In this note we give a lower bound for the second moment of a ratio of zeta functions summed over the non-trivial zeros of the Riemann zeta function that is half the size of the conjectured value. Our result is conditional on the assumption of the Riemann Hypothesis and that all the non-trivial zeros of the zeta function are simple.

In a recent paper, the authors introduced an infinite hierarchy of $q$-hypergeometric identities, of which the first three orders, $0$, $1$, and $2$, relate to the partition theorems of Euler, Lebesgue, and Capparelli, and stated a partition theorem at order 4 which lies beyond Capparelli's theorem. Here, we first state certain partition theorems that hold at all even orders beyond Capparelli and provide bijective proofs for these theorems. In doing so, we show that there is a fourfold infinite hierarchy of partition theorems that emanates from Capparelli's theorem, which is the base case. It is also shown that the equality of two of the four generating functions holds for all orders, odd and even. Lastly, a very general framework for the remaining two functions is constructed via the method of weighted words, encompassing all possible orders and yielding several infinite hierarchies with different dilations and translations.

We establish structure theorems for polarised flag type quotients arising in the study of the moduli space of supersingular abelian varieties. In particular, we prove a refined normal form for the top level of these flags and derive an explicit, computable descent criterion for quasi-polarisations. These results provide a complete classification of the first and last step of these flags. As an application, we compute polarised flag type quotients in dimensions $g\leq 5$, describing the fibers of the truncation morphisms as classification objects of supersingular abelian varieties.

We study general properties of the perfectoidization of finite algebras over a perfectoid ring, which helps to understand some precise and explicit descriptions. For example, we prove that if $A=R[t]/(m(t))$ where $m(t)$ is monic, $R$ is perfectoid and the discriminant $d$ of $m(t)$ is a non-zero divisor of $R$ satisfying a bounded torsion condition, then $dA_{\mathrm{pfd}}\subset A$. We also prove a density criterion reducing the construction of the perfectoidization to adjoining suitable $p$-power roots modulo $p$. In the second part of the paper, we compute perfectoidizations in several families of examples, including Kummer-type extensions and split finite algebras.

In this article, we relate any Farey triangle in the extended upper half-plane to a variant of Colmez--Fontaine's fundamental lemma in $p$-adic Hodge theory. In particular, their original fundamental lemma corresponds to the fundamental Farey triangle $(\frac{1}{0},\frac{1}{1},\frac{0}{1})$.

We show that for any $A\subseteq \mathbb{F}_q^n$ lacking non-trivial solutions to a translation-invariant linear equation of genus one, meaning that no nonempty proper subset of the coefficients sums to $0$, there is a set $B\subseteq \mathbb{F}_q^m$ in some higher dimension which also lacks non-trivial solutions, such that \[|B|^{1/m}>|A|^{1/n}.\] In particular, this implies that no fixed cap set in $\mathbb{F}_3^n$ gives an asymptotically optimal lower bound by direct products alone.

We prove two conjectural $q$-congruences proposed by Guo. The first is Conjecture 7.2 in Guo's work on $q$-analogues of two ``divergent'' Ramanujan-type supercongruences; it asserts a square-cyclotomic congruence for a truncated $q$-analogue of a Ramanujan-type sum when $n\equiv1\pmod4$. The second is Conjecture 4.1 in Guo's extension of Van Hamme's $(A.2)$ supercongruence; it gives divisibility modulo $[n]$ for a family of truncated basic hypergeometric sums with a parameter $s$. The proof of the first result relies on a known Watson-transformation congruence obtained by Guo. The proof of the second result is based on period decomposition at primitive roots of unity and a reflection cancellation inside residue blocks.

The classical p-adic Cauchy theorem asserts that formal solutions of ordinary p-adic differential equations are convergent. In this article we establish a geometric analogue of this result for arbitrary smooth rigid-analytic varieties. More precisely, we show that the horizontal sections functor defined using the overconvergent de Rham period structure sheaf agrees with Scholze's horizontal sections functor defined using OBdR. Equivalently, every formal solution arising from Scholze's construction is already overconvergent. As an application, we identify Scholze's horizontal sections functor with the de Rham functor for D-cap-modules on vector bundles with flat connection.

In this article, we present a new approach to studying multivariate period rings that is more consistent with classical theory and provides a clearer description of their structure. We also prove that the category of $B$-admissible representations forms a Tannakian subcategory of the category of representations of $G_{K,\Delta}$ by defining an analogue of $(F,G)$-regular rings, which is central to the classification of representations in multivariate $p$-adic Hodge theory.

The aim of this note is to present a simple strategy to answer negatively a decomposition question on tori posed by Colliot-Thélène and Sansuc in the article \textit{Principal Homogeneous Spaces under Flasque Tori: Applications} of 1987. We then deduce a torus $T$ over $\mathbb{Q}$ and a prime number $p$ such that $T(\mathbb{Z}_p)\,T(\mathbb{Q})\not=T(\mathbb{Q}_p)$, where $T(\mathbb{Z}_p)$ denotes the maximal compact subgroup of $T(\mathbb{Q}_p)$.

We establish explicit subconvex bounds for central values of Rankin--Selberg $L$-functions $L(1/2,\pi\times\pi')$ associated with pairs of unitary cuspidal automorphic representations of $\mathrm{GL}_2$ over a number field. Building on the spectral reciprocity framework of Michel and Venkatesh, we develop a refined, fully explicit form of spectral reciprocity that allows for precise control of conductors and local test vectors. As a consequence, we obtain an explicit subconvex bound, which, even over $F=\mathbb{Q}$, improves all previously known results. We further apply these bounds to several arithmetic problems. These include effective equidistribution of CM suborbits on quaternionic Shimura varieties, quantitative equidistribution of totally geodesic submanifolds, a uniform quantitative form of dihedral quantum unique ergodicity over number fields, and an application to distinguishing cuspidal automorphic representations.

