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Register a new userCM fields of Dyhedral type and the Colmez conjecture
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In this paper, we consider some CM fields which we call of dihedral type and compute the Artin $L$-functions associated to all CM types of these CM fields. As a consequence of this calculation, we see that the Colmez conjecture in this case is very closely related to understanding the log derivatives of certain Hecke characters of real quadratic fields. Recall that the `abelian case of the Colmez conjecture, proved by Colmez himself, amounts to understanding the log derivatives of Hecke characters of $Q$ (cyclotomic characters). In this paper, we also prove that the Colmez conjecture holds for `unitary CM types of signature $(n-1, 1)$ and holds on average for `unitary CM types of a fixed CM number field of signature $(n-r, r)$.
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Let $K$ be a quartic CM field, that is, a totally imaginary quadratic extension of a real quadratic number field. In a 1962 article titled On the classfields obtained by complex multiplication of abelian varieties, Shimura considered a particular family ${F_K(m) : m in mathbb{Z} >0 }$ of abelian extensions of $K$, and showed that the Hilbert class field $H_K$ of $K$ is contained in $F_K(m)$ for some positive integer m. We make this m explicit. We then give an algorithm that computes a set of defining polynomials for the Hilbert class field using the field $F_K(m)$. Our proof-of-concept implementation of this algorithm computes a set of defining polynomials much faster than current implementations of the generic Kummer algorithm for certain examples of quartic CM fields.
Let $K$ be a local function field of characteristic $l$, $mathbb{F}$ be a finite field over $mathbb{F}_p$ where $l e p$, and $overline{rho}: G_K rightarrow text{GL}_n (mathbb{F})$ be a continuous representation. We apply the Taylor-Wiles-Kisin method over certain global function fields to construct a mod $p$ cycle map $overline{text{cyc}}$, from mod $p$ representations of $text{GL}_n (mathcal{O}_K)$ to the mod $p$ fibers of the framed universal deformation ring $R_{overline{rho}}^square$. This allows us to obtain a function field analog of the Breuil--Mezard conjecture. Then we use the technique of close fields to show that our result is compatible with the Breuil-Mezard conjecture for local number fields in the case of $l e p$, obtained by Shotton.
In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the generalized $abc$-conjecture holds in K, then there exist at least $c,log X$ prime numbers $p leq X$ for which K is $p$-rational, here $c$ is some nonzero constant depending on K. The real quadratic case was recently suggested by Bockle-Guiraud-Kalyanswamy-Khare.
Bruinier and Yang conjectured a formula for an intersection number on the arithmetic Hilbert modular surface, CM(K).T_m, where CM(K) is the zero-cycle of points corresponding to abelian surfaces with CM by a primitive quartic CM field K, and T_m is the Hirzebruch-Zagier divisors parameterizing products of elliptic curves with an m-isogeny between them. In this paper, we examine fields not covered by Yangs proof of the conjecture. We give numerical evidence to support the conjecture and point to some interesting anomalies. We compare the conjecture to both the denominators of Igusa class polynomials and the number of solutions to the embedding problem stated by Goren and Lauter.
For a global function field K of positive characteristic p, we show that Artin conjecture for L-functions of geometric p-adic Galois representations of K is true in a non-trivial p-adic disk but is false in the full p-adic plane. In particular, we prove the non-rationality of the geometric unit root L-functions.
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