Mazur's isogeny theorem

Mazur's isogeny theorem states that if $p$ is a prime for which there exists an elliptic curve $E / \mathbb{Q}$ that admits a rational isogeny of degree $p$, then $p \in \{2,3,5,7,11,13,17,19,37,43,67,163 \}$. This result is one of the cornerstones of the theory of elliptic curves and plays a crucial role in the proof of Fermat's Last Theorem. In this expository paper, we overview Mazur's proof of this theorem, in which modular curves and Galois representations feature prominently.Comment: Expository paper, 16 page

A Unique Perfect Power Decagonal Number

Let $\mathcal{P}_s(n)$ denote the $n$th $s$-gonal number. We consider the equation $$\mathcal{P}_s(n) = y^m,$$ for integers $n,s,y,$ and $m$. All solutions to this equation are known for $m>2$ and $s \in \{3,5,6,8,20 \}$. We consider the case $s=10$, that of decagonal numbers. Using a descent argument and the modular method, we prove that the only decagonal number $>1$ expressible as a perfect $m$th power with $m>1$ is $\mathcal{P}_{10}(3) = 3^3$.Comment: 5 pages. Some typos correcte

Quadratic Points on Non-Split Cartan Modular Curves

In this paper we study quadratic points on the non-split Cartan modular curves $X_{ns}(p)$, for $p = 7, 11,$ and $13$. Recently, Siksek proved that all quadratic points on $X_{ns}(7)$ arise as pullbacks of rational points on $X_{ns}^+(7)$. Using similar techniques for $p=11$, and employing a version of Chabauty for symmetric powers of curves for $p=13$, we show that the same holds for $X_{ns}(11)$ and $X_{ns}(13)$. As a consequence, we prove that certain classes of elliptic curves over quadratic fields are modular.Comment: 18 pages. To appear in International Journal of Number Theory. Minor corrections and rewriting. Slightly improved sieving step and more computation details include

Frontières du caquet en forme de chanson

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal

MyD88-adaptor protein acts as a preventive mechanism for memory deficits in a mouse model of Alzheimer's disease

<p>Abstract</p> <p>Background</p> <p>Alzheimer's disease (AD) is an age-related neurodegenerative disorder associated with brain innate immune activation mainly mediated by microglia. These cells are known to be activated in the brain of AD patients and to produce inflammatory cytokines and neurotoxic molecules in response to Amyloid beta (Aβ). Activation of microglia can also promote Aβ clearance via Toll-like receptors (TLRs). Myeloid differentiation factor 88 (MyD88) is the adaptor molecule for most of these innate immune receptors, transducing the intracellular signal from TLRs to nucleus.</p> <p>Results</p> <p>Here, we report that more than 50% reduction in MyD88 expression in a mouse model of AD accelerated spatial learning and memory deficits. Brain of APP_swe/PS1-MyD88^+/-mice was characterized by a delay in accumulation of Aβ plaques and increased soluble levels of Aβ oligomers. Furthermore, inflammatory monocyte subset and brain IL-1β gene expression were significantly reduced in APP_swe/PS1 mice with impaired MyD88 signaling.</p> <p>Conclusions</p> <p>These data indicate that activation of MyD88 intracellular signaling pathway, likely by TLRs, acts as a natural innate immune mechanism to restrict disease progression of APP_swe/PS1 mice.</p

