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We initiate the study of a new class of polynomials which we call quasi-subfield polynomials. First, we show that this class of polynomials could lead to more efficient attacks for the elliptic curve discrete logarithm problem via the index calculus approach. Specifically, we use these polynomials to construct factor bases for the index calculus approach and we provide explicit complexity bounds. Next, we investigate the existence of quasi-subfield polynomials., SCOPUS: ar.j, info:eu-repo/semantics/published

Let P : ⋯ → C 2 → C 1 → P 1 be a Z p -cover of the projective line over a finite field of cardinality q and characteristic p which ramifies at exactly one rational point. We study the q-adic valuations of the reciprocal roots in C p of L-functions associated to characters of the Galois group of P . We show that for all covers P such that the genus of C n is a quadratic polynomial in p n for n large, the valuations of these reciprocal roots are uniformly distributed in the interval [ 0 , 1 ] . Furthermore, we show that for a large class of such covers P , the valuations of the reciprocal roots in fact form a finite union of arithmetic progressions.

We give a corrected necessary and sufficient condition for the genus stability in a Z p \mathbb {Z}_p -tower of function fields of characteristic p p .

Let $K$ be a function field over a finite field $k$ of characteristic $p$ and let $K_{\infty}/K$ be a geometric extension with Galois group $\mathbb{Z}_p$. Let $K_n$ be the corresponding subextension with Galois group $\mathbb{Z}/p^n\mathbb{Z}$ and genus $g_n$. In this paper, we give a simple explicit formula $g_n$ in terms of an explicit Witt vector construction of the $\mathbb{Z}_p$-tower. This formula leads to a tight lower bound on $g_n$ which is quadratic in $p^n$. Furthermore, we determine all $\mathbb{Z}_p$-towers for which the genus sequence is stable, in the sense that there are $a,b,c \in \mathbb{Q}$ such that $g_n=a p^{2n}+b p^n +c$ for $n$ large enough. Such genus stable towers are expected to have strong stable arithmetic properties for their zeta functions. A key technical contribution of this work is a new simplified formula for the Schmid-Witt symbol coming from local class field theory., 13 pages, this is a short version of arXiv:1607.00523

Let V be a finite-dimensional vector space over a field k and let W be a 1-dimensional k-vector space. Let < , >: V x V \to W be a symmetric bilinear form. Then < , > is called anisotropic if for all nonzero v \in V we have \neq 0. Motivated by a problem in algebraic number theory, we come up with a generalization of the concept of anisotropy to symmetric bilinear forms on finitely generated modules over artinian principal ideal rings. We will give many equivalent definitions of this concept of anisotropy. One of the definitions shows that one can check if a form is anisotropic by checking if certain forms on vector spaces are anisotropic. We will also discuss the concept of quasi-anisotropy of a symmetric bilinear form, which has no useful vector space analogue. Finally we will discuss the radical root of a symmetric bilinear form, which doesn't have a useful vector space analogue either. All three concepts have applications in algebraic number theory., 18 pages

In this article we study the following problem. Let $k$ be an infinite field and let $f \in k[x]$. Consider the evaluation map $f_k: k \to k$. Assume that $f_k$ is not surjective. Is $k \smallsetminus f_k(k)$ infinite? We give a positive answer to this question when $k$ is a perfect ample field. In fact, we prove that $|k \smallsetminus f_k(k)|=|k|$. This conclusion follows from a similar statement about finite morphisms between normal projective curves over perfect ample fields.

In this article we will discuss a new, mostly theoretical, method for solving (zero-dimensional) polynomial systems, which lies in between Gr\"obner basis computations and the heuristic first fall degree assumption and is not based on any heuristic. This method relies on the new concept of last fall degree. Let $k$ be a finite field of cardinality $q^n$ and let $k'$ be its subfield of cardinality $q$. Let $\mathcal{F} \subset k[X_0,\ldots,X_{m-1}]$ be a finite subset generating a zero-dimensional ideal. We give an upper bound of the last fall degree of the Weil descent system of $\mathcal{F}$, which depends on $q$, $m$, the last fall degree of $\mathcal{F}$, the degree of $\mathcal{F}$ and the number of solutions of $\mathcal{F}$, but not on $n$. This shows that such Weil descent systems can be solved efficiently if $n$ grows. In particular, we apply these results for multi-HFE and essentially show that multi-HFE is insecure. Finally, we discuss that the degree of regularity (or last fall degree) of Weil descent systems coming from summation polynomials to solve the elliptic curve discrete logarithm problem might depend on $n$, since such systems without field equations are not zero-dimensional., Comment: 16 pages, changed definition of tau and revised Section 5

In this article we discuss a version of the Chebotarev density for function fields over perfect fields with procyclic absolute Galois groups. Our version of this density theorem differs from other versions in two aspects: we include ramified primes and we do not have an error term., 6 pages, rewritten most of the paper

Let $\epsilon>0$. In this article we will present a deterministic algorithm which does the following. The input is a hyperelliptic curve $C$ of genus $g$ over a finite field $k$ of cardinality $q$ given by $y^2+h(x)y=f(x)$ such that the $x$-coordinate map is ramified at $\infty$. In time $O(g^{2+\epsilon} q^{1/2+\epsilon})$ the algorithm outputs a set of generators of the Picard group $\mathrm{Pic}^0_k(C)$. This extends results which others have obtained when $g=1$. In this article we introduce a combinatorial tool, the `shape parameter', which we use together with character sum estimates from class field theory to deduce the statement., Comment: 15 pages

Let $l$ be a finite field of cardinality $q$ and let $n$ be in $\mathbb{Z}_{\geq 1}$. Let $f_1,\ldots,f_n \in l[x_1,\ldots,x_n]$ not all constant and consider the evaluation map $f=(f_1,\ldots,f_n) \colon l^n \to l^n$. Set $\mathrm{deg}(f)=\max_i \mathrm{deg}(f_i)$. Assume that $l^n \setminus f(l^n)$ is not empty. We will prove \begin{align*} |l^n\setminus f(l^n)| \geq \frac{n(q-1)}{\mathrm{deg}(f)}. \end{align*} This improves previous known bounds., Comment: 6 pages

We give an algorithm for presenting S-unit groups of an order in a definite rational quaternion algebra B such that for every p ∈ S at which B splits, the localization of at p is maximal, and all left ideals of of norm p are principal. We then apply this to give presentations for projective S-unit groups of the Hurwitz order in Hamilton’s quaternions over the rational field . To our knowledge, this provides the first explicit presentations of an S-arithmetic lattice in a semisimple Lie group with S large. We also include some discussion and experimentation related to the congruence subgroup problem, which is open for S-units of the Hurwitz order when S contains at least two odd primes.

Let K be a number field and let A be an order in K. The trace map from K to Q induces a non-degenerate symmetric bilinear form : B x B \to Q/Z where B is a certain finite abelian group of size \Delta(A). In this article we discuss how one can obtain information about \mathcal{O}_K by purely looking at this symmetric bilinear form. The concepts of anisotropy and quasi-anisotropy, as defined in another article by the author, turn out to be very useful. We will for example show that under certain assumptions one can obtain \mathcal{O}_K directly from . In this article we will work in a more general setting than we have discussed above. We consider orders over Dedekind domains., Comment: 21 pages

Let R be a commutative ring. A not necessarily commutative R-algebra A is called futile if it has only finitely many R-subalgebras. In this article we relate the notion of futility to familiar properties of rings and modules. We do this by first reducing to the case where A is commutative. Then we refine the description of commutative futile algebras from Dobbs, Picavet and Picavet-L'hermite., 15 pages