We discuss several ways of packing a hyperbolic surface with circles (of either varying radii or all being congruent) or horocycles, and note down some observations related to their symmetries (or the absence thereof).

We use circulant matrices and hyperelliptic curves over finite fields to study some arithmetic properties of certain determinants involving Legendre symbols and kth power residues.

In this note, we use Dedekind’s eta function to prove a congruence relation between the number of representations by binary quadratic forms of discriminant $-31$ and Fourier coefficients of a weight $16$ cusp form. Our result is analogous to the classical result concerning Ramanujan’s tau function and binary quadratic forms of discriminant $-23$.

We prove that if $s\ge 2$ is a fixed integer, then the equation $ns^n+1=(b^m-1)/(b-1)$ has only finitely many positive integer solutions $(n,b,m)$ with $b\ge 2$ and $m\ge 3$. When $s=2$, it has no solution.

For a set A of positive integers and any positive integer n, let $R_{1}(A, n)$, $R_{2}(A,n)$ and $R_{3}(A,n)$ denote the number of solutions of $a+a^{\prime }=n$ with $a, a^{\prime }\in A$ and the additional restriction that $a<a^{\prime }$ for $R_{2}$ and $a\leq a^{\prime }$ for $R_{3}$. We consider Problem 6 of Erdős et al. [‘On additive properties of general sequences’, Discrete Math. 136 (1994), 75–99] about locally small and locally large values of $R_{1}, R_{2}$ and $R_{3}$.

Let $k\geq 2$ be an integer. We prove that the 2-automatic sequence of odious numbers $\mathcal {O}$ is a k-additive uniqueness set for multiplicative functions: if a multiplicative function f satisfies a multivariate Cauchy’s functional equation $f(x_1+x_2+\cdots +x_k)=f(x_1)+f(x_2)+\cdots +f(x_k)$ for arbitrary $x_1,\ldots ,x_k\in \mathcal {O}$, then f is the identity function $f(n)=n$ for all $n\in \mathbb {N}$.

Let $C$ be a smooth curve over a finite field of characteristic $p$ and let $M$ be an overconvergent $\mathbf {F}$-isocrystal over $C$. After replacing $C$ with a dense open subset, $M$ obtains a slope filtration. This is a purely $p$-adic phenomenon; there is no counterpart in the theory of lisse $\ell$-adic sheaves. The graded pieces of this slope filtration correspond to lisse $p$-adic sheaves, which we call geometric. Geometric lisse $p$-adic sheaves are mysterious, as there is no $\ell$-adic analogue. In this article, we study the monodromy of geometric lisse $p$-adic sheaves with rank one. More precisely, we prove exponential bounds on their ramification breaks. When the generic slopes of $M$ are integers, we show that the local ramification breaks satisfy a certain type of periodicity. The crux of the proof is the theory of $\mathbf {F}$-isocrystals with log-decay. We prove a monodromy theorem for these $\mathbf {F}$-isocrystals, as well as a theorem relating the slopes of $M$ to the rate of log-decay of the slope filtration. As a consequence of these methods, we provide a new proof of the Drinfeld–Kedlaya theorem for irreducible $\mathbf {F}$-isocrystals on curves.

In earlier work, the first named author generalized the construction of Darmon-style $\mathcal {L}$-invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial coefficient system and Steinberg at a fixed prime. In this paper, assuming that the Archimedean component of the group has discrete series we show that these automorphic $\mathcal {L}$-invariants can be computed in terms of derivatives of Hecke eigenvalues in $p$-adic families. Our proof is novel even in the case of modular forms, which was established by Bertolini, Darmon and Iovita. The main new technical ingredient is the Koszul resolution of locally analytic principal series representations by Kohlhaase and Schraen. As an application of our results we settle a conjecture of Spieß: we show that automorphic $\mathcal {L}$-invariants of Hilbert modular forms of parallel weight $2$ are independent of the sign character used to define them. Moreover, we show that they are invariant under Jacquet–Langlands transfer and, in fact, equal to the Fontaine–Mazur $\mathcal {L}$-invariant of the associated Galois representation. Under mild assumptions, we also prove the equality of automorphic and Fontaine–Mazur $\mathcal {L}$-invariants for representations of definite unitary groups of arbitrary rank. Finally, we study the case of Bianchi modular forms to show how our methods, given precise results on eigenvarieties, can also work in the absence of discrete series representations.

We show that the completed Hecke algebra of $p$-adic modular forms is isomorphic to the completed Hecke algebra of continuous $p$-adic automorphic forms for the units of the quaternion algebra ramified at $p$ and $\infty$. This gives an affirmative answer to a question posed by Serre in a 1987 letter to Tate. The proof is geometric, and lifts a mod $p$ argument due to Serre: we evaluate modular forms by identifying a quaternionic double-coset with a fiber of the Hodge–Tate period map, and extend functions off of the double-coset using fake Hasse invariants. In particular, this gives a new proof, independent of the classical Jacquet–Langlands correspondence, that Galois representations can be attached to classical and $p$-adic quaternionic eigenforms.

Consider three normalized cuspidal eigenforms of weight $2$ and prime level p. Under the assumption that the global root number of the associated triple product L-function is $+1$, we prove that the complex Abel–Jacobi image of the modified diagonal cycle of Gross–Kudla–Schoen on the triple product of the modular curve $X_0(p)$ is torsion in the corresponding Hecke isotypic component of the Griffiths intermediate Jacobian. The same result holds with the complex Abel–Jacobi map replaced by its étale counterpart. As an application, we deduce torsion properties of Chow–Heegner points associated with modified diagonal cycles on elliptic curves of prime conductor with split multiplicative reduction. The approach also works in the case of composite square-free level.

