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BEGIN:VEVENT
SUMMARY:Jackie Lang (Oxford)
DTSTART;VALUE=DATE-TIME:20201105T140000Z
DTEND;VALUE=DATE-TIME:20201105T150000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/1
DESCRIPTION:Title: E
isenstein congruences at prime-square level\nby Jackie Lang (Oxford) a
s part of Dublin Algebra and Number Theory Seminar\n\n\nAbstract\nCongruen
ces between modular forms have been studied by many mathematicians\, start
ing with some observations of Ramanujan. They have been exploited by numb
er theorists in the last 50 years to prove many deep arithmetic facts. We
will give a survey of examples of these congruences and some of their ari
thmetic applications. Having established the historical context\, we will
discuss some work in progress with Preston Wake where we study Eisenstein
congruences at prime-square level. We will end with an application to pr
oving nontriviality of class groups of a family of number fields.\n\nPassc
ode: The 3-digit prime numerator of Riemann zeta at -11\n
LOCATION:https://researchseminars.org/talk/UCDANT/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Wuthrich (Nottingham)
DTSTART;VALUE=DATE-TIME:20201112T140000Z
DTEND;VALUE=DATE-TIME:20201112T150000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/2
DESCRIPTION:Title: I
ntegrality of twisted L-values of elliptic curves\nby Chris Wuthrich (
Nottingham) as part of Dublin Algebra and Number Theory Seminar\n\n\nAbstr
act\nIn the context of the generalised Birch and Swinnerton-Dyer conjectur
e\, one considers the value at $s=1$ of the L-function of an elliptic curv
e $E/\\mathbb{Q}$ twisted by a Dirichlet character $\\chi$. When normalise
d with a period\, one obtains an algebraic number $\\mathscr{L}(E\,\\chi)$
. In joint work with Hanneke Wiersema\, we determine under what conditions
$\\mathscr{L}(E\,\\chi)$ is an algebraic integer.\n\nPasscode: The 3-digi
t prime numerator of Riemann zeta at -11\n
LOCATION:https://researchseminars.org/talk/UCDANT/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christopher Williams (Warwick)
DTSTART;VALUE=DATE-TIME:20201119T140000Z
DTEND;VALUE=DATE-TIME:20201119T150000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/3
DESCRIPTION:Title: p
-adic L-functions in higher dimensions\nby Christopher Williams (Warwi
ck) as part of Dublin Algebra and Number Theory Seminar\n\n\nAbstract\nThe
re are lots of theorems and conjectures relating special values of complex
analytic L-functions to arithmetic data\; for example\, celebrated exampl
es include the class number formula and the BSD conjecture. These conjectu
res predict a surprising (complex) bridge between the fields of analysis a
nd arithmetic. However\, these conjectures are extremely difficult to prov
e. Most recent progress has come from instead trying to build analogous $p
$-adic bridges\, constructing a $p$-adic version of the $L$-function and r
elating it to $p$-adic arithmetic data via "Iwasawa main conjectures". Fro
m the $p$-adic bridge\, one can deduce special cases of the complex bridge
\; this strategy has\, for example\, led to the current state-of-the-art r
esults towards the BSD conjecture.\n\nEssential in this strategy is the co
nstruction of a $p$-adic L-function. In this talk I will give an introduct
ion to $p$-adic L-functions\, focusing first on the $p$-adic analogue of t
he Riemann zeta function (the case of ${\\rm GL}_1$)\, then describing wha
t one expects in a more general setting. At the end of the talk I will sta
te some recent results from joint work with Daniel Barrera and Mladen Dimi
trov on the construction of $p$-adic L-functions for certain automorphic r
epresentations of ${\\rm GL}_{2n}$.\n\nPasscode: The 3-digit prime numerat
or of Riemann zeta at -11\n
LOCATION:https://researchseminars.org/talk/UCDANT/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Newton (King's College London)
