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SUMMARY:Peng Gao (BUAA)
DTSTART;VALUE=DATE-TIME:20201124T080000Z
DTEND;VALUE=DATE-TIME:20201124T090000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232347Z
UID:numsjtu/1
DESCRIPTION:Title:
The fourth moment of quadratic Hecke $L$-functions in $\\mathbb{Q}(i)$
\nby Peng Gao (BUAA) as part of SJTU number theory seminar\n\n\nAbstract\n
In this talk\, we study the fourth moment of central values of quadratic H
ecke $L$-functions in the Gaussian field. We show an asymptotic formula va
lid under the generalized Riemann hypothesis (GRH). We also present precis
e lower bounds unconditionally and upper bounds under GRH for higher momen
ts of the same family.\n
LOCATION:https://researchseminars.org/talk/numsjtu/1/
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BEGIN:VEVENT
SUMMARY:Chang Heon Kim (SKKU)
DTSTART;VALUE=DATE-TIME:20210128T070000Z
DTEND;VALUE=DATE-TIME:20210128T080000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232347Z
UID:numsjtu/2
DESCRIPTION:Title:
Hecke system of harmonic Maass functions and applications to modular curve
s of higher genera\nby Chang Heon Kim (SKKU) as part of SJTU number th
eory seminar\n\n\nAbstract\nThe unique basis functions $j_m$ of the form $
q^{-m}+O(q)$ for the space of weakly holomorphic modular functions on the
full modular group form a Hecke system. This feature was a critical ingred
ient in proofs of arithmetic properties of Fourier coefficients of modular
functions and denominator formula for the Monster Lie algebra.\n\nIn this
talk\, we consider the basis functions of the space of harmonic weak Maas
s functions of an arbitrary level\, which generalize $j_m$\, and show that
they form a Hecke system as well. As applications\, we\nestablish some di
visibility properties of Fourier coefficients of weakly holomorphic modula
r forms on modular curves of genus $\\ge1$. Furthermore\, we present a gen
eral duality relation that these modular forms\nsatisfy.\nThis is a joint
work with Daeyeol Jeon and Soon-Yi Kang.\n\nZoom ID: 955 492 12478\, passw
ord: 120205\n
LOCATION:https://researchseminars.org/talk/numsjtu/2/
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BEGIN:VEVENT
SUMMARY:Detchat Samart (Burapha University)
DTSTART;VALUE=DATE-TIME:20210407T070000Z
DTEND;VALUE=DATE-TIME:20210407T080000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232347Z
UID:numsjtu/3
DESCRIPTION:Title:
Mahler measures of algebraic varieties: Results and experiments\nby De
tchat Samart (Burapha University) as part of SJTU number theory seminar\n\
nLecture held in zoom: 658 593 56935.\n\nAbstract\nThe (logarithmic) Mahle
r measures of an $n$-variable polynomial $P$ is defined as the arithmetic
mean of $\\log |P|$ over the $n$-torus. Despite its purely analytic formul
ation\, Mahler measure is known to have a deep connection with the arithme
tic of the corresponding algebraic variety via $L$-functions\, thanks to w
ork of Deninger\, Boyd\, Bertin\, and many others. There are several known
results and conjectures in the literature\nrelating Mahler measures to sp
ecial $L$-values of Dirichlet series\, elliptic curves\, $K3$ surfaces\, a
nd modular forms. In this talk\, we will give a survey of recent results i
n this research direction from both\ntheoretical and experimental perspect
ives.\n\nzoom: 658 593 56935\, password: the first 6 digits of $\\zeta(3)$
\n
LOCATION:https://researchseminars.org/talk/numsjtu/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:YiFan Yang (National Taiwan University)
DTSTART;VALUE=DATE-TIME:20210106T070000Z
DTEND;VALUE=DATE-TIME:20210106T080000Z
DTSTAMP;VALUE=DATE-TIME:20230925T232347Z
UID:numsjtu/4
DESCRIPTION:Title:
Differential equations satisfied by modular forms\nby YiFan Yang (Nati
onal Taiwan University) as part of SJTU number theory seminar\n\n\nAbstrac
t\nA classical result known since the nineteenth century asserts that if $
F(z)$ is a modular form of weight $k$ and $t(z)$ is a nonconstant modular
function on a Fuchsian subgroup of $SL(2\,\\mathbb{R})$ of the first kind\
, then $F(z)\, zF(z)\,... z^kF(z)$\, as (multi-valued) functions of $t$\,
are solutions of a $k+1$-st order linear ordinary differential equations w
ith algebraic functions of t as coefficients. This result constitutes one
of the main sources of applications of modular forms to other branches of
mathematics. In this talk\, we will give a quick overview of this classica
l result and explain some of its applications in number theory.\n
LOCATION:https://researchseminars.org/talk/numsjtu/4/
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