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1. $$p^\infty $$-Selmer groups and rational points on CM elliptic curves

   https://doi.org/10.1007/s40316-022-00203-y  

   Burungale, Ashay; Castella, Francesc; Skinner, Christopher; Tian, Ye (July 2022, Annales mathématiques du Québec)

   Résumé Soit $$E/{\mathbb {Q}}$$ E / Q une courbe elliptique à multiplication complexe et p un nombre premier de bonne réduction ordinaire pour E . Nous montrons que si $${\mathrm{corank}}_{{\mathbb {Z}}_p}{\mathrm{Sel}}_{p^\infty }(E/{\mathbb {Q}})=1$$ corank Z p Sel p ∞ ( E / Q ) = 1 , alors E a un point d’ordre infini. Le point de non-torsion provient d’un point de Heegner, et donc $${{{\mathrm{ord}}}}_{s=1}L(E,s)=1$$ ord s = 1 L ( E , s ) = 1 , ce qui donne une réciproque à un théorème de Gross–Zagier, Kolyvagin, et Rubin dans l’esprit de [49, 54]. Pour $$p>3$$ p > 3 , cela donne une nouvelle preuve du résultat principal de [12], que notre approche étend à tous les nombres premiers. L’approche se généralise aux courbes elliptiques à multiplication complexe sur les corps totalement réels [4]. 

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2. A Duality Principle for Groups II: Multi-frames Meet Super-Frames

   https://doi.org/10.1007/s00041-020-09792-0  

   Balan, R.; Dutkay, D.; Han, D.; Larson, D.; Luef, F. (December 2020, Journal of Fourier Analysis and Applications)

   null (Ed.)

   Abstract The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that $$\{\pi _{\Lambda }(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda }(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ ( m , n ) g k } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})\oplus \cdots \oplus L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) ⊕ ⋯ ⊕ L 2 ( R d ) if and only if $$\cup _{i=1}^{k}\{\pi _{\Lambda ^{o}}(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ o ( m , n ) g i } m , n ∈ Z d is a Riesz sequence, and $$\cup _{i=1}^{k} \{\pi _{\Lambda }(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ ( m , n ) g i } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) if and only if $$\{\pi _{\Lambda ^{o}}(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda ^{o}}(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ o ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ o ( m , n ) g k } m , n ∈ Z d is a Riesz sequence, where $$\pi _{\Lambda }$$ π Λ and $$\pi _{\Lambda ^{o}}$$ π Λ o is a pair of Gabor representations restricted to a time–frequency lattice $$\Lambda $$ Λ and its adjoint lattice $$\Lambda ^{o}$$ Λ o in $${\mathbb {R}}\,^{d}\times {\mathbb {R}}\,^{d}$$ R d × R d . 

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3. Hermite Rational Function Interpolation with Error Correction

   https://doi.org/10.1007/978-3-030-60026-6_19  

   Kaltofen, Erich L.; Pernet, Clement; Yang, Zhi-Hong (September 2020, Proc. Computer Algebra in Scientific Computing (CASC) 2020)

   We generalize Hermite interpolation with error correction, which is the methodology for multiplicity algebraic error correction codes, to Hermite interpolation of a rational function over a field K from function and function derivative values. We present an interpolation algorithm that can locate and correct <= E errors at distinct arguments y in K where at least one of the values or values of a derivative is incorrect. The upper bound E for the number of such y is input. Our algorithm sufficiently oversamples the rational function to guarantee a unique interpolant. We sample (f/g)^(j)(y[i]) for 0 <= j <= L[i], 1 <= i <= n, y[i] distinct, where (f/g)^(j) is the j-th derivative of the rational function f/g, f, g in K[x], GCD(f,g)=1, g <= 0, and where N = (L[1]+1)+...+(L[n]+1) >= C + D + 1 + 2(L[1]+1) + ... + 2(L[E]+1) where C is an upper bound for deg(f) and D an upper bound for deg(g), which are input to our algorithm. The arguments y[i] can be poles, which is truly or falsely indicated by a function value infinity with the corresponding L[i]=0. Our results remain valid for fields K of characteristic >= 1 + max L[i]. Our algorithm has the same asymptotic arithmetic complexity as that for classical Hermite interpolation, namely soft-O(N). For polynomials, that is, g=1, and a uniform derivative profile L[1] = ... = L[n], our algorithm specializes to the univariate multiplicity code decoder that is based on the 1986 Welch-Berlekamp algorithm. 

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4. Intersection complexes and unramified 𝐿-factors

   https://doi.org/10.1090/jams/990  

   Sakellaridis, Yiannis; Wang, Jonathan (July 2022, Journal of the American Mathematical Society)

   Let X X be an affine spherical variety, possibly singular, and L + X \mathsf L^+X its arc space. The intersection complex of L + X \mathsf L^+X , or rather of its finite-dimensional formal models, is conjectured to be related to special values of local unramified L L -functions. Such relationships were previously established in Braverman–Finkelberg–Gaitsgory–Mirković for the affine closure of the quotient of a reductive group by the unipotent radical of a parabolic, and in Bouthier–Ngô–Sakellaridis for toric varieties and L L -monoids. In this paper, we compute this intersection complex for the large class of those spherical G G -varieties whose dual group is equal to G ˇ \check G , and the stalks of its nearby cycles on the horospherical degeneration of X X . We formulate the answer in terms of a Kashiwara crystal, which conjecturally corresponds to a finite-dimensional G ˇ \check G -representation determined by the set of B B -invariant valuations on X X . We prove the latter conjecture in many cases. Under the sheaf–function dictionary, our calculations give a formula for the Plancherel density of the IC function of L + X \mathsf L^+X as a ratio of local L L -values for a large class of spherical varieties. 

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5. Random walk on group extensions

   https://doi.org/10.1090/tran/9309  

   Golsefidy, S Alireza; Srinivas, Srivatsa (January 2025, Transactions of the American Mathematical Society)

   We study random walks on various group extensions. Under certain bounded generation and bounded scaled conditions, we estimate the spectral gap of a random walk on a quasi-random-by-nilpotent group in terms of the spectral gap of its projection to the quasi-random part. We also estimate the spectral gap of a random-walk on a product of two quasi-random groups in terms of the spectral gap of its projections to the given factors. Based on these results, we estimate the spectral gap of a random walk on the ${\mathbb{F}}_q$-points of a perfect algebraic group $\mathbb{G}$in terms of the spectral gap of its projections to the almost simple factors of the semisimple quotient of $\mathbb{G}$. These results extend a work of Lindenstrauss and Varjú and an earlier work of the authors. Moreover, using a result of Breuillard and Gamburd, we show that there is an infinite set $\mathcal{P}$of primes of density one such that, if $k$is a positive integer and $\mathbb{G}=\mathbb{U}⋊<\#comment/>({\mathrm{SL}}_2{)}_{\mathbb{Q}}^m$is a perfect group and $\mathbb{U}$is a unipotent group, then the family of all the Cayley graphs of $\mathbb{G}\left(\mathbb{Z}/\underset{i=1}{\overset{k}{\prod <\#comment/>}}{p}_i\mathbb{Z}\right)$, $p_i\in <\#comment/>\mathcal{P}$, is a family of expanders. 

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