Gauss congruences for rational functions in several variables

We investigate necessary as well as sufficient conditions under which the Laurent series coefficients $f_{\boldsymbol{n}}$ associated to a multivariate rational function satisfy Gauss congruences, that is $f_{\boldsymbol{m}p^r} \equiv f_{\boldsymbol{m}p^{r-1}}$ modulo $p^r$. For instance, we show that these congruences hold for certain determinants of logarithmic derivatives. As an application, we completely classify rational functions $P/Q$ satisfying the Gauss congruences in the case that $Q$ is linear in each variable.Comment: 20 page

Generalized class polynomials

The Hilbert class polynomial has as roots the j-invariants of elliptic curves whose endomorphism ring is a given imaginary quadratic order. It can be used to compute elliptic curves over finite fields with a prescribed number of points. Since its coefficients are typically rather large, there has been continued interest in finding alternative modular functions whose corresponding class polynomials are smaller. Best known are Weber's functions, that reduce the size by a factor of 72 for a positive density subset of imaginary quadratic discriminants. On the other hand, Br\"oker and Stevenhagen showed that no modular function will ever do better than a factor of 100.83. We introduce a generalization of class polynomials, with reduction factors that are not limited by the Br\"oker-Stevenhagen bound. We provide examples matching Weber's reduction factor. For an infinite family of discriminants, their reduction factors surpass those of all previously known modular functions by a factor at least 2.Comment: 28 pp. 5 fi

Dynamically affine maps in positive characteristic

We study fixed points of iterates of dynamically affine maps (a generalisation of Latt\`es maps) over algebraically closed fields of positive characteristic $p$. We present and study certain hypotheses that imply a dichotomy for the Artin-Mazur zeta function of the dynamical system: it is either rational or non-holonomic, depending on specific characteristics of the map. We also study the algebraicity of the so-called tame zeta function, the generating function for periodic points of order coprime to $p$. We then verify these hypotheses for dynamically affine maps on the projective line, generalising previous work of Bridy, and, in arbitrary dimension, for maps on Kummer varieties arising from multiplication by integers on abelian varieties.Comment: Lois van der Meijden co-authored Appendix B. 31 p

Navigating the Stars: Norway, the European Economic Area and the European Union. CEPS Paperback. February 2002

This study expertly assesses the evolving relationship between Norway and the European Union, the centrepiece of which is the European Economic Area (EEA). Faced with an increasingly outdated network of relationships with the EU, Norway finds itself marginalised from policy-making and subject instead to policy-taking. This report evaluates Norway’s position in relation to the ‘future of Europe’ debate as well as a range of hypothetical options that Norway may contemplate, focusing on several key policy areas including the single market, the macroeconomic agenda, justice and home affairs, and foreign security and defence policies

On the decisional Diffie-Hellman problem for class group actions on oriented elliptic curves

We show how the Weil pairing can be used to evaluate the assigned characters of an imaginary quadratic order $\mathcal{O}$ in an unknown ideal class $[\mathfrak{a}] \in \mathrm{Cl}(\mathcal{O})$ that connects two given $\mathcal{O}$-oriented elliptic curves $(E, \iota)$ and $(E', \iota') = [\mathfrak{a}](E, \iota)$. When specialized to ordinary elliptic curves over finite fields, our method is conceptually simpler and often somewhat faster than a recent approach due to Castryck, Sot\'akov\'a and Vercauteren, who rely on the Tate pairing instead. The main implication of our work is that it breaks the decisional Diffie-Hellman problem for practically all oriented elliptic curves that are acted upon by an even-order class group. It can also be used to better handle the worst cases in Wesolowski's recent reduction from the vectorization problem for oriented elliptic curves to the endomorphism ring problem, leading to a method that always works in sub-exponential time.Comment: 18 p

Horizontal racewalking using radical isogenies

We address three main open problems concerning the use of radical isogenies, as presented by Castryck, Decru and Vercauteren at Asiacrypt 2020, in the computation of long chains of isogenies of fixed, small degree between elliptic curves over finite fields. Firstly, we present an interpolation method for finding radical isogeny formulae in a given degree $N$, which by-passes the need for factoring division polynomials over large function fields. Using this method, we are able to push the range for which we have formulae at our disposal from $N \leq 13$ to $N \leq 37$ (where in the range $18 \leq N \leq 37$ we have restricted our attention to prime powers). Secondly, using a combination of known techniques and ad-hoc manipulations, we derive optimized versions of these formulae for $N \leq 19$, with some instances performing more than twice as fast as their counterparts from 2020. Thirdly, we solve the problem of understanding the correct choice of radical when walking along the surface between supersingular elliptic curves over $\mathbb{F}_p$ with $p \equiv 7 \bmod 8$; this is non-trivial for even $N$ and was settled for $N = 2$ and $N = 4$ only, in the latter case by Onuki and Moriya at PKC 2022. We give a conjectural statement for all even $N$ and prove it for $N \leq 14$. The speed-ups obtained from these techniques are substantial: using $16$-isogenies, the computation of long chains of $2$-isogenies over $512$-bit prime fields can be accelerated by a factor $3$, and the previous implementation of CSIDH using radical isogenies can be sped up by about $12\%$

Rich dynamics and functional organization on topographically designed neuronal networks in vitro

Neuronal cultures are a prominent experimental tool to understand complex functional organization in neuronal assemblies. However, neurons grown on flat surfaces exhibit a strongly coherent bursting behavior with limited functionality. To approach the functional richness of naturally formed neuronal circuits, here we studied neuronal networks grown on polydimethylsiloxane (PDMS) topographical patterns shaped as either parallel tracks or square valleys.We followed the evolution of spontaneous activity in these cultures along 20 days in vitro using fluorescence calcium imaging. The networks were characterized by rich spatiotemporal activity patterns that comprised from small regions of the culture to its whole extent. Effective connectivity analysis revealed the emergence of spatially compact functional modules that were associated with both the underpinned topographical features and predominant spatiotemporal activity fronts. Our results showthe capacity of spatial constraints tomold activity and functional organization, bringing new opportunities to comprehend the structure-function relationship in living neuronal circuits

Survey of 800+ data sets from human tissue and body fluid reveals xenomiRs are likely artifacts

miRNAs are small 22-nucleotide RNAs that can post-transcriptionally regulate gene expression. It has been proposed that dietary plant miRNAs can enter the human bloodstream and regulate host transcripts; however, these findings have been widely disputed. We here conduct the first comprehensive meta-study in the field, surveying the presence and abundances of cross-species miRNAs (xenomiRs) in 824 sequencing data sets from various human tissues and body fluids. We find that xenomiRs are commonly present in tissues (17%) and body fluids (69%); however, the abundances are low, comprising 0.001% of host human miRNA counts. Further, we do not detect a significant enrichment of xenomiRs in sequencing data originating from tissues and body fluids that are exposed to dietary intake (such as liver). Likewise, there is no significant depletion of xenomiRs in tissues and body fluids that are relatively separated from the main bloodstream (such as brain and cerebro-spinal fluids). Interestingly, the majority (81%) of body fluid xenomiRs stem from rodents, which are a rare human dietary contribution but common laboratory animals. Body fluid samples from the same studies tend to group together when clustered by xenomiR compositions, suggesting technical batch effects. Last, we performed carefully designed and controlled animal feeding studies, in which we detected no transfer of plant miRNAs into rat blood, or bovine milk sequences into piglet blood. In summary, our comprehensive computational and experimental results indicate that xenomiRs originate from technical artifacts rather than dietary intake