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Analytic Number Theory Graduate Student Seminar April 25, 2017 (04:00 PM PDT - 05:00 PM PDT)
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Location: MSRI: Baker Board Room
Speaker(s) Corina Panda (California Institute of Technology), Vinay Viswanathan (University of Bristol)
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4:00pm: Vinay Kumaraswamy

Title: On correlations between class numbers of imaginary quadratic fields

Abstract: Let h(-d) denote the class number of the imaginary quadratic field Q(\sqrt{-d}).  Moments of class numbers​ have been studied in the past, and are well understood. In this talk, I will speak about obtaining an asymptotic formula for the shifted sum \sum_{d \leq X} h(-d)h(-d-l), where l is a positive integer; the proof makes use of the smooth delta-symbol.



4:30pm: Corina Panda



A generalization of a theorem of Hecke for SL_2(F_p) to fundamental discriminants



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Let p > 3 be an odd prime, p ≡ 3 mod 4 and let π+, π− be the pair of cuspidal representations of SL2(Fp). It is well known by Hecke that the difference mπ+ − mπ− in the multiplicities of these two irreducible representations occurring in the space of weight 2 cusps forms with respect to the principal congruence subgroup Γ(p), equals the class number h(−p) of the imaginary quadratic field Q(\sqrt(−p)). We extend this result to all fundamental discriminants −D of imaginary quadratic fields Q( \sqrt(−D)). The proof uses the holomorphic Lefschetz number.


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