Let E be an elliptic curve defined over a number field F with good ordinary reduction at all primes above p, and let $F_\infty $ be a finitely ramified uniform pro-p extension of F containing the cyclotomic $\mathbb {Z}_p$-extension $F_{\operatorname {cyc}}$. Set $F^{(n)}$ be the nth layer of the tower, and $F^{(n)}_{\operatorname {cyc}}$ the cyclotomic $\mathbb {Z}_p$-extension of $F^{(n)}$. We study the growth of the rank of $E(F^{(n)})$ by analyzing the growth of the $\lambda $-invariant of the Selmer group over $F^{(n)}_{ \operatorname {cyc}}$ as $n\rightarrow \infty $. This method has its origins in work of A. Cuoco, who studied $\mathbb {Z}_p^2$-extensions. Refined estimates for growth are proved that are close to conjectured estimates. The results are illustrated in special cases.

We explicate the combinatorial/geometric ingredients of Arthur’s proof of the convergence and polynomiality, in a truncation parameter, of his noninvariant trace formula. Starting with a fan in a real, finite dimensional, vector space and a collection of functions, one for each cone in the fan, we introduce a combinatorial truncated function with respect to a polytope normal to the fan and prove the analogues of Arthur’s results on the convergence and polynomiality of the integral of this truncated function over the vector space. The convergence statements clarify the important role of certain combinatorial subsets that appear in Arthur’s work and provide a crucial partition that amounts to a so-called nearest face partition. The polynomiality statements can be thought of as far reaching extensions of the Ehrhart polynomial. Our proof of polynomiality relies on the Lawrence–Varchenko conical decomposition and readily implies an extension of the well-known combinatorial lemma of Langlands. The Khovanskii–Pukhlikov virtual polytopes are an important ingredient here. Finally, we give some geometric interpretations of our combinatorial truncation on toric varieties as a measure and a Lefschetz number.

In this paper, we give an axiomatization of the ordinal number system, in the style of Dedekind’s axiomatization of the natural number system. The latter is based on a structure $(N,0,s)$ consisting of a set N, a distinguished element $0\in N$ and a function $s\colon N\to N$. The structure in our axiomatization is a triple $(O,L,s)$, where O is a class, L is a class function defined on all s-closed ‘subsets’ of O, and s is a class function $s\colon O\to O$. In fact, we develop the theory relative to a Grothendieck-style universe (minus the power set axiom), as a way of bringing the natural and the ordinal cases under one framework. We also establish a universal property for the ordinal number system, analogous to the well-known universal property for the natural number system.

In this work we generalise the main result of [1] to the family of hyperelliptic curves with potentially good reduction over a p-adic field which have genus $g=({p-1})/{2}$ and the largest possible image of inertia under the $\ell$-adic Galois representation associated to its Jacobian. We will prove that this Galois representation factors as the tensor product of an unramified character and an irreducible representation of a finite group, which can be either equal to the inertia image (in which case the representation is easily determined) or a $C_2$-extension of it. In this second case, there are two suitable representations and we will describe the Galois action explicitly in order to determine the correct one.

Let $N\geq 1$ be squarefree with $(N,6)=1$. Let $c\phi _N(n)$ denote the number of N-colored generalized Frobenius partitions of n introduced by Andrews in 1984, and $P(n)$ denote the number of partitions of n. We prove

$$ \begin{align*}c\phi_N(n)= \sum_{d \mid N} N/d \cdot P\left( \frac{ N}{d^2}n - \frac{N^2-d^2}{24d^2} \right) + b(n),\end{align*} $$

$C(z) := (q;q)^N_\infty \sum _{n=1}^{\infty } b(n) q^n$

$S_{(N-1)/2} (\Gamma _0(N),\chi _N)$

$c\phi _N(n)$

$P(n)$

The Thue–Morse sequence is a prototypical automatic sequence found in diverse areas of mathematics, and in computer science. We study occurrences of factors w within this sequence, or more precisely, the sequence of gaps between consecutive occurrences. This gap sequence is morphic; we prove that it is not automatic as soon as the length of w is at least $2$, thereby answering a question by J. Shallit in the affirmative. We give an explicit method to compute the discrepancy of the number of occurrences of the block $\mathtt {01}$ in the Thue–Morse sequence. We prove that the sequence of discrepancies is the sequence of output sums of a certain base-$2$ transducer.

In 1946, Erdős and Niven proved that no two partial sums of the harmonic series can be equal. We present a generalisation of the Erdős–Niven theorem by showing that no two partial sums of the series $\sum _{k=0}^\infty {1}/{(a+bk)}$ can be equal, where a and b are positive integers. The proof of our result uses analytic and p-adic methods.

We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each $\epsilon>0$ there exists M such that every triangle-free graph G has an $\epsilon$-approximate homomorphism to a triangle-free graph F on at most M vertices (here an $\epsilon$-approximate homomorphism is a map $V(G) \to V(F)$ where all but at most $\epsilon \left\lvert{V(G)}\right\rvert^2$ edges of G are mapped to edges of F). One consequence of our results is that the least possible M in the triangle-free lemma grows faster than exponential in any polynomial in $\epsilon^{-1}$. We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal.

We demonstrate that every difference set in a finite Abelian group is equivalent to a certain ‘regular’ covering of the lattice $ A_n = \{ \boldsymbol {x} \in \mathbb {Z} ^{n+1} : \sum _{i} x_i = 0 \} $ with balls of radius $ 2 $ under the $ \ell _1 $ metric (or, equivalently, a covering of the integer lattice $ \mathbb {Z} ^n $ with balls of radius $ 1 $ under a slightly different metric). For planar difference sets, the covering is also a packing, and therefore a tiling, of $ A_n $. This observation leads to a geometric reformulation of the prime power conjecture and of other statements involving Abelian difference sets.