פעילויות השבוע

שדה המספרים הפיאדיים $\mathbb{Q}_p$ (עבור ראשוני $p$) הוא אובייקט מרכזי בתורת המספרים המודרנית, המופיע באופן טבעי. לדוגמה, לפי משפט הסה-מינקובסקי, למשוואה מהצורה $ax^2+by^2+cz^2+dw^2=0$ כאשר $a,b,c,d \in \mathbb{Z}$ יש פיתרון שלם שונה מאפס אם ורק אם יש לה פיתרון ממשי ופיתרון שלם פיאדי לכל ראשוני (ופתרונות אלו שונים מאפס).

בהרצאה זו נבנה את הפיאדיים (הכינו את סדרות $p$-הקושי שלכם), ונראה תכונות מעניינות של שדה זה, הנוגדות את האינטואיציה הממשית שלנו. בהתאם לזמן שיישאר, נראה שימושים ותוצאות בהם הפיאדיים משחקים תפקיד.

Artin‘s conjecture posits that the Brauer groups of smooth surfaces over finite fields are finite and have sizes that are perfect squares. The latter property was long expected to arise from the existence of a non-degenerate alternating symmetric bilinear pairing on the maximal non-divisible quotient of these Brauer groups, which is known to be finite. A natural candidate for such a pairing is the Artin-Tate pairing, which is easily shown to be anti-symmetric, but the alternating property is more subtle to establish.

In the case of surfaces over finite fields of odd characteristic, Feng‘s work confirmed that the Artin-Tate pairing is indeed alternating, resolving part of the conjecture in this setting. Feng‘s proof relies on a combination of etale Steenrod operations, Stiefel-Whitney classes, and a Wu formula connecting them.

In this talk, I will review Feng‘s argument and discuss ongoing joint work aimed at addressing the case of surfaces over finite fields of characteristic 2. Unlike the odd characteristic case, this scenario requires the use of syntomic cohomology in place of etale cohomology. A key aspect of our project involves developing a parallel theory of cohomology operations and characteristic classes for cohomology theories in equal characteristic. This work is a crucial step towards extending the resolution of Artin’s conjecture to surfaces over finite fields of characteristic 2.

In geometric group theory, finitely generated groups are studied from the geometric properties of their Cayley graphs. This point of view sees the group as a metric space, and allows one to apply methods of metric geometry, such as the metric compactification (also called the horofunction compactification). When considering the canonical action of the group on its boundary in such compactifications, it turns out that finite orbits are closely related to virtual homomorphisms. In this talk we will go over definitions, examples, known results and open problems for current and future research.