Seminare und Vorträge im WS 2022/2023

am Montag, 10. Oktober :

Oberseminar Zahlentheorie

Hasan Saad
Title: Explicit Sato-Tate type distribution for a family of \(K3\) surfaces
Abstract: In the 1960's, Birch proved that the traces of Frobenius for elliptic curves taken at random over a large finite field is modeled by the semicircular distribution (i.e. the usual Sato-Tate for non-CM elliptic curves). In analogy with Birch's result, a recent paper by Ono, the author, and Saikia proved that the limiting distribution of the normalized Frobenius traces \(A_{\lambda}(p)\) of a certain family of \( K3 \) surfaces \(X_\lambda\) with generic Picard rank \(19\) is the \(O(3)\) distribution. This distribution, which we denote by \(\frac{1}{4\pi}f(t)\), is quite different from the semicircular distribution. It is supported on \([-3,3]\) and has vertical asymptotes at \(t=\pm1\). Here we make this result explicit. We prove that if \(p\geq 5\) is prime and \(-3\leq a < b \leq 3 \), then \[ \left|\frac{\#\{\lambda\in\mathbb{F}_p :A_{\lambda}(p)\in[a,b]\}}{p}-\frac{1}{4\pi} \int_a^bf(t)dt\right|\leq \frac{110.84}{p^{1/4}}. \] As a consequence, we are able to determine when a finite field \(\mathbb{F}_p\) is large enough for the discrete histograms to reach any given height near \(t=\pm1\). To obtain these results, we make use of the theory of Rankin-Cohen brackets in the theory of harmonic Maass forms.

14:00 Online in ZOOM

am Montag, 31. Oktober :

Oberseminar Zahlentheorie

Valentin Blomer
Title: Analytic methods for simultaneous equidistribution
Abstract: Integral points on spheres of large radius D^(1/2) equidistribute (subject to appropriate congruence conditions), and so do Heegner points of large discriminant D on the modular curve. Both sets have roughly the same cardinality, and there is a natural way to associate with each point on the sphere a Heegner point. Do these pairs equidistribute in the product space of the sphere and the modular curve as D tends to infinity? Similarly, low-lying horocycles equidistribute on the modular curves as their length increases. If we run through a horocycle with two different speeds, do the pairs of points equidistribute in the product of two modular curves? I will explain how methods of analytic number theory can be used to shed light on this type of simultaneous equidistribution problem.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 07. November :

Oberseminar Zahlentheorie

Igor Shparlinski
Title: Bilinear forms with Kloosterman and Salie Sums and Moments of L-functions
Abstract: We present some recent results on bilinear forms with complete and incomplete Kloosterman and Salie sums. These results are of independent interest and also play a major role in bounding error terms in asymptotic formulas for moments of various L-functions. We then describe several results about non-correlation of Kloosterman and Salie sums between themselves and also with some classical number-theoretic functions such as the Mobius function, the divisor function and the sum of binary digits, etc. Some open problems will be outlined as well.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 21. November :

Oberseminar Zahlentheorie

Peter Stevenhagen
Title: Local imprimitivity of points on elliptic curves
Abstract: An element in a number field K that is primitive, i.e., not a k-th power for k>1, is a primitive root modulo infinitely many primes of K, at least under GRH. Primitive points on elliptic curves E/K may fail to have this reasonable property even in cases where E has inifinitely many primes of cyclic reduction. We discuss the anomalous behaviour of the Galois representations underlying the failure of this local-global principle.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 28. November :

Oberseminar Zahlentheorie

Nicolas Smoot
Title: Congruence Kernels and the Localization Method
Abstract: Since Ramanujan's groundbreaking work, the divisibility of partition numbers has become an enormously prolific subject. Such a simple question as when p(n) or its various extensions are divisible by, say, large powers of 5 can be an extremely deep problem; there can be a lot of variation with respect to the difficulty of proving such congruence families, and the methods developed to prove them often invoke very surprising relationships in analysis, algebra, topology, and algebraic geometry. We will discuss the application of the localization method to proving congruence families by sketching the proof of one recently discovered congruence family for a certain extension of p(n). In particular, we will discuss a critical aspect of such proofs, in which the associated generating functions of a given congruence family are closely related to the kernel of a certain linear mapping to a vector space over a finite field. We will then explain the importance of this approach to classifying congruence families.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 05. Dezember :

