# Göttingen-Hannover Number Theory Workshop

## Organizers

- Valentin Blomer (Bonn)
- Jörg Brüdern (Göttingen)
- Ulrich Derenthal (Hannover)
- Damaris Schindler (Göttingen, since 2020)

## 18th meeting (online, 12 July 2021)

- 16:00 Philipp Habegger (Universität Basel):
*Uniformity for the Number of Rational Points on a Curve, Part I**Abstract:*By Faltings's Theorem, formerly known as the Mordell Conjecture, a smooth projective curve of genus at least 2 that is defined over a number field K has at most finitely many K-rational points. Vojta later gave a new proof. Several authors, including Bombieri, David-Philippon, de Diego, Parshin, Rémond, and Vojta, obtained upper bounds for the number of K-rational points. I will discuss joint work with Vesselin Dimitrov and Ziyang Gao where we show that the number of K-rational points is bounded from above as a function of K, the genus, and the rank of the Mordell-Weil group of the curve's Jacobian. Our work uses Vojta's approach. I will discuss the new ingredient, an inequality for the Néron-Tate height in a family of abelian varieties, and how it relates to the general strategy. - 17:30 Ziyang Gao (CNRS and Institut de Mathématiques de Jussieu-Paris Rive Gauche):
*Uniformity for the Number of Rational Points on a Curve, Part II**Abstract:*I will continue reporting this joint work with Vesselin Dimitrov and Philipp Habegger. In the first talk, Philipp Habegger explained the history and the recent results on the number of rational points on curves of genus at least 2, especially the recent uniformity result of Dimitrov-Gao-Habegger. The framework of the proof was explained: apart from classical results the key point is a height inequality. The proof of this height inequality was also sketched. It remains to show the non-degeneracy of a particular subvariety of the universal abelian variety.

In this talk, I will focus on explaining how to answer this question of non-degeneracy. In particular, I will explain the bi-algebraic geometry associated with the universal abelian variety, the Ax-Schanuel theorem and its application to study the non-degeneracy. If time permits, I will briefly discuss another recent joint work (based on this one) with Tangli Ge and Lars Kühne about high dimensional subvarieties of abelian varieties, known as the Uniform Mordell-Lang Conjecture.

## 17th meeting (online, 14 June 2021)

- 16:00 Trevor Wooley (Purdue University):
*My PhD (thesis) problem*In 1988 Bob Vaughan suggested that the speaker think about a certain mean value estimate for exponential sums associated with simultaneous diagonal cubic and quadratic Diophantine equations. Only 33 years later the speaker has finally solved this problem. The talk will offer a tour of developments concerning mean values of exponential sums during this time period, the punchline being some new approaches going beyond recent work on decoupling and efficient congruencing.

Abstract: - 17:30 Masahiro Nakahara (University of Washington):
*Elliptic divisibility sequences and the elliptic sieve**Abstract:*Given an elliptic curve over Q in Weierstrass form, take a point P of infinite order. In the sequence P, 2P, 3P, ... the denominators that appear gives rise to a sequence called the elliptic divisibility sequence. We study periodicity for these sequences and use them to study rational points on conic bundles over elliptic curves via the elliptic sieve. This is joint work with Subham Bhakta, Daniel Loughran, and Simon Myerson.

## 16th meeting (Universität Göttingen, 22 January 2020)

- 14:15 Valentin Blomer (Bonn):
*A symplectic restriction problem* *Coffee*- 16:00 K. Soundararajan (Stanford):
*TBA* *Dinner*

## 15th meeting (Universität Göttingen, 14 June 2019)

*Lunch*- 14:00 Jan Steffen Müller (Groningen):
*Quadratic Chabauty and the split Cartan modular curve of level 13*I will discuss a method to compute the rational points on a curve of genus g>1 over the rationals whose Jacobian has Mordell-Weil rank equal to g, and which satisfies some additional conditions. This extends the method of Chabauty-Coleman by using p-adic heights to make certain aspects of Kim's non-abelian Chabauty program explicit for such curves. As an application, I will show how this technique can be used to compute the rational points on the split Cartan modular curve of level 13, which completes the classification of non-CM elliptic curves over the rationals with split Cartan level structure. This is joint work with J. Balakrishnan, N. Dogra, J. Tuitman and J. Vonk.

Abstract: *Coffee*- 15:30 Jörg Jahnel (Siegen):
*On integral points on open degree four del Pezzo surfaces*

## 14th meeting (Universität Hannover, 21 January 2019)

Talks in C311

- 14:30 Damaris Schindler (Utrecht):
*Diophantine inequalities for ternary diagonal forms**Abstract:*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. *Coffee*- 16:30 Kevin Destagnol (IST Austria):
*Rational points and prime values of polynomials**Abstract:*A famous conjecture due to Colliot-Thélène states that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation for smooth, proper, geometrically irreducible and rationally connected varieties. An interesting test-ground for this conjecture is the case of normic equations, namely smooth proper models of equations of the form*f*(*t*_{1},...,*t*)=_{n}*N*_{K/Q}(**x**) for a rational polynomial*f*and a number field*K*.

