In this study, we introduce multiple zeta functions with structures similar to those of symmetric functions such as the Schur P-, Schur Q-, symplectic and orthogonal functions in representation theory. Their basic properties, such as the domain of absolute convergence, are first considered. Then, by restricting ourselves to the truncated multiple zeta functions, we derive the Pfaffian expression of the Schur Q-multiple zeta functions, the sum formula for Schur P- and Schur Q-multiple zeta functions, the determinant expressions of symplectic and orthogonal Schur multiple zeta functions by making an assumption on variables. Finally, we generalize those to the quasi-symmetric functions.

We establish sharp upper bounds for shifted moments of quadratic Dirichlet L-function under the generalized Riemann hypothesis. Our result is then used to prove bounds for moments of quadratic Dirichlet character sums.

For a smooth affine group scheme G over the ring of p-adic integers and a cocharacter $\mu $ of G, we develop the deformation theory for G-$\mu $-displays over the prismatic site of Bhatt–Scholze, and discuss how our deformation theory can be interpreted in terms of prismatic F-gauges introduced by Drinfeld and Bhatt–Lurie. As an application, we prove the local representability and the formal smoothness of integral local Shimura varieties with hyperspecial level structure. We also revisit and extend some classification results of p-divisible groups.

We establish two new truncated identities for theta series. Their partition-theoretic interpretations are also given.

We study the rationality properties of the moduli space ${\mathcal{A}}_g$ of principally polarised abelian $g$-folds over $\mathbb{Q}$ and apply the results to arithmetic questions. In particular, we show that any principally polarised abelian 3-fold over ${\mathbb{F}}_p$ may be lifted to an abelian variety over $\mathbb{Q}$. This is a phenomenon of low dimension: assuming the Bombieri–Lang conjecture, we also show that this is not the case for abelian varieties of dimension at least 7. Concerning moduli spaces, we show that ${\mathcal{A}}_g$ is unirational over $\mathbb{Q}$ for $g\le 5$ and stably rational for $g=3$. This also allows us to make unconditional one of the results of Masser and Zannier about the existence of abelian varieties over $\mathbb{Q}$ that are not isogenous to Jacobians.

What proportion of integers $n \leq N$ may be expressed as $x^2 + dy^2$ for some $d \leq \Delta $, with $x,y$ integers? Writing $\Delta = (\log N)^{\log 2} 2^{\alpha \sqrt {\log \log N}}$ for some $\alpha \in (-\infty , \infty )$, we show that the answer is $\Phi (\alpha ) + o(1)$, where $\Phi $ is the Gaussian distribution function $\Phi (\alpha ) = \frac {1}{\sqrt {2\pi }} \int ^{\alpha }_{-\infty } e^{-x^2/2} dx$.

A consequence of this is a phase transition: Almost none of the integers $n \leq N$ can be represented by $x^2 + dy^2$ with $d \leq (\log N)^{\log 2 - \varepsilon }$, but almost all of them can be represented by $x^2 + dy^2$ with $d \leq (\log N)^{\log 2 + \varepsilon}\kern-1.5pt$.

Let s be a fixed positive integer constant and let $\varepsilon $ be a fixed small positive number. Then, provided that a prime p is large enough, we prove that, for any set ${\mathcal M}\subseteq \mathbb {F}_p^*$ of size $|{\mathcal M}|= \lfloor { p^{14/29}}\rfloor $ and integer $H=\lfloor {p^{14/29+\varepsilon }}\rfloor $, any integer $\lambda $ can be represented in the form

$$ \begin{align*} \frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2,3. \end{align*} $$

When $s=1$, we show that, for almost all primes p, if $|{\mathcal M}|= \lfloor p^{1/2}\rfloor $ and $H=\lfloor p^{1/2}(\log p)^{6+\varepsilon }\rfloor $, then any integer $\lambda $ can be represented in the form

$$ \begin{align*} \frac{m_1}{x_1}+\frac{m_2}{x_2}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2. \end{align*} $$

Given a polarised abelian variety over a number field, we provide totally explicit upper bounds for the cardinality of the rational points whose Néron-Tate height is less than a small threshold. These imply new estimates for the number of torsion points as well as the minimal height of a non-torsion point. Our bounds involve the Faltings height and dimension of the abelian variety together with the degrees of the polarisation and the number field but we also get a stronger statement where we use certain successive minima associated to the period lattice at a fixed archimedean place, in the spirit of a result of David for elliptic curves.

