Wintersemester 2012/2013

16.10.2012 Bill Duke (UCLA)
Mock-modular forms of weight one and Galois representations
anschl. Vorbesprechung
23.10.2012  Giovanni De Gaetano (HU Berlin)
A regularized determinant for the hyperbolic Laplacian on modular forms
Abstract: With the goal to generalize the work of T. Hahn, who was able to prove an arithmetic Riemann-Roch type formula for the Hodge bundle on a modular curve with log-singular metric, we define and compute a suitable metric for the determinant bundle of a power of the Hodge bundle. It will be a non-smooth analogue of the Quillen metric used by Gillet and Soulé to prove their arithmetic Riemann-Roch theorem.
30.10.2012 Niels Lindner (HU Berlin)
Cuspidal plane curves and their Alexander polynomials
Abstract: The Alexander polynomial is a useful invariant of the fundamental group of a curve in the complex projective plane. It is strongly connected with the singularities of the curve. If one restricts to curves whose singular points are either ordinary double points or ordinary cusps, one can use calculations on the Mordell-Weil rank of certain associated elliptic three-folds to obtain a nice description of the Alexander polynomial in terms of syzygies of the ideal of cusps.
06.11.2012 kein Seminar
13.11.2012 Alina C. Cojocaru (Univ. of Illinois at Chicago, z.Zt. Univ. Göttingen)
Frobenius fields for elliptic curves
Abstract: Let E be an elliptic curve defined over Q. For a prime p of good reduction for E, let πp be the p-Weil root of E and Q(πp) the associated imaginary quadratic field generated by πp. In 1976, Serge Lang and Hale Trotter formulated a conjectural asymptotic formula for the number of primes p < x for which Q(πp) is isomorphic to a fixed imaginary quadratic field. I will discuss progress on this conjecture, in particular an average result confirming the predicted asymptotic formula. The latter is joint work with Henryk Iwaniec and Nathan Jones.
20.11.2012 Martin Raum (ETH Zürich)
Computational methods, Jacobi forms, and linear equivalence of special divisors
Abstract: We will start by briefly discussing the influence of computations on mathematics. Some examples will make clear that computations have led to surprising insight into the area of modular forms, and continue doing so. After this general considerations, we will turn our attention to Jacobi forms. Their definition and their connection with ordinary modular forms will be explained in detail. A new algorithm allows us to compute Fourier expansions of Jacobi forms. We will translate this into information about linear equivalences of special divisors on modular varieties of orthogonal type.
27.11.2012 Jan Steffen Müller (Univ. Hamburg)
A p-adic BSD conjecture for modular abelian varieties
Abstract: In 1986 Mazur, Tate and Teitelbaum came up with a p-adic analogue of the conjecture of Birch and Swinnerton-Dyer for elliptic curves over the rationals. In this talk I will report on joint work with Jennifer Balakrishnan and William Stein on a generalization of this conjecture to the case of modular abelian varieties and primes p of good ordinary reduction. I will discuss the theoretical background that led us to the formulation of the conjecture, as well as numerical evidence supporting it in the case of modular abelian surfaces and the algorithms that we used to gather this evidence.
04.12.2012 Miguel Daygoro Grados (Univ. Peruana de Ciencias Aplicadas)
L-series of Elliptic Curves with Complex Multiplication
Abstract: L-series of elliptic curves are complex functions that carry arithmetical information of the curve. From the analytic point of view, it is desirable that those functions possess additional properties such as Euler product expression, analytic continuation to the entire complex plane and a functional equation. Remarkably, such properties hold for the L-series associated to elliptic curves with complex multiplication. In this talk, we will elaborate on the role played by complex multiplication in the context of L-series. Additionally, since the above properties remind us of the Prime Number Theorem in arithmetic progressions, we will discuss how the analytic methods used in Dirichlet's proof can be adjusted to yield a similar theorem for the L-series of elliptic curves.