Works of Colliot-Thélène-Sansuc and more recently of Harpaz-Wittenberg in the case of one variable polynomials and conditionnal to some conjectures regarding prime values of such polynomials indicate that getting information on prime values taken by*f*might be a way to tackle the study of the Brauer-Manin obstruction for some of those equations. I will explain in this talk how one can adapt Birch's circle method to tackle the problem of prime values of polynomials in (moderately) many variables and how to apply this result to derive new cases for the Hasse principle and weak approximation for normic equations. This is joint work with Efthymios Sofos. *Dinner*

## Workshop (Universität Göttingen, 19-23 November 2018)

## 13th meeting (Universität Göttingen, 15 January 2018)

- Jan-Christoph Schlage-Puchta (Rostock)
- Evelina Viada (Göttingen)

## 12th meeting (Universität Göttingen, 13 November 2017)

- 14:00 Florian Wilsch (Hannover): Counting integral points on a certain log Fano threefold
- 16:00 Raphael Steiner (Bristol): Equidistribution of rational points on shrinking sets of S
^{3}: A juxtaposition of an automorphic and a circle method approach

## 11th meeting (Universität Göttingen, 2 June 2017)

- 13:30 Valentin Blomer (Göttingen): Spectral reciprocity and moments of L-functions
- 15:15 Matthias Schütt (Hannover): Elliptic surfaces in geometry and arithmetic

## 10th meeting (Universität Hannover, 27 January 2017)

Talks in A410

- 14:00 Davide Lombardo (Hannover): Reductions of points of infinite order on algebraic groups
- 15:15 Valentin Blomer (Göttingen): The Manin-Peyre formula for a certain biprojective threefold
- 16:45 Ulrich Derenthal (Hannover): Manin's conjecture for a family of singular quartic del Pezzo surfaces

## 9th meeting (Universität Göttingen, 2 November 2016)

- 14:00 Valentin Blomer (Göttingen): Higher order divisor problems
- 15:00 Giuliano Gagliardi (Hannover): Manin's conjecture for some spherical threefolds

- 16:30 Harald Helfgott (Göttingen): The sieve of Eratosthenes in less space

## 8th meeting (Universität Göttingen, 6 June 2016)

- 14:00 Maryna Viazovska (HU Berlin):
*The sphere packing problem in dimensions 8 and 24*In this talk we will show that the sphere packing problem in dimensions 8 and 24 can be solved by a linear programing method. In 2003 N. Elkies and H. Cohn proved that the existence of a real function satisfying certain constrains leads to an upper bound for the sphere packing constant. Using this method they obtained almost sharp estimates in dimensions 8 and 24. We will show that functions providing exact bounds can be constructed explicitly as certain integral transforms of modular forms. Therefore, we solve the sphere packing problem in dimensions 8 and 24.

Abstract: - 15:00 Elisa Sedunova (Göttingen):
*On the Bombieri-Pila method over function fields* - 16:30 Emmanuel Kowalski (ETH Zürich):
*Mixing algebra and analysis over finite fields**Abstract:*Algebraic geometry is the source of a very rich class of "special functions" over finite fields, that contains and generalizes in some sense the classical additive and multiplicative characters. These functions can be used for a form of harmonic analysis where the algebraic and analytic aspects can be very closely intertwined. The talk will survey some of results and also problems and conjectures in this direction, for instance, to de-randomization problems. (Based on joint works with É. Fouvry, Ph. Michel and W. Sawin).

## 7th meeting (Universität Hannover, 8 March 2016)

Talks in A410

- 11:00 Christopher Frei (Graz):
*The Hasse norm principle for abelian extensions**Abstract:*Let L/K be a normal extension of number fields. The Hasse norm principle is a local-global principle for norms. It is satisfied if any element x of K is a norm from L whenever it is a norm locally at every place. For any fixed abelian Galois group G, we investigate the density of G-extensions violating the Hasse norm principle, when G-extensions are counted in order of their discriminant. This is joint work with Dan Loughran and Rachel Newton. *12:00 Lunch*- 14:30 Kathrin Bringmann (Köln):
*Fourier coefficients of meromorphic modular forms* *15:30 Coffee*- 16:00 Michael Stoll (Bayreuth):
*The Generalized Fermat Equation*x^{2}+ y^{3}= z^{11}*Abstract:*Generalizing Fermat's original problem, equations of the form x^{p}+ y^{q}= z^{r}, to be solved in coprime integers, have been quite intensively studied. It is conjectured that there are only finitely many solutions in total for all triples (p,q,r) such that 1/p + 1/q + 1/r < 1 (the `hyperbolic case'). The case (p,q) = (2,3) is of special interest, since several solutions are known. To solve it completely in the hyperbolic case, one can restrict to r = 8,9,10,15,25 or a prime ≥ 7. The cases r = 7,8,9,10,15 have been dealt with by various authors. In joint work with Nuno Freitas and Bartosz Naskrecki, we are now able to solve the case r = 11 and prove that the only solutions (up to signs) are (x,y,z) = (1,0,1), (0,1,1), (1,-1,0), (3,-2,1). We use Frey curves to reduce the problem to the determination of the sets of rational points satisfying certain conditions on certain twists of the modular curve X(11). A study of local properties of mod-11 Galois representations cuts down the number of twists to be considered. The main new ingredient is the use of the `Selmer group Chabauty' techniques developed recently by the speaker to finish the determination of the relevant rational points.

