Lecture -01: Introduction to modular forms and modular curves
Speaker: Anna-Maria Pippich (University of Konstanz, Germany)
Date & Time: 13th March 2024 from 10:30 AM to 11:30 AM (IST)
Abstract: In the first lecture, we will give an introduction to the classical theory of modular forms for the full modular group motivated by the consideration of generating functions. In particular, we will discuss Eisenstein series and theta series together with their arithmetic significance. Moreover, we will present the construction of the fundamental domain for the modular group providing a realization of the modular curve as a compact Riemann surface. We will end with the presentation of Kronecker’s limit formula.
Lecture -02: Introduction to Arakelov geometry
Speaker: Jurg Kramer (Humboldt University, Berlin)
Date & Time: 13th March 2024 from 12:00 PM to 01:00 PM (IST)
Abstract: In the second lecture, we will give an introduction to the arithmetic intersection theory on arithmetic surfaces motivated by the notion of height which measures the arithmetic complexity of rational points on smooth projective curves defined over number fields. We will see that Green’s functions on the associated compact Riemann surfaces will play a crucial role. We end our lecture by stating the arithmetic Riemann–Roch theorem in the setting under consideration.
Lecture -03: Sup-norm bounds of automorphic forms
Speaker: Anilatmaja Aryasomayajula (IISER Tirupati)
Date & Time: 13th March 2024 from 02:30 PM to 03:30 PM (IST)
Abstract: In this lecture, we discuss sup-norm bounds for automorphic forms, which arise naturally in the analytic theory of Arakelov theory. We will give a survey on sup-norm bounds for classical cusp forms, for Hilbert and Picard modular cusp forms, as well as for Siegel modular cusp forms. We describe a method emanating from complex geometry to derive sup-norm bounds for automorphic forms. We end our lecture with a brief discussion on the amplification technique of Sarnak and Iwaniec by which one can derive sub-convexity estimates for Hecke eigenforms.
Lecture -04: Arakelov geometry for log-singular metrics
Speaker: Jurg Kramer (Humboldt University, Berlin)
Date & Time: 13th March 2024 from 04:00 PM to 05:00 PM (IST)
Abstract: In this lecture, we will present a crucial modification of classical arithmetic intersection theory on arithmetic surfaces that allows to incorporate line bundles equipped with logarithmically singular metrics into the theory. Only by means of this modification we will be able to study arithmetic intersection theory for the line bundle of modular forms equipped the Petersson metric, which becomes logarithmically singular at the cups of the modular curves under consideration.
Lecture -05: Modular curves and regularized determinants
Speaker: Anna-Maria Pippich (University of Konstanz, Germany)
Date & Time: 15th March 2024 from 02:30 PM to 03:30 PM (IST)
Abstract: On the basis of the preceding lectures, we will be able to discuss the necessary modifications to be done to formulate an arithmetic Riemann–Roch theorem for the line bundle of modular forms on modular curves equipped with the logarithmically singular Petersson metric. By lecture 4, the right-hand side is well-defined. Therefore, the main emphasis of this lecture is the necessary adaption of the left-hand side which amounts to a suitable regularization of the determinant of the hyperbolic Laplacian using the Selberg zeta function.
Lecture -06: Arakelov geometry in higher dimensions
Speaker: Jurg Kramer (Humboldt University, Berlin)
Date & Time: 15th March 2024 from 04:00 PM to 05:00 PM (IST)
Abstract: In this final lecture, we will present the generalization of arithmetic intersection theory to higher dimensional arithmetic varieties, which will allow us to generalize the notion of height from points to higher dimensional cycles. Moreover, we will discuss also the modifications to be carried out to take into account line bundles equipped with hermitian metrics that become logarithmically singular along a divisor with normal crossings. If time permits, we will give an outlook to the most recent generalizations of arithmetic intersection theory on mixed Shimura varieties.
Zoom link: https://icts-res-in.zoom.us/j/91209079566?pwd=UlVOOFJ6djdZSGdTbDFwVi9JS2Vndz09
Meeting ID: 912 0907 9566
Passcode: 131315