Let F be a non-archimedean locally compact field of residual characteristic p, let $G=\operatorname {GL}_{r}(F)$ and let $\widetilde {G}$ be an n-fold metaplectic cover of G with $\operatorname {gcd}(n,p)=1$. We study the category $\operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$ of complex smooth representations of $\widetilde {G}$ having inertial equivalence class $\mathfrak {s}=(\widetilde {M},\mathcal {O})$, which is a block of the category $\operatorname {Rep}(\widetilde {G})$, following the ‘type theoretical’ strategy of Bushnell-Kutzko.

Precisely, first we construct a ‘maximal simple type’ $(\widetilde {J_{M}},\widetilde {\lambda }_{M})$ of $\widetilde {M}$ as an $\mathfrak {s}_{M}$-type, where $\mathfrak {s}_{M}=(\widetilde {M},\mathcal {O})$ is the related cuspidal inertial equivalence class of $\widetilde {M}$. Along the way, we prove the folklore conjecture that every cuspidal representation of $\widetilde {M}$ could be constructed explicitly by a compact induction. Secondly, we construct ‘simple types’ $(\widetilde {J},\widetilde {\lambda })$ of $\widetilde {G}$ and prove that each of them is an $\mathfrak {s}$-type of a certain block $\operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$. When $\widetilde {G}$ is either a Kazhdan-Patterson cover or Savin’s cover, the corresponding blocks turn out to be those containing discrete series representations of $\widetilde {G}$. Finally, for a simple type $(\widetilde {J},\widetilde {\lambda })$ of $\widetilde {G}$, we describe the related Hecke algebra $\mathcal {H}(\widetilde {G},\widetilde {\lambda })$, which turns out to be not far from an affine Hecke algebra of type A, and is exactly so if $\widetilde {G}$ is one of the two special covers mentioned above.

We leave the construction of a ‘semi-simple type’ related to a general block $\operatorname {Rep}_{\mathfrak {s}}(\widetilde {G})$ to a future phase of the work.

In this paper, we complete the proof of the conjecture of Gross and Zagier concerning algebraicity of higher Green functions at a single CM point on the product of modular curves. The new ingredient is an analogue of the incoherent Eisenstein series over a real quadratic field, which is constructed as the Doi-Naganuma theta lift of a deformed theta integral on hyperbolic 1-space.

Arthur packets have been defined for pure real forms of symplectic and special orthogonal groups following two different approaches. The first approach, due to Arthur, Moeglin, and Renard uses harmonic analysis. The second approach, due to Adams, Barbasch, and Vogan uses microlocal geometry. We prove that the two approaches produce essentially equivalent Arthur packets. This extends previous work of the authors and J. Adams for the quasisplit real forms.

In the early 1900s, Maillet [Introduction a la theorie des nombres transcendants et des proprietes arithmetiques des fonctions (Gauthier–Villars, Paris, 1906)] proved that the image of any Liouville number under a rational function with rational coefficients is again a Liouville number. The analogous result for quadratic Liouville matrices in higher dimensions turns out to fail. In fact, using a result by Kleinbock and Margulis [‘Flows on homogeneous spaces and Diophantine approximation on manifolds’, Ann. of Math. (2) 148(1) (1998), 339–360], we show that among analytic matrix functions in dimension $n\ge 2$, Maillet’s invariance property is only true for Möbius transformations with special coefficients. This implies that the analogue in higher dimensions of an open question of Mahler on the existence of transcendental entire functions with Maillet’s property has a negative answer. However, extending a topological argument of Erdős [‘Representations of real numbers as sums and products of Liouville numbers’, Michigan Math. J. 9 (1962), 59–60], we prove that for any injective continuous self-mapping on the space of rectangular matrices, many Liouville matrices are mapped to Liouville matrices. Dropping injectivity, we consider setups similar to Alniaçik and Saias [‘Une remarque sur les $G_{\delta }$-denses’, Arch. Math. (Basel) 62(5) (1994), 425–426], and show that the situation depends on the matrix dimensions $m,n$. Finally, we discuss extensions of a related result by Burger [‘Diophantine inequalities and irrationality measures for certain transcendental numbers’, Indian J. Pure Appl. Math. 32 (2001), 1591–1599] to quadratic matrices. We state several open problems along the way.

