Arithmetic duality theorems over a local field $k$ are delicate to prove if $\text{char}\,k>0$. In this case, the proofs often exploit topologies carried by the cohomology groups $H^{n}(k,G)$ for commutative finite type $k$-group schemes $G$. These ‘Čech topologies’, defined using Čech cohomology, are impractical due to the lack of proofs of their basic properties, such as continuity of connecting maps in long exact sequences. We propose another way to topologize $H^{n}(k,G)$: in the key case when $n=1$, identify $H^{1}(k,G)$ with the set of isomorphism classes of objects of the groupoid of $k$-points of the classifying stack $\mathbf{B}G$ and invoke Moret-Bailly’s general method of topologizing $k$-points of locally of finite type $k$-algebraic stacks. Geometric arguments prove that these ‘classifying stack topologies’ enjoy the properties expected from the Čech topologies. With this as the key input, we prove that the Čech and the classifying stack topologies actually agree. The expected properties of the Čech topologies follow, and these properties streamline a number of arithmetic duality proofs given elsewhere.

We examine the tail distributions of integer partition ranks and cranks by investigating tail moments, which are analogous to the positive moments introduced by Andrews et al. [‘The odd moments of ranks and cranks’, J. Combin. Theory Ser. A120(1) (2013), 77–91].

It is well known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular units. This answers a question raised by W. Zudilin in a recent work on Mahler measures. Further, we give the list of all elliptic curves $E$ of conductor up to 1000 parametrized by modular units supported in the rational torsion subgroup of $E$. Finally, we raise several open questions.

Let $m$ be a positive integer and $p$ a prime number. We prove the orthogonality of some character sums over the finite field $\mathbb{F}_{p^{m}}$ or over a subset of a finite field and use this to construct some new approximately mutually unbiased bases of dimension $p^{m}$ over the complex number field $\mathbb{C}$, especially with $p=2$.

We prove an analog of the Yomdin–Gromov lemma for $p$-adic definable sets and more broadly in a non-Archimedean definable context. This analog keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of $p$-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over $\mathbb{C}(\!(t)\!)$, in analogy to results by Pila and Wilkie, and by Bombieri and Pila, respectively. Along the way we prove, for definable functions in a general context of non-Archimedean geometry, that local Lipschitz continuity implies piecewise global Lipschitz continuity.

We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a principal series representation, induced from a certain exceptional character. The global exceptional representation is obtained as the multi-residue of an Eisenstein series: it is an automorphic representation, and it decomposes as the restricted tensor product of local exceptional representations. As in the case of the small representation of $\mathit{SO}_{2n+1}$ of Bump, Friedberg, and Ginzburg, exceptional representations enjoy the vanishing of a large class of twisted Jacquet modules (locally), or Fourier coefficients (globally). Consequently they are useful in many settings, including lifting problems and Rankin–Selberg integrals. We describe one application, to a calculation of a co-period integral.

We prove that formal Fourier Jacobi expansions of degree two are Siegel modular forms. As a corollary, we deduce modularity of the generating function of special cycles of codimension two, which were defined by Kudla. A second application is the proof of termination of an algorithm to compute Fourier expansions of arbitrary Siegel modular forms of degree two. Combining both results enables us to determine relations of special cycles in the second Chow group.

Let $g\geq 2$ be a fixed integer. Let $\mathbb{N}$ denote the set of all nonnegative integers and let $A$ be a subset of $\mathbb{N}$. Write $r_{2}(A,n)=\sharp \{(a_{1},a_{2})\in A^{2}:a_{1}+a_{2}=n\}.$ We construct a thin, strongly minimal, asymptotic $g$-adic basis $A$ of order two such that the set of $n$ with $r_{2}(A,n)=2$ has density one.

Van Wamelen [Math. Comp. 68 (1999) no. 225, 307–320] lists 19 curves of genus two over $\mathbf{Q}$ with complex multiplication (CM). However, for each curve, the CM-field turns out to be cyclic Galois over  $\mathbf{Q}$ , and the generic case of a non-Galois quartic CM-field did not feature in this list. The reason is that the field of definition in that case always contains the real quadratic subfield of the reflex field.

We extend Van Wamelen’s list to include curves of genus two defined over this real quadratic field. Our list therefore contains the smallest ‘generic’ examples of CM curves of genus two.

We explain our methods for obtaining this list, including a new height-reduction algorithm for arbitrary hyperelliptic curves over totally real number fields. Unlike Van Wamelen, we also give a proof of our list, which is made possible by our implementation of denominator bounds of Lauter and Viray for Igusa class polynomials.

We study elliptic curves over quadratic fields with isogenies of certain degrees. Let $n$ be a positive integer such that the modular curve $X_{0}(n)$ is hyperelliptic of genus ${\geqslant}2$ and such that its Jacobian has rank  $0$ over  $\mathbb{Q}$ . We determine all points of $X_{0}(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with a finite number of exceptions up to $\overline{\mathbb{Q}}$ -isomorphism, every elliptic curve over a quadratic field  $K$ admitting an $n$ -isogeny is $d$ -isogenous, for some $d\mid n$ , to the twist of its Galois conjugate by a quadratic extension $L$ of  $K$ . We determine $d$ and  $L$ explicitly, and we list all exceptions. As a consequence, again with a finite number of exceptions up to $\overline{\mathbb{Q}}$ -isomorphism, all elliptic curves with $n$ -isogenies over quadratic fields are in fact $\mathbb{Q}$ -curves.