DTSTART;VALUE=DATE-TIME:20201126T140000Z
DTEND;VALUE=DATE-TIME:20201126T150000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/4
DESCRIPTION:Title: S
ymmetric power functoriality for modular forms\nby James Newton (King'
s College London) as part of Dublin Algebra and Number Theory Seminar\n\n\
nAbstract\nOne prediction of the Langlands program is that all 'reasonable
' L-functions should arise from automorphic forms. For example\, the modul
arity theorem of Wiles\, Breuil\, Conrad\, Diamond and Taylor identifies t
he Hasse-Weil L-function of an elliptic curve defined over the rationals w
ith the L-function of a modular form. More generally\, the symmetric power
L-functions of elliptic curves should be the L-functions of higher rank a
utomorphic forms. This prediction is closely related to the arithmetic of
the elliptic curve (e.g. the Sato-Tate conjecture). I will discuss this ci
rcle of ideas\, including some recent work with Jack Thorne in which we pr
ove automorphy of these symmetric power L-functions.\n
LOCATION:https://researchseminars.org/talk/UCDANT/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Preston Wake (Michigan State)
DTSTART;VALUE=DATE-TIME:20210211T140000Z
DTEND;VALUE=DATE-TIME:20210211T150000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/6
DESCRIPTION:Title: T
ame derivatives and the Eisenstein ideal\nby Preston Wake (Michigan St
ate) as part of Dublin Algebra and Number Theory Seminar\n\n\nAbstract\nAs
was made famous by Mazur\, the mod-5 Galois representation associated to
the elliptic curve $X_0(11)$ is reducible. Less famously\, but also noted
by Mazur\, the mod-25 Galois representation is reducible. We'll explain wh
y this mod-5 reducibility is to be expected\, but why this mod-25 reducibi
lity is surprising. We'll also discuss the analytic and algebraic signific
ance of the characters that appear in the mod-25 representation.\n
LOCATION:https://researchseminars.org/talk/UCDANT/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chi-Yun Hsu (UCLA)
DTSTART;VALUE=DATE-TIME:20210225T140000Z
DTEND;VALUE=DATE-TIME:20210225T150000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/7
DESCRIPTION:Title: P
artial classicality of Hilbert modular forms\nby Chi-Yun Hsu (UCLA) as
part of Dublin Algebra and Number Theory Seminar\n\n\nAbstract\nOverconve
rgent Hilbert modular forms are defined over a strict neighborhood of the
ordinary locus of the Hilbert modular variety. The philosophy of classical
ity theorems is that when the valuation of $U_p$-eigenvalues are small eno
ugh (called a small slope condition)\, an overconvergent Hecke eigenform i
s automatically classical\, namely\, it can be defined over the whole Hilb
ert modular variety. On the other hand\, we can define partially classical
forms as forms defined over a strict neighborhood of a “partially ordin
ary locus”. We show that under a weaker small slope condition\, an overc
onvergent form is automatically partially classical. We adapt Kassaei’s
method of analytic continuation.\n
LOCATION:https://researchseminars.org/talk/UCDANT/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefano Vigni (University of Genoa)
DTSTART;VALUE=DATE-TIME:20210401T130000Z
DTEND;VALUE=DATE-TIME:20210401T140000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/8
DESCRIPTION:Title: O
n Shafarevich–Tate groups and analytic ranks in Hida families of modula
r forms\nby Stefano Vigni (University of Genoa) as part of Dublin Alge
bra and Number Theory Seminar\n\n\nAbstract\nShafarevich-Tate groups and a
nalytic ranks (that is\, vanishing orders of L-functions) play a major rol
e in the study of the arithmetic of elliptic curves\, abelian varieties\,
and more generally higher (even) weight modular forms. In this talk\, I wi
ll describe results on the behaviour of these arithmetic invariants when t
he modular forms they are attached to vary in a so-called Hida family. In
