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1. Transversality of holomorphic maps into hyperquadrics

   https://doi.org/10.1007/s00208-025-03134-5  

   Huang, Xiaojun; Zhu, Weixia (June 2025, Mathematische Annalen)

   Abstract We study holomorphic mapsFfrom a smooth Levi non-degenerate real hypersurface$$ M_{\ell }\subset {\mathbb {C}}^n $$ $M_{\ell }\subset C^n$ into a hyperquadric$$ {\mathbb {H}}_{\ell '}^N $$ $H_{{\ell }^{\prime }}^N$ with signatures$$ \ell \le (n-1)/2 $$ $\ell \le (n-1)/2$ and$$ \ell '\le (N-1)/2,$$ ${\ell }^{\prime }\le (N-1)/2,$ respectively. Assuming that$$ N - n < n - 1,$$ $N-n<n-1,$ we prove that if$$ \ell = \ell ',$$ $\ell ={\ell }^{\prime },$ thenFis either CR transversal to$$ {\mathbb {H}}_{\ell }^N $$ $H_{\ell }^N$ at every point of$$ M_{\ell },$$ $M_{\ell },$ or it maps a neighborhood of$$ M_{\ell } $$ $M_{\ell }$ in$$ {\mathbb {C}}^n $$ $C^n$ into$$ {\mathbb {H}}_{\ell }^N.$$ $H_{\ell }^N.$ Furthermore, in the case where$$ \ell ' > \ell ,$$ ${\ell }^{\prime }>\ell ,$ we show that ifFis not CR transversal at$$0\in M_\ell ,$$ $0\in M_{\ell },$ then it must be transversally flat. The latter is best possible. 

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2. Ramanujan’s partition generating functions modulo $$\ell $$

   https://doi.org/10.1007/s11139-025-01241-0  

   Bringmann, Kathrin; Craig, William; Ono, Ken (November 2025, The Ramanujan Journal)

   Abstract For the partition functionp(n), Ramanujan proved the striking identities$$\begin{aligned} \begin{aligned} \mathcal {P}_5(q):=\sum _{n\ge 0} p(5n+4)q^n&=5\prod _{n\ge 1} \frac{\left( q^5;q^5\right) _{\infty }^5}{(q;q)_{\infty }^6},\\ \mathcal {P}_{7}(q):=\sum _{n\ge 0} p(7n+5)q^n&=7\prod _{n\ge 1}\frac{\left( q^7;q^7\right) _{\infty }^3}{(q;q)_{\infty }^4}+49q \prod _{n\ge 1}\frac{\left( q^7;q^7\right) _{\infty }^7}{(q;q)_{\infty }^8}, \end{aligned} \end{aligned}$$ $\begin{array}{c}\begin{array}{cc}{P}_5\left(q\right):=\sum _{n\ge 0}p(5n+4)q^n& =5\prod _{n\ge 1}\frac{{\left(q^5,;,q^5\right)}_{\infty }^5}{{(q;q)}_{\infty }^6},\\ P_7\left(q\right):=\sum _{n\ge 0}p(7n+5)q^n& =7\prod _{n\ge 1}\frac{{\left(q^7,;,q^7\right)}_{\infty }^3}{{(q;q)}_{\infty }^4}+49q\prod _{n\ge 1}\frac{{\left(q^7,;,q^7\right)}_{\infty }^7}{{(q;q)}_{\infty }^8},\end{array}\end{array}$ where$$(q;q)_{\infty }:=\prod _{n\ge 1}(1-q^n).$$ ${(q;q)}_{\infty }:={\prod }_{n\ge 1}(1-q^n).$ As these identities imply his celebrated congruences modulo 5 and 7, it is natural to seek, for primes$$\ell \ge 5,$$ $\ell \ge 5,$ closed form expressions of the power series$$ \mathcal {P}_{\ell }(q):=\sum _{n\ge 0} p(\ell n-\delta _{\ell })q^n\pmod {\ell }, $$ $P_{\ell }\left(q\right):=\sum _{n\ge 0}p(\ell n-{\delta }_{\ell })q^n(mod\ell ),$ where$$\delta _{\ell }:=\frac{\ell ^2-1}{24}.$$ ${\delta }_{\ell }:=\frac{{\ell }^2-1}{24}.$ In this paper, we prove that$$ \mathcal {P}_{\ell }(q)\equiv c_{\ell } \dfrac{\mathcal {T}_{\ell }(q)}{\left( q^\ell ; q^\ell \right) _\infty } \pmod {\ell }, $$ $P_{\ell }\left(q\right)\equiv c_{\ell }\frac{T_{\ell }\left(q\right)}{{\left(q^{\ell },;,q^{\ell }\right)}_{\infty }}(mod\ell ),$ where$$c_{\ell }\in \mathbb Z$$ $c_{\ell }\in Z$ is explicit and$${\mathcal {T}}_{\ell }(q)$$ $T_{\ell }\left(q\right)$ is the generating function for the Hecke traces of$$\ell $$ $\ell$ -ramified values of special Dirichlet series for weight$$\ell -1$$ $\ell -1$ cusp forms on$$\textrm{SL}_2(\mathbb Z)$$ ${\text{SL}}_2\left(Z\right)$ . This is a new proof of Ramanujan’s congruences modulo 5, 7, and 11, as there are no nontrivial cusp forms of weight 4, 6, and 10. 

