September 2023: University of Reading

Thomas Bloom (Oxford): 

Title: Odd moments and adding fractions

Abstract: In 1986 Montgomery and Vaughan proved an asymptotic formula for the even moments of the distribution of reduced residues in short intervals of integers. For odd moments they proved only a small improvement over the 'trivial' bound. In joint work with Vivian Kuperberg we prove essentially optimal upper bounds for the odd moments of this distribution, confirming a conjecture of Montgomery and Vaughan. As an application we prove (probably) optimal upper bounds for the average of the singular series when counting prime tuples of odd length. Our main new tool is a new upper bound for the number of solutions to $\sum_{1\leq i\leq k} \frac{a_i}{q_i}=0$ when $k$ is odd, an elementary self-contained problem of independent interest.

Vaidehee Thatte (KCL):

Title: Understanding the Defect via Ramification Theory

Abstract: Classical ramification theory deals with complete discrete valuation fields k((X)) with perfect residue fields k. Invariants such as the Swan conductor capture important information about extensions of these fields. Many fascinating complications arise when we allow non-discrete valuations and imperfect residue fields k. Particularly in positive residue characteristic, we encounter the mysterious phenomenon of the defect (or ramification deficiency). The occurrence of a non-trivial defect is one of the main obstacles to long-standing problems, such as obtaining resolution of singularities in positive characteristic.

Degree p extensions of valuation fields are building blocks of the general case. In this talk, we will present a generalization of ramification invariants for such extensions and discuss how this leads to a better understanding of the defect. If time permits, we will briefly discuss their connection with some recent work (joint with K. Kato) on upper ramification groups.

Harry Schmdit (Basel/Warwick):

Title:  Effective Zilber-Pink in Poincaré bi-extensions. 

Abstract: It is well-known that an elliptic curve $E$ is isomorphic to its dual. The Poincaré bi-extension of $E$ is a three-dimensional variety $P$ that projects to $E\times \hat{E} = E^2$ and for which each fibre above a point in $\hat{E}$ is a multiplicative extension of $E$. If $E$ has complex multiplication, then $P$ is a mixed Shimura variety and the work of Bertrand and Edixhoven gives an appealing classification of the special sub-varieties of $P$. In joint work with Gareth Jones, we are investigating effective and uniform versions of the André-Oort and Zilber-Pink conjecture for $P$. The latter is connected with the so-called relative Manin-Mumford conjecture that, for $P$, was proven by Bertrand, Masser, Pillay, and Zannier. I will go a bit deeper into the history of this conjecture and its counterexample by Bertrand before explaining the new methods used by Jones and me to obtain effective versions of Zilber-Pink for $P$. Among those methods is an effective version of the Pila-Wilkie counting theorem for Pfaffian functions recently obtained by Binyamini, Jones, Thomas, and the speaker.

Sebastian Eterovic (Leeds): 

Title: Multiplicative Relations Among Differences of Singular Moduli

Abstract: A singular modulus is the $j$-invariant of an elliptic curve with complex multiplication; as such the arithmetic properties of these numbers are of great interest. In particular, there are important results concerning the differences of singular moduli, and also multiplicative dependencies of singular moduli. In joint work with Vahagn Aslanyan and Guy Fowler we show that for every positive integer $n$ there are a finite set $S$ and finitely many algebraic curves $T_1,\ldots,T_k$ with the following property: if $(x_1,\ldots,x_n,y)$ is a tuple of pairwise distinct singular moduli satisfying a multiplicative dependency of the form $(x_1-y)^{a_1}\cdots (x_n-y)^{a_n}=1$, where $a_1,\ldots,a_n$ are non-zero integers, then $(x_1, \ldots, x_n, y) \in S \cup T_1 \cup \ldots \cup T_k$.