particular\, our results provide some evidence for a conjecture of \nGreen
berg predicting that the analytic ranks of even weight modular forms in a
Hida family should be as small as allowed by the functional equation\, wit
h at most finitely many exceptions.\n
LOCATION:https://researchseminars.org/talk/UCDANT/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chan-Ho Kim (KIAS)
DTSTART;VALUE=DATE-TIME:20210128T140000Z
DTEND;VALUE=DATE-TIME:20210128T150000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/9
DESCRIPTION:Title: R
efined applications of Kato's Euler systems\nby Chan-Ho Kim (KIAS) as
part of Dublin Algebra and Number Theory Seminar\n\n\nAbstract\nIn modern
number theory\, one of the most interesting goals is to understand the ari
thmetic meaning of special values of L-functions of various arithmetic obj
ects (e.g. Birch and Swinnerton-Dyer conjecture and Bloch-Kato's Tamagawa
number conjecture). Iwasawa theory is the most successful way at present t
o achieve this aim\, and many important results are based on the theory of
Euler systems. We will discuss more refined applications of Kato's Euler
systems for modular forms beyond their standard applications.\n
LOCATION:https://researchseminars.org/talk/UCDANT/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christina Roehrig (Universität zu Köln)
DTSTART;VALUE=DATE-TIME:20210204T140000Z
DTEND;VALUE=DATE-TIME:20210204T150000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/10
DESCRIPTION:Title:
Siegel theta series for indefinite quadratic forms\nby Christina Roehr
ig (Universität zu Köln) as part of Dublin Algebra and Number Theory Sem
inar\n\n\nAbstract\nIn this talk\, we will give an insight into the field
of Siegel modular forms. As they occur as a generalization of elliptic mod
ular forms\, some results can be transferred from the well-known theory de
veloped for these functions. We examine a result by Vignéras\, who showe
d that there is a quite simple way to determine whether a certain theta-se
ries admits modular transformation properties. To be more specific\, she s
howed that solving a differential equation of second order serves as a cri
terion for modularity. We generalize this result for Siegel theta-series.\
n\nIn order to do so\, we construct Siegel theta-series for indefinite qua
dratic forms by considering functions that solve an $n\\times n$-system of
partial differential equations. These functions do not only give examples
of Siegel theta-series\, but we can even determine a basis of Schwartz f
unctions that generate series which transform like modular forms.\n
LOCATION:https://researchseminars.org/talk/UCDANT/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yukako Kezuka (MPI (Bonn))
DTSTART;VALUE=DATE-TIME:20210422T130000Z
DTEND;VALUE=DATE-TIME:20210422T140000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/11
DESCRIPTION:Title:
Tamagawa number divisibility of central L-values\nby Yukako Kezuka (MP
I (Bonn)) as part of Dublin Algebra and Number Theory Seminar\n\n\nAbstrac
t\nIn this talk\, I will report on some recent progress on the conjecture
of Birch and Swinnerton-Dyer for elliptic curves $E$ of the form $x^3+y^3=
N$ for cube-free positive integers $N$. They are cubic twists of the Ferma
t elliptic curve $x^3+y^3=1$\, and admit complex multiplication by the rin
g of integers of $\\mathbb{Q}(\\sqrt{-3})$. First\, I will explain the Tam
agawa number divisibility satisfied by the central $L$-values\, and exhibi
t a curious relation between the $3$-part of the Tate$-$Shafarevich group
of $E$ and the number of prime divisors of $N$ which are inert in $\\mathb
b{Q}(\\sqrt{-3})$. I will then explain my joint work with Yongxiong Li\, s
tudying in more detail the cases when $N=2p$ or $2p^2$ for an odd prime nu
mber $p$ congruent to $2$ or $5$ modulo $9$. For these curves\, we establi
sh the $3$-part of the Birch$-$Swinnerton-Dyer conjecture and a relation b