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3. Restricted spaces of holomorphic sections vanishing along subvarieties

   https://doi.org/10.1007/s00209-025-03706-w  

   Coman, Dan; Marinescu, George; Nguyên, Viêt-Anh (May 2025, Mathematische Zeitschrift)

   Abstract LetXbe a compact normal complex space of dimensionnandLbe a holomorphic line bundle onX. Suppose that$$\Sigma =(\Sigma _1,\ldots ,\Sigma _\ell )$$ $\Sigma =({\Sigma }_1,\dots ,{\Sigma }_{\ell })$ is an$$\ell $$ $\ell$ -tuple of distinct irreducible proper analytic subsets ofX,$$\tau =(\tau _1,\ldots ,\tau _\ell )$$ $\tau =({\tau }_1,\dots ,{\tau }_{\ell })$ is an$$\ell $$ $\ell$ -tuple of positive real numbers, and let$$H^0_0(X,L^p)$$ $H_0^0(X,L^p)$ be the space of holomorphic sections of$$L^p:=L^{\otimes p}$$ $L^p:=L^{\otimes p}$ that vanish to order at least$$\tau _jp$$ ${\tau }_{j}p$ along$$\Sigma _j$$ ${\Sigma }_j$ ,$$1\le j\le \ell $$ $1\le j\le \ell$ . If$$Y\subset X$$ $Y\subset X$ is an irreducible analytic subset of dimensionm, we consider the space$$H^0_0 (X|Y, L^p)$$ $H_0^0\left(X\right|Y,L^p)$ of holomorphic sections of$$L^p|_Y$$ $L^p{|}_Y$ that extend to global holomorphic sections in$$H^0_0(X,L^p)$$ $H_0^0(X,L^p)$ . Assuming that the triplet$$(L,\Sigma ,\tau )$$ $(L,\Sigma ,\tau )$ is big in the sense that$$\dim H^0_0(X,L^p)\sim p^n$$ $dim{H}_0^0(X,L^p)\sim p^n$ , we give a general condition onYto ensure that$$\dim H^0_0(X|Y,L^p)\sim p^m$$ $dim{H}_0^0\left(X\right|Y,L^p)\sim p^m$ . WhenLis endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces$$H^0_0(X|Y,L^p)$$ $H_0^0\left(X\right|Y,L^p)$ converge to a certain equilibrium current onY. We apply this to the study of the equidistribution of zeros inYof random holomorphic sections in$$H^0_0(X|Y,L^p)$$ $H_0^0\left(X\right|Y,L^p)$ as$$p\rightarrow \infty $$ $p\to \infty$ . 

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4. Sunflowers in Set Systems with Small VC-Dimension

   https://doi.org/10.1007/s00493-025-00186-8  

   Balogh, József; Bernshteyn, Anton; Delcourt, Michelle; Ferber, Asaf; Pham, Huy Tuan (December 2025, Combinatorica)

   Abstract A family ofrdistinct sets$$\{A_1,\ldots , A_r\}$$ $\{A_1,\dots ,A_r\}$ is anr-sunflower if for all$$1 \leqslant i < j\leqslant r$$ $1⩽i<j⩽r$ and$$1 \leqslant i' < j'\leqslant r$$ $1⩽i^{\prime }<j^{\prime }⩽r$ , we have$$A_i\cap A_j = A_{i'}\cap A_{j'}$$ $A_i\cap A_j=A_{i^{\prime }}\cap A_{j^{\prime }}$ . Erdős and Rado conjectured in 1960 that every family$$\mathcal {H}$$ $H$ of$$\ell $$ $\ell$ -element sets of size at least$$K(r)^\ell $$ $K{\left(r\right)}^{\ell }$ contains anr-sunflower, whereK(r) is some function that depends only onr. We prove that if$$\mathcal {H}$$ $H$ is a family of$$\ell $$ $\ell$ -element sets of VC-dimension at mostdand$$|\mathcal H| > (C r(\log d+\log ^*\ell ))^\ell $$ $\left|H\right|>{\left(Cr(logd+{log}^{\ast }\ell )\right)}^{\ell }$ for some absolute constant$$C > 0$$ $C>0$ , then$$\mathcal {H}$$ $H$ contains anr-sunflower. This improves a recent result of Fox, Pach, and Suk. When$$d=1$$ $d=1$ , we obtain a sharp bound, namely that$$|\mathcal H| > (r-1)^\ell $$ ${\left|H\right|>(r-1)}^{\ell }$ is sufficient. Along the way, we establish a strengthening of the Kahn–Kalai conjecture for set families of bounded VC-dimension, which is of independent interest. 

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5. Relating elliptic curve point-counting and solutions of quadratic forms with congruence conditions

   https://doi.org/10.1007/s11139-026-01341-5  

   Mondal, Koustav (March 2026, The Ramanujan Journal)

   Abstract In this paper, we analyze the theta series associated to the quadratic form$$Q(\vec {x}) :=x_1^2+x_2^2+x_3^2+x_4^2$$ $Q\left(\overrightarrow{x}\right):=x_1^2+x_2^2+x_3^2+x_4^2$ with congruence conditions on$$x_i$$ $x_i$ modulo 2, 3, 4 and 6. By employing special operators on modular, non-holomorphic Eisenstein series of weight 2, we construct a basis for Eisenstein space for levels$$2^k, k\le 7$$ $2^k,k\le 7$ ,$$3^{\ell }, \ell \le 3$$ $3^{\ell },\ell \le 3$ andp, for odd primep. Using the relation between the trace of Frobenius on an elliptic curve and the Fourier coefficients of the cusp form part of theta series corresponding toQ, we establish relation between the number of integer solutions to the equation$$Q(\vec {x}) = p$$ $Q\left(\overrightarrow{x}\right)=p$ and the number of$$\mathbb {F}_p$$ $F_p$ -rational points on the associated elliptic curve under certain congruence conditions onp. 

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