BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Abhishek Saha (Queen Mary University of London) DTSTART:20230523T130000Z DTEND:20230523T140000Z DTSTAMP:20260421T153833Z UID:UEAPS/8 DESCRIPTION:Title: Th e Manin constant\, the modular degree\, and Fourier expansions at cusps\nby Abhishek Saha (Queen Mary University of London) as part of ANTLR sem inar\n\nLecture held in SCI 3.05.\n\nAbstract\nLet f be a normalized newfo rm of weight k for $\\Gamma_0(N)$. It is a natural question to try to unde rstand the size (in a $p$-adic sense) of the "denominators" of the Fourier expansions of f at a cusp of $X_0(N)$. The problem is easy if N is square -free but is delicate when N is highly square-full. I will talk about rece nt joint work with Kȩstutis Česnavičius and Michael Neururer where we s olve this problem using representation-theoretic techniques. Roughly speak ing\, we reduce the problem to bounding $p-$adic valuations of local Whitt aker newforms and then use a "basic identity" (a consequence of the Jacque t-Langlands local functional equation) to reduce to $p$-adic properties of local epsilon factors of representations of $\\GL_2(\\Q_p)$. \n\nA key a pplication of our result is to understand the Manin constant c of an ellip tic curve E over the rationals. The integer c scales the differential dete rmined by the normalized newform f associated to E into the pullback of a N\\'{e}ron differential under a minimal modular parametrization. Manin con jectured that c equals 1 or -1 for optimal parametrizations. We prove that c divides the degree of the parametrization under a minor assumption at t he primes 2 and 3. For this result\, we establish a certain integrality pr operty of $\\omega_f$ that follows from the Manin conjecture. We expect th at this integrability property we prove here will be necessary for any fur ther progress towards Manin's conjecture.\n LOCATION:https://researchseminars.org/talk/UEAPS/8/ END:VEVENT END:VCALENDAR