Our goal is to show that both the fast and slow versions of the triangle map (a type of multi-dimensional continued fraction algorithm) in dimension n are ergodic, resolving a conjecture of Messaoudi, Noguiera, and Schweiger [Ergodic properties of triangle partitions. Monatsh. Math. 157 (2009), 283–299]. This particular type of higher dimensional multi-dimensional continued fraction algorithm has recently been linked to the study of partition numbers, with the result that the underlying dynamics has combinatorial implications.

Using an explicit Eichler–Shimura–Harder isomorphism, we establish the analog of Manin’s rationality theorem for Bianchi periods and hence special values of L-functions of Bianchi cusp forms. This gives a new short proof of a result of Hida in the case of Euclidean imaginary quadratic fields. In particular, we give an explicit proof using the space of Bianchi period polynomials constructed by Karabulut and describe the action of Hecke operators.

It is conjectured that for any fixed relatively prime positive integers a,b and c all greater than 1 there is at most one solution to the equation $a^x+b^y=c^z$ in positive integers x, y and z, except for specific cases. We develop the methods in our previous work which rely on a variety from Baker’s theory and thoroughly study the conjecture for cases where c is small relative to a or b. Using restrictions derived from the hypothesis that there is more than one solution to the equation, we obtain a number of finiteness results on the conjecture. In particular, we find some, presumably infinitely many, new values of c with the property that for each such c the conjecture holds true except for only finitely many pairs of a and b. Most importantly we prove that if $c=13$ then the equation has at most one solution, except for $(a,b)=(3,10)$ or (10,3) each of which gives exactly two solutions. Further, our study with the help of the Schmidt Subspace Theorem among others more, brings strong contributions to the study of Pillai’s type Diophantine equations, notably a general and satisfactory result on a well-known conjecture of M. Bennett on the equation $a^x-b^y=c$ for any fixed positive integers a,b and c with both a and b greater than 1. Some conditional results are presented under the abc-conjecture as well.

We show that the set of Liouville numbers has a rich set-theoretic structure: it can be partitioned in an explicit way into an uncountable collection of subsets, each of which is dense in the real line. Furthermore, each of these partitioning subsets can be similarly partitioned, and the process can be repeated indefinitely.

We formulate Guo–Jacquet type fundamental lemma conjectures and arithmetic transfer conjectures for inner forms of $GL_{2n}$. Our main results confirm these conjectures for division algebras of invariant $1/4$ and $3/4$.

We show that smooth numbers are equidistributed in arithmetic progressions to moduli of size $x^{66/107-o(1)}$. This overcomes a longstanding barrier of $x^{3/5-o(1)}$ present in previous works of Bombieri, Friedlander and Iwaniec, Fouvry and Tenenbaum, Drappeau, and Maynard. We build on Drappeau’s variation of Linnik’s dispersion method and on exponential sum manipulations of Maynard, ultimately relying on optimized Deshouillers–Iwaniec-type estimates for sums of Kloosterman sums.

We show that each local field $\mathbb{F}_q(\!(t)\!)$ of characteristic $p > 0$ is characterised up to isomorphism within the class of all fields of imperfect exponent at most 1 by (certain small quotients of) its absolute Galois group together with natural axioms concerning the p-torsion of its Brauer group. This complements previous work by Efrat and Fesenko, who analysed fields whose absolute Galois group is isomorphic to that of a local field of characteristic p.

We will give a precise and explicit asymptotic estimate for the characteristic of the Riemann zeta function $\zeta $ with an error term of order $O(\frac {\log r}{r})$ and a corresponding asymptotic estimate for the number of fixed points of $\zeta $.

For an even positive integer n, we study rank-one Eisenstein cohomology of the split orthogonal group $\mathrm {O}(2n+2)$ over a totally real number field $F.$ This is used to prove a rationality result for the ratios of successive critical values of degree-$2n$ Langlands L-functions associated to the group $\mathrm {GL}_1 \times \mathrm {O}(2n)$ over F. The case $n=2$ specializes to classical results of Shimura on the special values of Rankin–Selberg L-functions attached to a pair of Hilbert modular forms.

Let $t\geq 2$ and $k\geq 1$ be integers. A t-regular partition of a positive integer n is a partition of n such that none of its parts is divisible by t. Let $b_{t,k}(n)$ denote the number of hooks of length k in all the t-regular partitions of n. In this article, we prove some inequalities for $b_{t,k}(n)$ for fixed values of k. We prove that for any $t\geq 2$, $b_{t+1,1}(n)\geq b_{t,1}(n)$, for all $n\geq 0$. We also prove that $b_{3,2}(n)\geq b_{2,2}(n)$ for all $n>3$, and $b_{3,3}(n)\geq b_{2,3}(n)$ for all $n\geq 0$. Finally, we state some problems for future works.