Motivated in part by anomaly cancellation for integral charge spectra in chiral gauge theory, we study the symmetric locus in the ideal degree-three Prouhet-Tarry-Escott problem. A symmetric integer solution is one whose entries are paired about a common center $c\in \frac12\mathbb Z$. This symmetry reduces the problem to a sum-of-two-squares equation, $x^2+y^2=u^2+v^2$, in integer variables, subject to the appropriate parity conditions. Thus the problem is governed by representations as sums of two squares. For the full symmetric locus, let $N_{\mathrm{sym}}(H)$ denote the number of nontrivial symmetric integer solutions of height at most $H$, counted with unordered multiset conventions and summed over the admissible centers. Then \begin{align*} N_{\mathrm{sym}}(H) = \frac{4\log 2}{3\pi^2}H^3\log H+O(H^3). \end{align*} The logarithmic enhancement comes from the second moment of the sum-of-two-squares representation function. In particular, the symmetric locus is larger than one would expect from the naive $H^3$ degree-weighted box-counting scale alone. This asymptotic identifies a large arithmetically structured subfamily of the ideal degree-three solution space, and suggests that paired anomaly-free integral charge spectra reflect a fundamental number-theoretic structure.

We show how to use divisors on the projectivized Hodge bundle to construct special vector-valued modular forms and then apply invariant theory to construct all vector-valued Siegel modular forms of level two and degree two. Thus we construct all modular forms in terms of certain basic modular forms that are intimately connected to the moduli of curves of genus two.

We show that up to isomorphism there are exactly twenty pairs $(C,E)$, where $C$ is a genus-$2$ curve over ${\mathbf C}$, where $E$ is an elliptic curve over ${\mathbf C}$, and where for every integer $n>1$ there is a map of degree $n$ from $C$ to $E$. We also show that for every genus-$2$ curve $C$, there is an integer $n$ with $1 < n \le 59$ such that there is no minimal degree-$n$ map from $C$ to an elliptic curve.

We investigate the mean value of the first moment of primitive quartic $L$-functions over $\mathbb{F}_q(T)$ in the non-Kummer setting. Specifically, we study the sum \begin{equation*} \sum_{\substack{\chi\ primitive\ quartic\\ \chi^2 primitive\\ genus(\chi)=g}}L_q(\frac{1}{2}, \chi), \end{equation*} where $L_q(s,\chi)$ denotes the $L$-function associated with primitive quartic character $\chi$. Using double Dirichlet series, we derive an error term of size $q^{(\frac{3}{5}+\varepsilon)g}$.

We give an equivalent condition for the validity of the Hasse norm principle for finite separable extensions of prime squared degree of global fields. Our theorem recovers the result of Drakokhrust--Platonov, which claims that the Hasse norm principle holds for adequate extensions of prime squared degree.

This article establishes a Riemann-Hilbert correspondence in rigid-analytic geometry. We construct an explicit solution functor and prove that it is fully faithful on Ardakov-Wadsley's coadmissible D-cap-modules. For vector bundles with flat connection, our functor is canonically identified with Scholze's horizontal sections functor.

Motivated by the theory of p-adic differential equations and p-adic geometric representation theory, we introduce overconvergent variants of Fontaine's classical period rings. In particular, we study the positive overconvergent de Rham period ring, which is the stalk of the structure sheaf of the analytic Fargues-Fontaine curve at infinity. Our main results include the computation of the Galois cohomology of these overconvergent period rings, as well as the cohomology of the associated period sheaves and period structure sheaves.

We define the adelic hypergeometric function of special Gaussian type by means of a tower of hypergeometric curves. This function takes values in an adelic completed group ring and interpolates all the hypergeometric functions of the same type over all finite fields. It specializes at the unit argument to the adelic beta function of Ihara and Anderson. We prove some transformation formulas and a summation formula for the adelic hypergeometric function, which are known classically for complex hypergeometric functions.

In this paper, we characterize the vanishing of twisted central $L$-values attached to newforms of square-free level in terms of certain polynomials of quadratic forms introduced by Zagier and the action of finitely many Hecke operators thereon. To be more precise, we establish that a twisted central $L$-value attached to a newform vanishes if and only if a certain explicitly computable polynomial is constant. We describe these constants explicitly in two different ways. One of the descriptions involves the generalized Hurwitz class numbers, which were introduced by Pei and Wang in $2003$. We provide some numerical examples and conclude by offering some questions for future work.

This note reports some advances in the Equivariant Minimal Model Program (EMMP) for non-isomorphic surjective endomorphisms and their applications in complex and arithmetic dynamics.

Let $\pi$ be a cuspidal automorphic representation of a general linear group over the rational numbers. We establish a subconvex bound for the standard $L$-function of $\pi$ in the $t$-aspect. More generally, we address the spectral aspect in the case of uniform parameter growth.

Reasons for the emergence of Chebyshev's bias were investigated. The Deep Riemann Hypothesis (DRH) enables us to reveal that the bias is a natural phenomenon for achieving a well-balanced disposition of the whole sequence of primes, in the sense that the Euler product converges at the center. By means of a weighted counting function of primes, the authors succeed in expressing magnitudes of the deflection by a certain asymptotic formula under the assumption of DRH, which provides a new formulation of Chebyshev's bias. For any Galois extension of global fields and for any element $σ$ in the Galois group, we have established a criterion of the bias of primes whose Frobenius elements are equal to $σ$ under the assumption of DRH. As an application we have obtained a bias toward non-splitting and non-principle primes in abelian extensions under DRH. In positive characteristic cases, DRH is known, and all these results hold unconditionally.