliott H; Loss, Michael P (January 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We consider the inequality $$f \geqslant f\star f$$ for real functions in $$L^1({\mathbb{R}}^d)$$ where $$f\star f$$ denotes the convolution of $$f$$ with itself. We show that all such functions $$f$$ are nonnegative, which is not the case for the same inequality in $L^p$ for any $$1 < p \leqslant 2$$, for which the convolution is defined. We also show that all solutions in $$L^1({\mathbb{R}}^d)$$ satisfy $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x \leqslant \tfrac 12$$. Moreover, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$, then $$f$$ must decay fairly slowly: $$\int _{{\mathbb{R}}^{\textrm{d}}}|x| f(x)\ \textrm{d}x = \infty $$, and this is sharp since for all $r< 1$, there are solutions with $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$ and $$\int _{{\mathbb{R}}^{\textrm{d}}}|x|^r f(x)\ \textrm{d}x <\infty $$. However, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x =: a < \tfrac 12$$, the decay at infinity can be much more rapid: we show that for all $$a<\tfrac 12$$, there are solutions such that for some $$\varepsilon>0$$, $$\int _{{\mathbb{R}}^{\textrm{d}}}e^{\varepsilon |x|}f(x)\ \textrm{d}x < \infty $$. 

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3. Gröbner bases, symmetric matrices, and type C Kazhdan–Lusztig varieties

   https://doi.org/10.1112/jlms.12856  

   Escobar, Laura; Fink, Alex; Rajchgot, Jenna; Woo, Alexander (February 2024, Journal of the London Mathematical Society)

   Abstract We study a class of combinatorially defined polynomial ideals that are generated by minors of a generic symmetric matrix. Included within this class are the symmetric determinantal ideals, the symmetric ladder determinantal ideals, and the symmetric Schubert determinantal ideals of A. Fink, J. Rajchgot, and S. Sullivant. Each ideal in our class is a type C analog of a Kazhdan–Lusztig ideal of A. Woo and A. Yong; that is, it is the scheme‐theoretic defining ideal of the intersection of a type C Schubert variety with a type C opposite Schubert cell, appropriately coordinatized. The Kazhdan–Lusztig ideals that arise are exactly those where the opposite cell is 123‐avoiding. Our main results include Gröbner bases for these ideals, prime decompositions of their initial ideals (which are Stanley–Reisner ideals of subword complexes), and combinatorial formulas for their multigraded Hilbert series in terms of pipe dreams. 

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4. Non-vanishing of class group L -functions for number fields with a small regulator

   https://doi.org/10.1112/S0010437X20007472  

   Khayutin, Ilya (November 2020, Compositio Mathematica)

   null (Ed.)

   Let $$E/\mathbb {Q}$$ be a number field of degree $$n$$ . We show that if $$\operatorname {Reg}(E)\ll _n |\!\operatorname{Disc}(E)|^{1/4}$$ then the fraction of class group characters for which the Hecke $$L$$ -function does not vanish at the central point is $$\gg _{n,\varepsilon } |\!\operatorname{Disc}(E)|^{-1/4-\varepsilon }$$ . The proof is an interplay between almost equidistribution of Eisenstein periods over the toral packet in $$\mathbf {PGL}_n(\mathbb {Z})\backslash \mathbf {PGL}_n(\mathbb {R})$$ associated to the maximal order of $$E$$ , and the escape of mass of the torus orbit associated to the trivial ideal class. 

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5. On ℓ p -Gaussian–Grothendieck Problem

   https://doi.org/10.1093/imrn/rnab311  

   Chen, Wei-Kuo; Sen, Arnab (November 2021, International Mathematics Research Notices)

   Abstract For $$p\geq 1$$ and $$(g_{ij})_{1\leq i,j\leq n}$$ being a matrix of i.i.d. standard Gaussian entries, we study the $$n$$-limit of the $$\ell _p$$-Gaussian–Grothendieck problem defined as $$\begin{align*} & \max\Bigl\{\sum_{i,j=1}^n g_{ij}x_ix_j: x\in \mathbb{R}^n,\sum_{i=1}^n |x_i|^p=1\Bigr\}. \end{align*}$$The case $p=2$ corresponds to the top eigenvalue of the Gaussian orthogonal ensemble; when $$p=\infty $$, the maximum value is essentially the ground state energy of the Sherrington–Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In the present work, we focus on the cases $$1\leq p<2$$ and $$2<p<\infty .$$ For the former, we compute the limit of the $$\ell _p$$-Gaussian–Grothendieck problem and investigate the structure of the set of all near optimizers along with stability estimates. In the latter case, we show that this problem admits a Parisi-type variational representation and the corresponding optimizer is weakly delocalized in the sense that its entries vanish uniformly in a polynomial order of $$n^{-1}$$. 

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