etween the ideal class group of $\\mathbb{Q}(\\sqrt[3]{p})$ and the $2$-Se
lmer group of $E$\, which can be used to study non-triviality of the $2$-p
art of their Tate$-$Shafarevich group.\n
LOCATION:https://researchseminars.org/talk/UCDANT/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simeon Ball (UPC)
DTSTART;VALUE=DATE-TIME:20210325T153000Z
DTEND;VALUE=DATE-TIME:20210325T163000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/12
DESCRIPTION:Title:
Additive codes over finite fields\nby Simeon Ball (UPC) as part of Dub
lin Algebra and Number Theory Seminar\n\n\nAbstract\nIf $A$ is an abelian
group then we define an additive code to be a code $C$ with the property t
hat for all $u\,v \\in C$\, we have $u+v \\in C$. If $A$ is a finite field
then $C$ is linear over some subfield of $A$\, so we take $A={\\mathbb F}
_{q^h}$ and assume that $C$ is linear over ${\\mathbb F}_q$.\n\nI will spe
nd the first part of the talk talking about the geometry of linear\, addit
ive and quantum stabiliser codes. \n\nThe second part of the talk (joint w
ork with Michel Lavrauw and Guillermo Gamboa) will concern additive MDS co
des. An {\\em MDS code} $C$ is a subset of $A^n$ of size $|A|^k$ in which
any two elements of $C$ differ in at least $n-k+1$ coordinates. In other w
ords\, the minimum (Hamming) distance $d$ between any two elements of $C$
is $n-k+1$. \n\n\n\nThe trivial upper bound on the length $n$ of a $k$-dim
ensional additive MDS code over ${\\mathbb F}_{q^h}$ is\n$$\nn \\leqslant
q^h+k-1.\n$$\n\n\nThe classical example of an MDS code is the Reed-Solomon
code\, which is the evaluation code of all polynomials of degree at most
$k-1$ over ${\\mathbb F}_{q^h}$. The Reed-Solomon code is linear over ${\\
mathbb F}_{q^h}$ and has length $q^h+1$.\n\nThe MDS conjecture states (exc
epting two specific cases) that an MDS code has length at most $q^h+1$. In
other words\, there are no better MDS codes than the Reed-Solomon codes.\
n\nWe use geometrical and computational techniques to classify all additiv
e MDS codes over ${\\mathbb F}_{q^h}$ for $q^h \\in \\{4\,8\,9\\}$. We als
o classify the longest additive MDS codes over ${\\mathbb F}_{16}$ which a
re linear over ${\\mathbb F}_4$. These classifications not only verify the
MDS conjecture for additive codes in these cases but also confirm there a
re no additive non-linear MDS codes that perform as well as their linear c
ounterparts. \n\nIn this talk\, I will cover the main geometrical theorem
that allows us to obtain this classification and compare these classificat
ions with the classifications of {\\bf all} MDS codes of alphabets of size
at most $8$\, obtained previously by Alderson (2006)\, Kokkala\, Krotov a
nd Östergård (2015) and Kokkala and Östergård (2016).\n
LOCATION:https://researchseminars.org/talk/UCDANT/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frank Garvan (University of Florida)
DTSTART;VALUE=DATE-TIME:20210304T140000Z
DTEND;VALUE=DATE-TIME:20210304T150000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/13
DESCRIPTION:Title:
The spt and unimodal sequence conjectures\nby Frank Garvan (University
of Florida) as part of Dublin Algebra and Number Theory Seminar\n\n\nAbst
ract\nIn 2012 Bryson\, Ono\, Pitman\, and Rhoades showed how the generatin
g functions\nfor certain strongly unimodal sequences are related to quantu
m modular\nand mock modular forms. They proved some parity results and con
jectured\nsome mod 4 congruences for the coefficients of these generating
functions.\nIn 2016 Kim\, Lim and Lovejoy obtained similar results for odd
-balanced\nunimodal sequences and made similar mod 4 conjectures. In 2017\
nthe speaker made some similar conjectures for the Andrews spt-function.\n
\n \nIn this talk\, we outline how to prove these conjectures.\nThis invo
lves a connection between the Hurwitz class number function\nand Ramanujan
's mock theta functions.\n \nThis is joint work with Rong Chen (Shanghai).