In 2006, F. Luca and I. E. Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) proved that there are only finitely many pairs (n, m) of positive integers which satisfy the Diophantine equation |τ(n!)|=m!, where τ is the Ramanujan function. In this paper, we follow the same approach of Luca and Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) to determine all solutions of the above equation. The proof of our main theorem uses linear forms in two logarithms and arithmetic properties of the Ramanujan function.

We give upper and lower bounds for the number of rational points on Prym varieties over finite fields. Moreover, we determine the exact maximum and minimum number of rational points on Prym varieties of dimension 2.

Suppose E is an elliptic curve over $\Bbb Q$, and p>3 is a split multiplicative prime for E. Let q ≠ p be an auxiliary prime, and fix an integer m coprime to pq. We prove the generalised Mazur–Tate–Teitelbaum conjecture for E at the prime p, over number fields $K\subset \Bbb Q\big(\mu_{{q^{\infty}}},\;\!^{q^{\infty}\!\!\!\!}\sqrt{m}\big)$ such that p remains inert in $K\cap\Bbb Q(\mu_{{q^{\infty}}})^+$. The proof makes use of an improved p-adic L-function, which can be associated to the Rankin convolution of two Hilbert modular forms of unequal parallel weight.

Let $\text{pod}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\text{pod}_{-4}(n)$ including the following infinite family of congruences: for any integers ${\it\alpha}\geq 1$ and $n\geq 0$,

$$\begin{eqnarray}\text{pod}_{-4}\biggl(3^{{\it\alpha}+1}n+\frac{5\cdot 3^{{\it\alpha}}+1}{2}\biggr)\equiv 0~(\text{mod}~9).\end{eqnarray}$$

$\text{pod}_{-4}(n)$

In this paper, by using the theory of reproducing kernel Hilbert spaces and the pair correlation formula constructed by Chandee et al. [‘Simple zeros of primitive Dirichlet $L$-functions and the asymptotic large sieve’, Q. J. Math.65(1) (2014), 63–87], we prove that at least 93.22% of low-lying zeros of primitive Dirichlet $L$-functions are simple in a proper sense, under the assumption of the generalised Riemann hypothesis.

In 1981 Beck and Fiala proved an upper bound for the discrepancy of a set system of degree d that is independent of the size of the ground set. In the intervening years the bound has been decreased from 2d − 2 to 2d − 4. We improve the bound to 2d − log*d.

In previous work, Ohno conjectured, and Nakagawa proved, relations between thecounting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of ‘extra functional equations’ involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In the present paper, we generalize their result by proving a similar identity relating certain degree-$\ell$ fields to Galois groups $D_{\ell }$ and $F_{\ell }$, respectively, for any odd prime $\ell$; in particular, we give another proof of the Ohno–Nakagawa relation without appealing to binary cubic forms.

We study the growth of $\unicode[STIX]{x0428}$ and $p^{\infty }$-Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth types are usually exponential, as in the ‘positive ${\it\mu}$-invariant’ setting in the Iwasawa theory of elliptic curves. The towers we consider are $p$-adic and $l$-adic Lie extensions for $l\neq p$, in particular cyclotomic and other $\mathbb{Z}_{l}$-extensions.

We use a relative trace formula on $\text{GL}(2)$ to compute a sum of twisted modular $L$-functions anywhere in the critical strip, weighted by a Fourier coefficient and a Hecke eigenvalue. When the weight $k$ or level $N$ is sufficiently large, the sum is nonzero. Specializing to the central point, we show in some cases that the resulting bound for the average is as good as that predicted by the Lindelöf hypothesis in the $k$ and $N$ aspects.

The approximation constant ${\it\lambda}_{k}({\it\zeta})$ is defined as the supremum of ${\it\eta}\in \mathbb{R}$ such that the estimate $\max _{1\leqslant j\leqslant k}\Vert {\it\zeta}^{j}x\Vert \leqslant x^{-{\it\eta}}$ has infinitely many integer solutions $x$. Here $\Vert .\Vert$ denotes the distance to the closest integer. We establish a connection on the joint spectrum $({\it\lambda}_{1}({\it\zeta}),{\it\lambda}_{2}({\it\zeta}),\ldots )$, which will lead to various improvements of known results on the individual spectrum of the approximation constants ${\it\lambda}_{k}({\it\zeta})$ as well. In particular, for given $k\geqslant 1$ and ${\it\lambda}\geqslant 1$, we construct ${\it\zeta}$ in the Cantor set with ${\it\lambda}_{k}({\it\zeta})={\it\lambda}$. Moreover, we establish an estimate for the uniform approximation constants $\widehat{{\it\lambda}}_{k}({\it\zeta})$, which enables us to determine classical approximation constants for Liouville numbers.