Oberseminar Zahlentheorie

Campbell Wheeler
Title: Difference equations and modularity
Abstract: Difference equations are natural analogues of differential equations. Recently, Borel resummation was described in this setting. This can be used to construct a meromorphic basis of solutions to a difference equation. I will discuss a class of q-difference equations we call modular difference equations that give rise to quantum modular forms and how this can lead to complete descriptions of Stokes phenomenon.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 12. Dezember :

Oberseminar Zahlentheorie

Herbert Gangl
Title: Zagier's Polylogarithm Conjecture revisited
Abstract: Instigated by work of Borel and Bloch, Zagier formulated his Polylogarithm Conjecture in the late eighties and proved it for weight 2. After a flurry of activity and advances at the time, notably by Goncharov who provided not only a proof for weight 3 but set out a vast program with a plethora of conjectural statements for attacking it, progress seemed to be stalled for a number of years. More recently, a solution to one of Goncharov's central conjectures in weight 4 has been given. Moreover, by adopting a new point of view, Goncharov and Rudenko provided a proof of the original conjecture in weight 4. In this impressionist talk I intend to give a rough idea of the developments from the early days on, avoiding most of the technical bits, and also hint at a number of recent results for higher weight (joint with S.Charlton and D.Radchenko).

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 19. Dezember :

Oberseminar Zahlentheorie

Axel Kleinschmidt
Title: Non-holomorphic modular functions from string theory at genus one
Abstract: By briefly reviewing some basic ingredients from string theory scattering amplitudes, I will motivate certain types of non-holomorphic functions on the upper half plane that have good transformation properties under the modular group SL(2,Z). These functions satisfy interesting systems of differential and algebraic relations, are closely related to F. Brown's iterated Eisenstein integrals and contain interesting (single-valued) periods. The functions will be presented in different equivalent versions. Mainly based on work with Daniele Dorigoni and Oliver Schlotterer.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 09. Januar :

Oberseminar Zahlentheorie

Florian Munkelt
Title: On the number of rational points close to a compact manifold
Abstract: The fundamental question of number theory to count the rational points on an algebraic variety can be approached from the perspective of studying rational points close to manifolds. Groundbreaking work for the case of hypersurfaces has been done by Huang and was generalized by Schindler and Yamagishi for manifolds in arbitrary codimension. The validity of these results rely on certain curvature conditions imposed on the manifold. We establish an asymptotic formula for the number of rational points within a given distance to a manifold and with bounded denominators under a relaxed curvature condition, generalizing previous results. We are also able to recover a slightly weaker analogue of Serre's dimension growth conjecture for compact submanifolds.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 23. Januar :

Oberseminar Zahlentheorie

Peter Paule
Title: Holonomic Functions and Modular Forms: A Computer Algebra Bridge
Abstract: Holonomic functions and sequences satisfy linear differential and difference equations, respectively, with polynomial coefficients. It has been estimated that holonomic functions cover about 60 percent of the functions contained in the 1964 "Handbook" by Abramowitz and Stegun. A recent estimate says that holonomic sequences constitute about 20 percent of Sloane's OEIS database. The study of these ubiquitous objects traces back to the time of Gauss (at least).

Also tracing back to the time of Gauss (at least) are highly non-holonomic objects: modular functions and modular forms with q-series representations arising, for instance, as generating functions of partitions of various kinds.

Using computer algebra, the talk connects these two different worlds. Applications concern partition congruences, Fricke–Klein relations, irrationality proofs a la Beukers, or approximations to pi studied by Ramanujan and the Borweins. As a major ingredient to a "first guess, then prove" strategy, a new algorithm for proving differential equations for modular forms is used. The results presented arose in joint work with Silviu Radu (RISC).

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 30. Januar :

Oberseminar Zahlentheorie

Markus Neuhauser
Title: Bounds on the zeros of recursively defined polynomials
Abstract: The talk presents some results motivated by Lehmer's conjecture that Ramanujan's tau function constituted by the Fourier coefficients of the 24th power of Dedekind's eta function never vanishes. Generally, when Dedekind's eta function is raised to a power with exponent x it turns out that the Fourier coefficients are polynomials in x. They satisfy a recurrence relation. Even in a more general form we can provide a bound on x outside of which these never vanish. In some cases, including Dedekind's eta function, this seems to be the best possible. In its general form it provides relations for example to orthogonal polynomials and Eisenstein series. The talk is based on joint work mainly with B. Heim and with R. Tröger and A. Weisse.

14:00 Seminarraum 3 des Mathematischen Instituts

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