\n
LOCATION:https://researchseminars.org/talk/UCDANT/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giada Grossi (Paris 13)
DTSTART;VALUE=DATE-TIME:20210408T130000Z
DTEND;VALUE=DATE-TIME:20210408T140000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/14
DESCRIPTION:Title:
The p-part of BSD for rational elliptic curves at Eisenstein primes\nb
y Giada Grossi (Paris 13) as part of Dublin Algebra and Number Theory Semi
nar\n\n\nAbstract\nLet $E$ be an elliptic curve over the rationals and $p$
an odd prime such that E admits a rational $p$-isogeny satisfying some as
sumptions. In joint work with F. Castella\, J. Lee\, and C. Skinner\, we s
tudy the anticyclotomic Iwasawa theory for $E/K$ for some suitable quadrat
ic imaginary field $K$. I will give a general introduction to Iwasawa theo
ry and to how it can be used to obtain results about the Birch--Swinnerton
-Dyer conjecture. In particular\, I will talk about how our results\, comb
ined with complex and $p$-adic Gross-Zagier formulae\, allow us to prove a
$p$-converse to the theorem of Gross--Zagier and Kolyvagin and the $p$-pa
rt of the Birch--Swinnerton-Dyer formula in analytic rank 1 for elliptic c
urves as above.\n
LOCATION:https://researchseminars.org/talk/UCDANT/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pieter Moree (MPI Bonn)
DTSTART;VALUE=DATE-TIME:20210923T130000Z
DTEND;VALUE=DATE-TIME:20210923T140000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/15
DESCRIPTION:Title:
Euler-Kronecker constants and cusp forms\nby Pieter Moree (MPI Bonn) a
s part of Dublin Algebra and Number Theory Seminar\n\n\nAbstract\nRamanuja
n\, in a manuscript not published during his lifetime\, made a very precis
e conjecture for how many integers $n\\le x$ the Ramanujan tau function $\
\tau(n)$ is not divisible by 691 (and likewise for some other primes). The
$\\tau(n)$ are the Fourier coefficients of the Delta function\, which is
a cusp form for the full modular group. Rankin proved that Ramanujan's cla
im is asymptotically correct. However\, the speaker showed in 2004 that th
e second-order behavior predicted by Ramanujan does not match reality. The
proof makes use of high precision computation of constants akin to the Eu
ler-Mascheroni constant called Euler-Kronecker constants.\n\nRecently the
author\, joint with Ciolan and Languasco\, studied the analogue of Ramanuj
an's conjecture for the exceptional primes\, as classified by Serre and Sw
innerton-Dyer\, for the 5 cups forms akin to the Delta function for which
the space of cusp forms is 1-dimensional. The tool for this is a high-prec
ision evaluation of the number of integers $n\\le x$ for which a prescrib
ed integer $q$ does not divide the $k$th sum of divisors function\, sharpe
ning earlier work of Rankin and Scourfield. In my talk\, I will report on
generalities on Euler-Kronecker constants and the above work\, with ample
of historical material.\n
LOCATION:https://researchseminars.org/talk/UCDANT/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eva Bayer-Fluckiger (EPFL)
DTSTART;VALUE=DATE-TIME:20210930T130000Z
DTEND;VALUE=DATE-TIME:20210930T140000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/16
DESCRIPTION:Title:
Isometries of lattices\, knot theory and K3 surfaces\nby Eva Bayer-Flu
ckiger (EPFL) as part of Dublin Algebra and Number Theory Seminar\n\n\nAbs
tract\nWe give necessary and sufficient conditions for an integral polynom
ial to be the characteristic polynomial of an isometry of some even\, unim
odular lattice of given signature. This result has applications in knot th
eory (existence of knots with given Alexander polynomial and Milnor signat
ures) as well as to K3 surfaces (existence of K3 surfaces having an automo
rphism with given dynamical degree).\n