La comédie de destruction

Celui qui rit du comique s’empêche d’en comprendre la véritable nature, remarquait Aby Warburg lorsqu’il évoquait, dans sa conférence sur le rituel du serpent, l’irruption des clowns au milieu des danseurs katchinas à laquelle il avait assisté lors de son séjour en territoire Hopi durant l’hiver 1896. Parce qu’il reste confiné dans le registre de la culture populaire, le burlesque cinématographique des premiers temps échappe encore aux conventions sentimentales et morales qui viendront en édulcorer la force au cours des années 1920. Brutal, violent, improvisé, il recueille les principes d’une poétique négative animée par une dynamique de destruction qui affecte à la fois les figures et les éléments de la composition. La comédie n’est pas l’autre de la tragédie : elle en révèle le sens. À la manière du satyre nietzschéen s’avançant sur la scène tragique, l’acteur comique révèle au spectateur, dans un éclat de rire, la profondeur de « l’effroyable nuit » et la nullité du principe d’individuation.The one who laughs at the comical prevents them from understanding its true nature, as Aby Warburg noted when he mentioned, in his lecture on the serpent ritual, the arrival of clowns in the midst of the Kachina dancers that he saw during his sojourn in Hopi territory during the winter of 1896. Since it remained restricted to the field of popular culture, early farcical films escaped the sentimental and moral conventions that would dilute their strength during the 1920s. Brutal, violent, and improvised, they embraced the principles of a negative poetics animated by a dynamic of destruction that affected both the characters and the composition’s elements. Comedy is not the inverse of tragedy: it reveals its meaning. Like the Nietzschean satyr entering the tragic scene, the comic actor reveals to the spectator, in a burst of laughter, the depth of “the terrible night” and the foolishness of the principle of individuation.Wer über das Komische lacht, wird dessen eigentliches Wesen nicht begreifen können, stellte Aby Warburg fest, als er in seinem Vortrag über das Schlangenritual das Auftauchen der Clowns inmitten der Katsina-Tänzer kommentierte, deren Ritual er während seines Aufenthalts im Hopi-Reservat im Winter 1896 miterlebt hatte. Da es sich lediglich in den eng gesteckten Grenzen der Populärkultur bewegte, entkam das burleske Kino der ersten Zeit noch den emotionalen und moralischen Konventionen, die seine ursprüngliche Kraft in den 1920er Jahren einfärbten. Brutal, gewalttätig, improvisiert, es enthält alle Prinzipien einer negativen Poetik, die von der Dynamik der Destruktion angetrieben wird und sich auf Figuren und Elemente der Komposition auswirkt. Die Komödie ist nicht das Gegenstück zur Tragödie: sie legt ihren Sinn frei. So wie Nietzsches Satyr sich in die tragische Szene wagt, so zeigt der Komödiant dem Zuschauer in einem Lachen die Tiefe der „furchtbaren Nacht“ und die Nichtigkeit des Individuationsprinzips.Colui che ride del comico non riesce a comprenderne la vera natura, notava Aby Warburg quando evocava, nella sua conferenza sul rituale del serpente, l’irruzione dei clowns tra i danzatori katchinas alla quale aveva assistito durante un suo soggiorno in territorio Hopi durante l’inverno 1896. Poiché resta confinato nell’ambito della cultura popolare, il burlesco cinematografico dei primi tempi sfugge ancora alle convenzioni sentimentali e morali che vorranno edulcorarne la forza nel corso degli anni Venti. Brutale, violento, improvvisato, esso raccoglie i principi di una poetica negativa animata da una dinamica di distruzione che intacca, allo stesso tempo, le figure e gli elementi della composizione. La commedia non è l’altro lato della tragedia: essa ne rivela invece il senso. Alla maniera del satiro nietzschiano che avanza sulla scena tragica, l’attore comico rivela allo spettatore, in uno scoppio di risa, la profondità della spaventosa notte e la nullità di un principio di individuazione.Aquel que ríe de lo cómico se imposibilita comprender su verdadera naturaleza, remarcaba Aby Warburg cuando evocaba, en su conferencia sobre el ritual de la serpiente, la irrupción de los payasos en medio de los bailarines katchinas a la cual él había asistido durante su paso por el territorio Hopi en el invierno de 1896. Confinado al registro de la cultura popular, el burlesco cinematográfico iniciático escapa todavía a las convenciones sentimentales y morales que vendrán a edulcorar su fuerza a partir de 1920. Brutal, violento, improvisado, recoge los principios de una poética negativa animada por una dinámica de la destrucción que afecta a la vez a las figuras y a los elementos de la composición. La comedia no es lo otro de la tragedia: ella revela su sentido. A la manera de un sátiro nietzscheano avanzando sobre la escena trágica, el actor cómico revela al espectador, en un estallido de risa, la profundidad del “horror de la noche” y la estupidez del principio de individuación