Given any finite collection $\mathcal {G}$ of translation-invariant linear equations of the form

$$ \begin{align*} \sum_{i=1}^{v} m_i x_i = m_0 x_0, \end{align*} $$

where $(m_0, m_1, \ldots , m_v) \in \mathbb {N}^{v+1}$, $m_0 = \sum _{i=1}^{v} m_i$ and $v \ge 2$, we estimate lower and upper bounds of the supremum of the Hausdorff dimension of sets on the real line that uniformly avoid nontrivial zeros of any f in $\mathcal {G}$.

We construct an fpqc gerbe $\mathcal {E}_{\dot {V}}$ over a global function field F such that for a connected reductive group G over F with finite central subgroup Z, the set of $G_{\mathcal {E}_{\dot {V}}}$-torsors contains a subset $H^{1}(\mathcal {E}_{\dot {V}}, Z \to G)$ which allows one to define a global notion of (Z-)rigid inner forms. There is a localization map $H^{1}(\mathcal {E}_{\dot {V}}, Z \to G) \to H^{1}(\mathcal {E}_{v}, Z \to G)$, where the latter parametrizes local rigid inner forms (cf. [8, 6]) which allows us to organize local rigid inner forms across all places v into coherent families. Doing so enables a construction of (conjectural) global L-packets and a conjectural formula for the multiplicity of an automorphic representation $\pi $ in the discrete spectrum of G in terms of these L-packets. We also show that, for a connected reductive group G over a global function field F, the adelic transfer factor $\Delta _{\mathbb {A}}$ for the ring of adeles $\mathbb {A}$ of F serving an endoscopic datum for G decomposes as the product of the normalized local transfer factors from [6].

We consider Shimura varieties associated to a unitary group of signature $(n-s,s)$ where n is even. For these varieties, we construct smooth p-adic integral models for $s=1$ and regular p-adic integral models for $s=2$ and $s=3$ over odd primes p which ramify in the imaginary quadratic field with level subgroup at p given by the stabilizer of a $\pi $-modular lattice in the hermitian space. Our construction, which has an explicit moduli-theoretic description, is given by an explicit resolution of a corresponding local model.

We give a short proof of the anticyclotomic analogue of the “strong” main conjecture of Kurihara on Fitting ideals of Selmer groups for elliptic curves with good ordinary reduction under mild hypotheses. More precisely, we completely determine the initial Fitting ideal of Selmer groups over finite subextensions of an imaginary quadratic field in its anticyclotomic $\mathbb {Z}_p$-extension in terms of Bertolini–Darmon’s theta elements.

Let $A\ \mathrm{and}\ B$ be subsets of $(\mathbb {Z}/p^r\mathbb {Z})^2$. In this note, we provide conditions on the densities of A and B such that $|gA-B|\gg p^{2r}$ for a positive proportion of $g\in SO_2(\mathbb {Z}/p^r\mathbb {Z})$. The conditions are sharp up to constant factors in the unbalanced case, and the proof makes use of tools from discrete Fourier analysis and results in restriction/extension theory.

We classify finite groups that act faithfully by symplectic birational transformations on an irreducible holomorphic symplectic (IHS) manifold of $OG10$ type. In particular, if X is an IHS manifold of $OG10$ type and G a finite subgroup of symplectic birational transformations of X, then the action of G on $H^2(X,\mathbb {Z})$ is conjugate to a subgroup of one of 375 groups of isometries. We prove a criterion for when such a group is determined by a group of automorphisms acting on a cubic fourfold, and apply it to our classification. Our proof is computer aided, and our results are available in a Zenodo dataset.

In the early 2000s, Ramakrishna asked the question: for the elliptic curve

\[E\;:\; y^2 = x^3 - x,\]

$a_p(E)$

$E/\mathbb{Q}$

We construct explicit generating series of arithmetic extensions of Kudla’s special divisors on integral models of unitary Shimura varieties over CM fields with arbitrary split levels and prove that they are modular forms valued in the arithmetic Chow groups. This provides a partial solution to Kudla’s modularity problem. The main ingredient in our construction is S. Zhang’s theory of admissible arithmetic divisors. The main ingredient in the proof is an arithmetic mixed Siegel-Weil formula.

Brun’s constant is the summation of the reciprocals of all twin primes, given by

$$ \begin{align*}B=\sum_{p \in P_2}{\bigg( \frac{1}{p} + \frac{1}{p+2}\bigg)}.\end{align*} $$

While rigorous unconditional bounds on B are known, we present the first rigorous bound on Brun’s constant under the assumption of GRH, yielding $B < 2.1594$.

We determine the geometric monodromy groups attached to various families, both one-parameter and multi-parameter, of exponential sums over finite fields, or, more precisely, the geometric monodromy groups of the $\ell $-adic local systems on affine spaces in characteristic $p> 0$ whose trace functions are these exponential sums. The exponential sums here are much more general than we previously were able to consider. As a byproduct, we determine the number of irreducible components of maximal dimension in certain intersections of Fermat surfaces. We also show that in any family of such local systems, say parameterized by an affine space S, there is a dense open set of S over which the geometric monodromy group of the corresponding local system is a fixed known group.