Thursday, May 31 |
07:30 - 09:00 |
Breakfast (Restaurant at your assigned hotel) |
09:00 - 10:00 |
Roger Heath-Brown: Manin's conjecture for a bi-projective variety ↓ This is joint work with Tim Browning. The talk concerns the variety X1Y21+X2Y22+X3Y23+X4Y24=0 in P3×P3. The height of a point (x,y) is given by H(x)3H(y)2, and Manin's conjecture predicts asymptotically cBlogB points of height at most B. To obtain this one must exclude points on the subvariety x1x2x3x4=0; but in order to achieve the Peyre constant one must exclude an infinite number of subvarieties in which x1x2x3x4 is a square.
By combining the circle method with lattice point counting techniques we are able to prove Manin's conjecture for this example, and the talk will give an overview of the various ingredients, and the way that they fit together. (Conference Room San Felipe) |
10:10 - 10:40 |
Damaris Schindler: Diophantine inequalities for ternary diagonal forms ↓ We discuss small solutions to ternary diagonal inequalities of any degree where all of the variables are assumed to be of size P. We study this problem on average over a one-parameter family of forms and discuss a generalization of work of Bourgain on generic ternary diagonal quadratic forms to higher degree. In particular we discuss how these Diophantine inequalities are related to counting rational points close to varieties. (Conference Room San Felipe) |
10:40 - 11:10 |
Coffee Break (Conference Room San Felipe) |
11:10 - 11:40 |
Martin Bright: A uniform bound on the Brauer groups of certain log K3 surfaces ↓ There has been much interest recently in bounding the Brauer groups of K3 surfaces over number fields. On the other hand, the arithmetic of integral points on log K3 surfaces appears to share some features with that of rational points on K3 surfaces. Some of the simplest examples of log K3 surfaces are the open surfaces obtained by starting with a projective del Pezzo surface and removing a smooth anticanonical divisor. We use Merel's boundedness of torsion on elliptic curves to prove boundedness of the Brauer groups of such log K3 surfaces over a number field. This is joint work with Julian Lyczak. (Conference Room San Felipe) |
11:50 - 12:50 |
David Zureick-Brown: Counting points, counting fields, and heights on stacks ↓ A folklore conjecture is that the number Nd(K,X) of degree-d extensions of K with discriminant at most d is on order cdX. In the case K=Q, this is easy for d=2, a theorem of Davenport and Heilbronn for d=3, a much harder theorem of Bhargava for d=4 and 5, and completely out of reach for d>5. More generally, one can ask about extensions with a specified Galois group G; in this case, a conjecture of Malle holds that the asymptotic growth is on order Xa(logX)b for specified constants a,b.
The form of Malle's conjecture is reminiscent of the Batyrev--Manin conjecture, which says that the number of rational points of height at most X on a Batyrev-Manin variety also grows like Xa(logX)b for specified constants a,b. What's more, an extension of Q with Galois group G is a rational point on a Deligne-Mumford stack called BG, the classifying stack of G. A natural reaction is to say "the two conjectures is the same; to count number fields is just to count points on the stack BG with bounded height?" The problem: there is no definition of the height of a rational point on a stack. I'll explain what we think the right definition is, and explain how it suggests a heuristic which has both the Malle conjecture and the Batyrev--Manin conjecture as special cases.
This is joint with Jordan Ellenberg and Matt Satriano. (Conference Room San Felipe) |
13:00 - 14:30 |
Lunch (Restaurant Hotel Hacienda Los Laureles) |
15:20 - 15:50 |
Pankaj Vishe: Quartic forms in 30 variables ↓ We will prove that smooth Quartic hypersurfaces satisfy the Hasse Principle as long as they are defined over at least 30 variables. The key tool here is employing Kloosterman's version of circle method. This is a joint work with Oscar Marmon (U Lund). (Conference Room San Felipe) |
15:50 - 16:20 |
Coffee Break (Conference Room San Felipe) |
16:20 - 16:50 |
Rachel Newton: Arithmetic of rational points and zero-cycles on Kummer varieties ↓ Yongqi Liang has shown that for rationally connected varieties over a number field K, sufficiency of the Brauer-Manin obstruction to the existence of rational points over all finite extensions of K implies sufficiency of the Brauer-Manin obstruction to the existence of zero-cycles of degree 1 over K. I will discuss joint work with Francesca Balestrieri where we extend Liang's result to Kummer varieties. (Conference Room San Felipe) |
17:00 - 17:30 |
Otto Overkamp: Finite descent obstruction and non-Abelian reciprocity ↓ For a nice algebraic variety X over a number field F, one of the central problems of Diophantine Geometry is to locate precisely the set X(F) inside X(AF), where AF denotes the ring of adèles of F. One approach to this problem is provided by the finite descent obstruction, which is defined to be the set of adelic points which can be lifted to twists of torsors for finite étale group schemes over F on X. More recently, Kim proposed an iterative construction of another subset of X(AF) which contains the set of rational points. We compare the two constructions. Our main result shows that the two approaches are equivalent. (Conference Room San Felipe) |
19:00 - 21:00 |
Dinner (Restaurant Hotel Hacienda Los Laureles) |