For a family of real quadratic fields ${\left\{K_n=\mathbb{Q}\left(\sqrt{f\left(n\right)}\right)\right\}}_{n\in \mathbb{N}}$, a Dirichlet character $\chi$ modulo $q$, and prescribed ideals $\left\{{\mathfrak{b}}_n\subset K_n\right\}$, we investigate the linear behavior of the special value of the partial Hecke $L$-function $L_{K_n}\left(s,{\chi }_n:=\chi \circ N_{K_n},{\mathfrak{b}}_n\right)$ at $s=0$. We show that for $n=qk+r$, $L_{K_n}\left(0,{\chi }_n,{\mathfrak{b}}_n\right)$ can be written as

$$\frac{1}{12q^2}\left(A_{\chi }\left(r\right)+k{B}_{\chi }\left(r\right)\right),$$

where $A_{\chi }\left(r\right),B_{\chi }\left(r\right)\in \mathbb{Z}\left[\chi \left(1\right),\chi \left(2\right),\dots ,\chi \left(q\right)\right]$ if a certain condition on ${\mathfrak{b}}_n$ in terms of its continued fraction is satisfied. Furthermore, we write $A_{\chi }\left(r\right)$ and $B_{\chi }\left(r\right)$ explicitly using values of the Bernoulli polynomials. We describe how the linearity is used in solving the class number one problem for some families and recover the proofs in some cases.

We present a method to compute the geometric Picard rank of a $K3$ surface over $\mathbb{Q}$. Contrary to a widely held belief, we show that it is possible to verify Picard rank $1$ using reduction at a single prime.

We prove that every projective embedding of a connected scheme determined by the complete linear series of a sufficiently ample line bundle is defined by the $2\times 2$ minors of a $1$-generic matrix of linear forms. Extending the work of Eisenbud, Koh and Stillman for integral curves, we also provide effective descriptions for such determinantally presented ample line bundles on products of projective spaces, Gorenstein toric varieties, and smooth varieties.

We prove that for an odd prime $p$, a finite group $G$ with no element of order $2p$ has a $p$-block of defect zero if it has a non-Abelian Sylow $p$-subgroup or more than one conjugacy class of involutions. For $p=2$, we prove similar results using elements of order $3$ in place of involutions. We also illustrate (for an arbitrary prime $p$) that certain pairs $\left(Q,y\right)$, with a $p$-regular element $y$ and $Q$ a maximal $y$-invariant $p$-subgroup, give rise to $p$-blocks of defect zero of $N_G\left(Q\right)∕Q$, and we give lower bounds for the number of such blocks which arise. This relates to the weight conjecture of J. L. Alperin.

We parametrize quartic commutative algebras over any base ring or scheme (equivalently finite, flat degree-4 $S$-schemes), with their cubic resolvents, by pairs of ternary quadratic forms over the base. This generalizes Bhargava’s parametrization of quartic rings with their cubic resolvent rings over $\mathbb{Z}$ by pairs of integral ternary quadratic forms, as well as Casnati and Ekedahl’s construction of Gorenstein quartic covers by certain rank-2 families of ternary quadratic forms. We give a geometric construction of a quartic algebra from any pair of ternary quadratic forms, and prove this construction commutes with base change and also agrees with Bhargava’s explicit construction over $\mathbb{Z}$.

We study the Coleman maps for a crystalline representation $V$ with non-negative Hodge–Tate weights via Perrin-Riou’s $p$-adic “regulator” or “expanded logarithm” map ${\mathcal{ℒ}}_V$. Denote by $\mathcal{\mathscr{H}}\left(\Gamma \right)$ the algebra of ${\mathbb{Q}}_p$-valued distributions on $\Gamma =Gal\left({\mathbb{Q}}_p\left({\mu }_{p^{\infty }}\right)∕{\mathbb{Q}}_p\right)$. Our first result determines the $\mathcal{\mathscr{H}}\left(\Gamma \right)$-elementary divisors of the quotient of ${\mathbb{D}}_{cris}\left(V\right)\otimes {\left({\mathbb{B}}_{rig,{\mathbb{Q}}_p}^{+}\right)}^{\psi =0}$ by the $\mathcal{\mathscr{H}}\left(\Gamma \right)$-submodule generated by ${\left({\phi }^{\ast }\mathbb{N}\left(V\right)\right)}^{\psi =0}$, where $\mathbb{N}\left(V\right)$ is the Wach module of $V$. By comparing the determinant of this map with that of ${\mathcal{ℒ}}_V$ (which can be computed via Perrin-Riou’s explicit reciprocity law), we obtain a precise description of the images of the Coleman maps. In the case when $V$ arises from a modular form, we get some stronger results about the integral Coleman maps, and we can remove many technical assumptions that were required in our previous work in order to reformulate Kato’s main conjecture in terms of cotorsion Selmer groups and bounded $p$-adic $L$-functions.

On donne une borne supérieure explicite en fonction de $K$, $S$, $g$ pour la hauteur de Faltings de la jacobienne d’une courbe $C$ de genre $g$, définie sur un corps de nombres $K$ et ayant bonne réduction en dehors d’un ensemble fini $S$ de places de $K$, pourvu que $C$ puisse s’écrire comme un revêtement cyclique de degré premier de la droite projective. La preuve repose sur le fait que les birapports des points de branchement du revêtement sont des $S$-unités, donc de hauteur bornée, et donnent un modèle plan de $C$.

We give an explicit upper bound in terms of $K$, $S$, $g$ for the Faltings height of the jacobian of a curve $C$ of genus $g$, defined over a number field $K$ and with good reduction outside a finite set $S$ of places of $K$ under the condition that $C$ can be written as a cyclic cover of prime order of the projective line. The proof rests on the fact that the cross ratios of the branch points of the cover are $S$-units, thus of bounded height, and give a plane model of $C$.