For a finite abelian group G and a splitting field K of G, let (G,K) denote the largest integer l ∈ ℕ for which there is a sequence $S=g₁·...·g_l$ over G such that $(X^{g₁}-a₁)·...·(X^{g_l}-a_l)\ne 0\in K\left[G\right]$ for all $a₁,...,a_l\in K^{\times }$. If (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (G,K). Equality holds for a variety of groups, and a conjecture of W. Gao et al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups G for which (G) -...

In this paper, we give an explicit description of the de Rham and $p$-adic polylogarithms for elliptic curves using the Kronecker theta function. In particular, consider an elliptic curve $E$ defined over an imaginary quadratic field $𝕂$ with complex multiplication by the full ring of integers ${𝒪}_{𝕂}$ of $𝕂$. Note that our condition implies that $𝕂$ has class number one. Assume in addition that $E$ has good reduction above a prime $p\ge 5$ unramified in ${𝒪}_{𝕂}$. In this case, we prove that the specializations of the $p$-adic elliptic...

If $a$ and $b$ are positive integers with $a\le b$ and $a^2\equiv a\mathrm{mod}b$, then the set$$M_{a,b}=\{x\in \mathbb{N}:x\equiv a\mathrm{mod}b\text{or}x=1\}$$is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid $M$ with units $M^{\times }$ and any $x\in M\setminus M^{\times }$ we say that $t\in \mathbb{N}$ is a factorization length of $x$ if and only if there exist irreducible elements $y_1,...,y_t$ of $M$ and $x=y_1\cdots y_t$. Let $ℒ\left(x\right)=\{t_1,...,t_j\}$ be the set of all such lengths (where $t_i\<t_{i+1}$ whenever $i\<j$). The Delta-set of the element $x$ is defined as the set of gaps in $ℒ\left(x\right)$: $\Delta \left(x\right)=\{t_{i+1}-t_i:1\le i\<k\}$ and the Delta-set of the monoid $M$ is given by ${\bigcup }_{x\in M\setminus M^{\times }}\Delta \left(x\right)$. We consider the $\Delta \left(M\right)$ when $M=M_{a,b}$ is an ACM with...