Göttingen Junior Number Theory Seminar

Summer 2014

30.04.2014 - Christopher Ambrose: Über Artins Primitvwurzelvermutung

Abstract: Artin's primitive root conjecture states for any integer a, neither $0$, $±1$ nor a perfect square, the existence of infinitely many primes p for which a generates a subgroup of index $1$ in $({ℤ}{/}{p ℤ})^{*}$ . This motivates the question how order and index of integers are distributed in $({ℤ}{/}{pℤ})^{*}$ as $p$ varies. In this talk we focus on the average behaviour of index and order in a larger class of families of finite abelian groups. More precisely we consider suitable families of multiplicative groups of residual rings of algebraic integers on the one hand, and of $𝔽_p$ -rational points of elliptic curves over $𝔽_p$ on the other hand. We prove, partly conditionally on fairly standard hypotheses, asymptotic formulae in some cases, and mention general obstacles which prevent such formulae in the remaining cases.

07.05.2014 - Oscar Marmon: The dimension growth conjecture and thin sets

Abstract: The density of solutions to a given Diophantine equation is intimately connected to the geometry of the corresponding algebraic variety. In many cases, however, it is desirable to have bounds that require as little geometric information as possible. We will discuss an influential conjecture by Heath-Brown in this direction. In particular, I will try to present Serre's proof of one of the first results in this spirit, due to Cohen. The proof uses the large sieve, and will lead us to study so called thin subsets of $ℤ^n$.

14.05.2014 - Lasse Grimmelt: Die Selberg-Delange-Method

Abstract: Die Methode beantwortet vollständig die Frage nach einer Asymptotik für $∑ ↙{n≤x}a_n$, wenn die zugehörige Dirichlet-Reihe eine Singularität der Form $(s-1)z$ mit komplexen $z$ hat. Es wird im Vortrag nicht auf alle technischen Details eingegangen werden (siehe dafür Tenenbaum S. 180 ff.), sondern die Struktur dargelegt und insbesondere die Benutzung von Hankel's formular zum Auswerten der dabei auftretenden Integrale erklärt werden. (Dadurch lässt sich bspw. verstehen, dass das bei Landaus zwei quadrate Satz auftauchende $√π$ mit der Gamma funktion an $1/2$ zu tun hat. S. 51 Brüdern.) Wenn es die Zeit zulässt werde ich noch kurz den Zusammenhang zu meinem letzten Vortrag darstellen und zeigen, dass die Methode auch geeignet ist Singularitäten der Form $(s - 1)z \log(s - 1)^j$ mit ganzzahligem $j$, zu behandeln, sofern man immernoch eine analytische Fortsetzung in ein leicht größeres Gebiet hat. Es gibt dazu auch ein Paper von Delange: "Generalisation du Theoreme de Ikehara" (s. Tenenbaum S. 243 f.) der nur die holomorphie entlang der Achse mit Realteil 1 benötigt, dessen Methoden aber auch sehr viel umständlicher sind.

21.05.2014 - Fabian Dehnert: Hyperbolas, Boxes and Manin's conjecture

Abstract: We give a general overview over Manin's conjecture concerning the distribution of rational points on certain projective varieties and thereby focus on the special cases where the circle method may be applied. As mathematical part we prove a 'baby' verision of a hyperbola type method developed by Brüdern and Blomer to approach problems in biprojective settings reducing a hyperbola summation condition to an ordinary box count.

28.05.2014 - Lutz Helfrich: The circle method over number fields

Abstract: The circle method is usually used to count the number of integer solutions, but the method is versatile enough to be used also in other situations, e.g. for counting solutions over an algebraic number field. In this talk I'll give an brief introduction into the number field setting and the differences to the "normal" setting.

04.06.2014 - Julia Brandes: An introduction to Vinogradov's Mean Value Theorem

Abstract: The strongest bounds for $G(k)$ and $Ğ (k)$ in Waring's problem are currently attained by studying a certain mean value of exponential sums. I will show how powerful this method is in giving bounds for Waring's problem and give an outline of the classical approach to the proof. Finally, I will sketch the main ideas of the recent breakthrough by Trevor Wooley.

11.06.2014 - Alexander Adam: On the Hurwitz Zeta function and its connection to the DFT

Abstract: I will talk about a special property of the Hurwitz Zeta function on rational values for the second argument. With its help we can understand for e.g. the functional equation of Dirichlet $L$-series from a different view. I also want to show a multidimensional functional equation involving the Hurwitz Zeta function on rational arguments.

18.06.2014 - Julia Brandes: Vinogradov's Mean Value Theorem II: Efficient congruencing

Abstract: For a number of applications in analytic number theory it is useful to have strong bounds on the number of solutions to the system of equations $$x_1^j+…+x_s^j=y_1^j+…+y_s^j,1≤j≤k.$$ I will give an outline of a recent breakthrough of Trevor Wooley in estimating the number of solutions to such systems, giving bounds that are not far from optimal. This is a follow-up to my talk of two weeks ago, but nonetheless I will try to make it reasonably self-contained.

25.06.2014 - Deniz Balakci: Introduction to automorphic forms on $GL_3$

Abstract: N/A

02.07.2014 - Jan Malec: Squareful Numbers In Arithmetic Progressions

Abstract: I will present the results of my "Diplomarbeit". I calculated the number of squareful numbers in an arithmetic progression by using methods from analytic number theory (Dirichlet Series, Perron's formula, character relations).

9.07.2014 - Deniz Balakci: Automorphic forms on $GL(3)$ II: $L$-functions, Hecke operators and Eisenstein series

Abstract: In the second part of my talk we will -analog to the $GL(2)$ case- use the Fourier coefficients to introduce the $L$-function of a Maass cuspform. Further we will introduce Hecke operators and then we establish the analog stuff of Hecke eigenfunctions, Euler products of the $L$-function, functional equation and so on, like in the $GL(2)$ case. But it will turn out that there are differences to the classical $GL(2)$ theory, for example the Hacke operators are only normal not selfadjoint. Finally if time permits we will discuss Eisenstein series and the spectral problem, hence the decomposition of $L(SL_{3}(ℤ) \\ h^{3})$ into invariant subspaces.

16.07.2014 - Lasse Grimmelt: Introduction to the circle method

Abstract: I will give a short introduction to the basic ideas of the Hardy-Littlewood circle-method. The method is made to deal with so called additive problems. I will explain the underlying principle of using the orthogonality relation of the e function to translate a additive counting problem into an integral. Then I will explain what major and minor arcs are and how this decomposition helps to evaluate the integral. The result will be that there are terms called singular series and singular integral in the appearing asymptotics, I will try to give a basic interpretation of these terms and their relation to the Hasse principle. If there is time I will shortly mention other applications of the method like ternary Goldbach. I will follow the notes of the ANT II course by Brüdern and Davenports "Analytical methods for diaophantine equations and diophantine inequalities" and will focus on the application of the circle method on warring's problem. As 60 minutes are far too less to go into any details there will probably be no full proofs.

23.07.2014 - Maximilian Schmidt: A Short Introduction to Quaternion Algebras

Abstract: As the title says, I'm going to give a short (as in: not very advanced, mostly for time-related reasons) introduction to quaternion algebras, including those over fields of characteristic 2.

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