11.12.2012 Tony Varilly-Alvarado (Rice University z.Zt. EPFL Lausanne)
Failure of the Hasse principle on general K3 surfaces
Abstract: Transcendental elements of the Brauer group of an algebraic variety, i.e., Brauer classes that remain nontrivial after extending the ground field to an algebraic closure, are quite mysterious from an arithmetic point of view. These classes do not arise for curves or surfaces of negative Kodaira dimension. In 1996, Harari constructed the a 3-fold with a transcendental Brauer-Manin obstruction to the Hasse principle. Until recently, his example was the only one of its kind. We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general K3 surface X of degree 2 over Q, together with a two-torsion Brauer class α that is unramified at every finite prime, but ramifies at real points of X. Motivated by Hodge theory, the pair (X,α) is constructed from a double cover of P2 × P2 ramified over a hypersurface of bi-degree (2, 2). This is joint work with Brendan Hassett.
18.12.2012 Özlem Imamoglu (ETH Zürich, z.Zt. HU Berlin)
On weakly harmonic Maass forms and their Fourier coefficients
Abstract: Harmonic Maass forms have been studied extensively in last several years due to their connection to several arithmetic problems. Most notably in the half integral weight case, it can be shown that the Ramanujan's mock theta functions and generating functions of traces of singular moduli can be understood in terms of harmonic Maass forms. A simpler example is provided by the non-holomorphic Eisenstein series of weight 2. In this talk, after giving a short overview of the subject, I will show how to construct a distinguished basis of such forms in the case of weight 2 and study their relation of the regularized inner products of modular functions.
08.01.2013 Damian Rössler (Toulouse)
A direct proof of the equivariant Gauss-Bonnet formula on abelian schemes
Abstract: We shall present a proof of the relative equivariant Gauss-Bonnet formula for abelian schemes, which does not rely on the Grothendieck-Riemann-Roch theorem. This proof can be carried through in the arithmetic setting (i.e. in Arakelov theory) and leads to interesting analytic questions.
15.01.2013 Jürg Kramer (HU Berlin)
Towards arithmetic intersections on mixed Shimura varieties
22.01.2013 Maryna Viazovska (Univ. Köln)
CM Values of Higher Green's Functions
Abstract: Higher Green functions are real-valued functions of two variables on the upper half plane which are bi-invariant under the action of a congruence subgroup, have logarithmic singularity along the diagonal, and satisfy the equation Δ f = k(1 - k)f, where Δ is the hyperbolic Laplace operator and k is a positive integer. The significant arithmetic properties of these functions were disclosed in the paper of B. Gross and D. Zagier "Heegner points and derivatives of L-series" (1986). In particular, it was conjectured that higher Green's functions have "algebraic" values at CM points. In this talk we will present a proof of the conjecture for any pair of CM points lying in the same quadratic imaginary field. Moreover, we give an explicit factorization formula for the algebraic number obtained (up to powers of ramified primes).
29.01.2013 kein Seminar
05.02.2013 Anil Aryasomayajula (HU Berlin)
Towards an extension of a key identity to Hilbert modular surfaces
Abstract: In 2006, J. Jorgenson and J. Kramer came up with a beautiful identity relating the canonical and hyperbolic volume forms on a Riemann surface. In this talk, we report the progress of the on-going work in collaboration with Nahid Walji, towards an extension of this formula to Hilbert modular surfaces.
12.02.2013 Vladimir Dokchitser (Cambridge)
Parity of ranks of elliptic curves
Abstract: It is in general very difficult to compute ranks of elliptic curves over number fields, even if equipped with any conjectures that are available. On the other hand, the parity of the rank is (conjecturally) very easy to determine -- it is given as a sum of purely local terms, which have a reasonably simple classification. Since "odd rank" implies "non-zero rank" implies "the curve has infinitely many points", this leads to a number of (conjectural!) arithmetic phenomena. The second part of the talk will concern the "parity conjecture" - that the parity of the rank that is predicted by the Birch-Swinnerton-Dyer conjecture agrees with the prediction of the Shafarevich-Tate conjecture.
04.03.2013 Aprameyo Pal (Univ. Heidelberg)
Functional equation of characteristic elements of Abelian varieties over function fields
Abstract: In this talk we apply methods from the Number field case of Perrin-Riou & Zabradi in the Function field set up. In Zl- and GL2-case (l≠ p), we prove algebraic functional equations of the Pontryagin dual of Selmer group which give further evidence of the Main conjectures of Iwasawa Theory. We also prove some parity conjectures in commutative and non-commutative cases. As consequence, we also get results on the growth behaviour of Selmer groups in extension of Function fields