We prove new fundamental lemma and arithmetic fundamental lemma identities for general linear groups over quaternion division algebras. In particular, we verify the transfer conjecture and the arithmetic transfer conjecture from Li and Mihatsch (2023, Preprint, arXiv:2307.11716) in cases of Hasse invariant $1/2$.

We study linear random walks on the torus and show a quantitative equidistribution statement, under the assumption that the Zariski closure of the acting group is semisimple.

We construct skew corner-free subsets of $[n]^2$ of size $n^2\exp(\!-O(\sqrt{\log n}))$, thereby improving on recent bounds of the form $\Omega(n^{5/4})$ obtained by Pohoata and Zakharov. We also prove that any such set has size at most $O(n^2(\log n)^{-c})$ for some absolute constant $c \gt 0$. This improves on the previously best known upper bound $O(n^2(\log\log n)^{-c})$, coming from Shkredov’s work on the corners theorem.

We consider integral models of Hilbert modular varieties with Iwahori level structure at primes over p, first proving a Kodaira–Spencer isomorphism that gives a concise description of their dualizing sheaves. We then analyze fibres of the degeneracy maps to Hilbert modular varieties of level prime to p and deduce the vanishing of higher direct images of structure and dualizing sheaves, generalizing prior work with Kassaei and Sasaki (for p unramified in the totally real field F). We apply the vanishing results to prove flatness of the finite morphisms in the resulting Stein factorizations, and combine them with the Kodaira–Spencer isomorphism to simplify and generalize the construction of Hecke operators at primes over p on Hilbert modular forms (integrally and mod p).

Let $e$ and $q$ be fixed co-prime integers satisfying $1\lt e\lt q$. Let $\mathscr {C}$ be a certain family of deformations of the curve $y^e=x^q$. That family is called the $(e,q)$-curve and is one of the types of curves called plane telescopic curves. Let $\varDelta$ be the discriminant of $\mathscr {C}$. Following pioneering work by Buchstaber and Leykin (BL), we determine the canonical basis $\{ L_j \}$ of the space of derivations tangent to the variety $\varDelta =0$ and describe their specific properties. Such a set $\{ L_j \}$ gives rise to a system of linear partial differential equations (heat equations) satisfied by the function $\sigma (u)$ associated with $\mathscr {C}$, and eventually gives its explicit power series expansion. This is a natural generalisation of Weierstrass’ result on his sigma function. We attempt to give an accessible description of various aspects of the BL theory. Especially, the text contains detailed proofs for several useful formulae and known facts since we know of no works which include their proofs.

Bergelson and Richter [‘Dynamical generalizations of the prime number theorem and disjointness of additive and multiplicative semigroup actions’, Duke Math. J. 171(15) (2022), 3133–3200] established a new dynamical generalisation of the prime number theorem (PNT) and the PNT for arithmetic progressions. Let $h\ge 1, k\ge 2$. Mirsky [‘Note on an asymptotic formula connected with r-free integers’, Quart. J. Math. Oxford Ser. 18 (1947), 178–182] showed that the numbers n such that $n+l_1,\ldots , n+l_h$ are k-free have a natural density for any given nonnegative integers $l_1,\ldots , l_h$. In this note, we show that the Bergelson–Richter theorem holds for the numbers studied by Mirsky.

Arithmetic-geometric mean sequences were already studied over the real and complex numbers, and recently, Griffin et al. [‘AGM and jellyfish swarms of elliptic curves’, Amer. Math. Monthly 130(4) (2023), 355–369] considered them over finite fields $\mathbb {F}_q$ for $q \equiv 3 \pmod 4$. We extend the definition of arithmetic-geometric mean sequences over $\mathbb {F}_q$ to $q \equiv 5 \pmod 8$. We explain the connection of these sequences with graphs and explore the properties of the graphs in the case where $q \equiv 5 \pmod 8$.