Bill Readings : penser le dissensus dans les ‘ruines’ de l’université

Ce mémoire présente une interprétation et une explication critique des propositions principales de l’ouvrage posthume de Bill Readings, The University in Ruins (Harvard, 1996). L’auteur vise à introduire la pensée de Readings en tant que translation du différend lyotardien vers les débats autour de la crise des curricula et des « guerres culturelles » américaines. Deuxièmement, ce travail mobilise le dialogue qui l’a opposé au projet pédagogique de Gerald Graff, lequel trouvait dans la notion de « Conflit » elle-même le ciment d’une nouvelle communauté universitaire. La communauté du « dissensus » chez Readings apparaît en comparaison dans toute sa radicalité. En renonçant à l’idéologie de l’émancipation du sujet caractéristique de la modernité, Readings envisageait la scène de l’enseignement depuis un mode éthique hétéronome, et caractérisait la pédagogie comme relation asymétrique et infinie à autrui. Enfin, il s’agit d’exposer comment la communauté du dissensus préserve comme aporie la question de l’identité collective autant que celle de la nature du lien social dans l’université « déréférentialisée ». Pour ce faire, ce mémoire propose sur ces idées un éclairage nouveau en les rapprochant de la pensée dialogique et anthropologique bakhtiniennes.This master’s thesis offers a critical interpretation and exploration of the main proposals of Bill Readings’ posthumous book, The University in Ruins (Harvard, 1996). I situate Readings’ provocative thoughts as a translation of Lyotard’s notion of differend into American pedagogical debates within the literary crisis upon the curricula and the so-called “Culture Wars”. The study expands on the dialogue which opposed Readings to Graff, where the latter sought to posit the notion of “Conflict” as the locus of a renewed academic community. In contrast to Graff, the radicalism of Readings’ “community of dissensus” becomes more salient. While renouncing the emancipatory ideals of Modernity, Readings suggested an understanding of education as a space of heteronomous ethics, and pedagogy as an infinite and asymmetrical relationship to others. Finally, I show that the “community of dissensus” maintains as a permanent question the aporetic nature of its identity as well as of the nature of the social bond within the “post-historical” university. By doing so, this work provides an original contribution which underlines a kinship between Readings’ most controversial statements and Mikhail Bakhtin’s dialogism and anthropological ideas

Pourvoyeur d’irréalité, Fantasmagorie, Émile Cohl, 1908

Dans le cinéma des premiers temps se fait jour, de façon fugitive, une conception ornementale de l’image en mouvement que le cinéma narratif, fondé sur les ressorts de l’illusionnisme photographique, devait éclipser et dont Fantasmagorie, le premier film de Cohl réalisé en 1908, pourrait être l’archétype.In early cinema there appears, fleetingly, an ornamental conception of the moving image, which narrative cinema would soon eclipse and whose archetype might be Fantasmagorie, Cohl’s first film, made in 1908

Fermat's last theorem and modular curves over real quadratic fields

We study the Fermat equation xn+yn=zn over quadratic fields Q(d√) for squarefree d with 26≤d≤97. By studying quadratic points on the modular curves X0(N), d-regular primes, and working with Hecke operators on spaces of Hilbert newforms, we extend work of Freitas and Siksek to show that for most squarefree d in this range there are no non-trivial solutions to this equation for n≥4

Brancusi filma

Abordando a inédita produção de filmes e as fotografias de Brancusi, o autor destaca como tais meios eram usados pelo artista como instrumentos de reflexão, que dialogavam com sua produção de esculturas. Na análise, coloca-se o ateliê do artista como espaço de trabalho em transformação, substituindo a ideia de apresentação das obras, pela de sua representação. As relações entre os meios são discutidas a partir de três processos de transferência: a fotografia como filme, o filme como fotografia e o filme como escultura