## 6th meeting (Universität Göttingen, 1 February 2016)

*13:30 Welcome and Coffee*- 15:00 Lasse Grimmelt (Göttingen):
*Representation of squares by cubic forms* *16:00 Coffee*- 16:30 Jörg Brüdern (Göttingen):
*Differenced cubes and diophantine approximation*

## 5th meeting (Universität Göttingen, 7 December 2015)

Talks in Sitzungszimmer, Coffee in Schlauch.

*13:00 Welcome and Coffee*- 13:30 Jim Parks (Hannover):
*An asymptotic result for the number of amicable pairs of elliptic curves on average* - 14:30 Berke Topacogullari (Göttingen):
*On a certain additive divisor problem* *15:30 Coffee*- 16:00 Jörg Brüdern (Göttingen):
*Zhao's trick*

## 4th meeting (Universität Hannover, 9 July 2015)

*10:30 in A451: coffee*- 11:00 in B305: Lilian Matthiesen (Hannover):
*Multiplicative functions and nilsequences*We discuss bounds on the correlation of multiplicative functions with polynomial nilsequences and their applications to evaluating linear correlations among themselves.

Abstract: *12:00: lunch**13:00 in A451: coffee*- 14:00 in B305: Jörg Brüdern (Göttingen):
*Sums of two like powers*

*Abstract:*We discuss recent joint work with Trevor Wooley concerning the distribution of sums of two like powers in short intervals. *15:00 in A451: coffee**15:45 in G116: more coffee*- 16:30 in F309: Yonatan Harpaz (École Normale Superieure):
*The Hasse principle for generalized Kummer varieties*The descent-fibration method of Swinnerton-Dyer can be used to study rational points on Kummer varieties by analyzing the variation of the 2-Selmer group of the associated abelian variety under quadratic twists. In a joint work with Alexei Skorobogatov we apply this method to the case where the Galois action on the 2-torsion has a large image. Under a mild additional assumption we prove that the Hasse principle holds for generalized Kummer varieties assuming the finiteness of all relevant Shafarevich-Tate groups. Our calculations are inspired by the work of Mazur and Rubin.

Abstract:

## 3rd meeting (Universität Göttingen, 24 April 2015)

- Julia Brandes (Göttingen):
*Simultaneous additive equations: repeated and differing degrees* - Dan Loughran (Hannover):
*The Hasse principle for lines on cubic surfaces* - Vita Kala (Göttingen):
*Number fields without n-ary universal quadratic forms*

## 2nd meeting (Universität Göttingen, 6 February 2015)

- Marta Pieropan (Hannover):
*Rational points as lattice points* - Berke Topacogullari (Göttingen):
*The shifted convolution of divisor functions* - Olivier Robert (St. Etienne):
*On the abc conjecture and nuclear numbers*

## 1st meeting (Universität Hannover, 21 November 2014)

- Péter Maga (Göttingen):
*Subconvexity for supnorms of automorphic forms on PGL(n)*As it was proved by Sarnak, the supnorm of eigenfunctions of the Laplacian on a compact symmetric Riemannian manifold can be estimated from above by an appropriate power (given in terms of some invariants of the space) of their Laplace eigenvalue. Examples show that Sarnak's exponent is sharp in some cases. However, when the space has also arithmetic symmetries (i.e. Hecke operators) and we restrict to joint eigenfunctions of the Laplacian and the Hecke operators, one might expect a better exponent. We prove that a better exponent exists for automorphic forms on PGL(n,R). Joint result with Valentin Blomer.

Abstract: - Christopher Frei (Hannover):
*Forms of differing degrees over number fields*Consider a system of m forms of degree d in n variables over the integers. A classical result by Birch uses the circle method to provide an asymptotic formula for the number of integer solutions to this system in a homogeneously expanding box, as long as n is large compared to m and d. An analogous result over arbitrary number fields was proved by Skinner. In joint work with M. Madritsch, we extend Skinner's techniques to a recent generalization of Birch's theorem by Browning and Heath-Brown, where they allow the forms to have differing degrees.

Abstract: - Oscar Marmon (Göttingen):
*Random Thue and Fermat equations*We consider Thue equations of the form ax

Abstract:^{k}+by^{k}= 1, and assuming the truth of the abc-conjecture, we show that almost all locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion holds true for Fermat equations ax^{k}+by^{k}+cz^{k}= 0 of degree at least six. This is joint work with Rainer Dietmann.