LOCATION:https://researchseminars.org/talk/UCDANT/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jehanne Dousse (CNRS\, Lyon)
DTSTART;VALUE=DATE-TIME:20211104T140000Z
DTEND;VALUE=DATE-TIME:20211104T150000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/17
DESCRIPTION:Title:
Cylindric partitions\, q-difference equations and Rogers-Ramanujan type id
entities\nby Jehanne Dousse (CNRS\, Lyon) as part of Dublin Algebra an
d Number Theory Seminar\n\n\nAbstract\nCylindric partitions\, which were i
ntroduced by Gessel and Krattenthaler in 1997\, can be seen as generalisat
ions of integer partitions involving periodicity conditions. Since the 198
0s and the founding work of Lepowsky and Wilson on Rogers-Ramanujan identi
ties\, several connections between characters of Lie algebras and partitio
n identities have emerged. In particular\, Andrews\, Schilling and Warnaa
r discovered in 1998 a family of partition identities related to character
s of A_2. Recently\, Corteel and Welsh established a q-difference equatio
n satisfied by generating functions for cylindric partitions and used it t
o reprove the A_2 Rogers-Ramanujan identities of Andrews\, Schilling and W
arnaar. We build on this technique to discover and prove a new family of A
_2 Rogers-Ramanujan identities and give explicitly positive expressions fo
r certain characters. \nThis is joint work with Sylvie Corteel and Ali Unc
u.\n
LOCATION:https://researchseminars.org/talk/UCDANT/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuma Mizuno (Chiba University)
DTSTART;VALUE=DATE-TIME:20211118T120000Z
DTEND;VALUE=DATE-TIME:20211118T130000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/18
DESCRIPTION:Title:
Nahm's problem and cluster algebras\nby Yuma Mizuno (Chiba University)
as part of Dublin Algebra and Number Theory Seminar\n\n\nAbstract\nNahm\,
Terhoeven\, and Zagier studied the intersection of the set of q-hypergeom
etric series and the set of modular functions\, and they found that the pr
oblem of finding an element of this intersection is related to the dilogar
ithm function and algebraic K-theory. In this talk\, I will explain this p
roblem is also related to the periodicity of T-systems (and Y-systems)\, w
hich are difference equations that appear in the theory of cluster algebra
s. I will give a systematic construction of q-hypergeometric series that a
re expected to be modular using the theory of cluster algebras.\n
LOCATION:https://researchseminars.org/talk/UCDANT/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesc Castella (UCSB)
DTSTART;VALUE=DATE-TIME:20211202T140000Z
DTEND;VALUE=DATE-TIME:20211202T150000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/19
DESCRIPTION:Title:
Iwasawa theory of elliptic curves at Eisenstein primes\nby Francesc Ca
stella (UCSB) as part of Dublin Algebra and Number Theory Seminar\n\n\nAbs
tract\nIn an influential paper from 2000\, Greenberg and Vatsal introduced
a method for studying the cyclotomic Iwasawa theory of elliptic curves E
over Q at Eisenstein primes (i.e.\, primes p for which E admits a rational
p-isogeny). Combined with Kato's work\, their results had important impli
cations towards the Birch and Swinnerton-Dyer conjecture in rank 0. In thi
s talk\, I will try to convey the main ideas of their method\, and then mo
ve on to explain recent joint work with G. Grossi and C. Skinner (partly i
n progress) in which we develop the method of Greenberg-Vatsal in the anti
-cyclotomic setting\, leading to new applications towards the Birch and Sw
innerton-Dyer conjecture in rank 1.\n
LOCATION:https://researchseminars.org/talk/UCDANT/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Scott Ahlgren (UIUC)
DTSTART;VALUE=DATE-TIME:20211028T130000Z
DTEND;VALUE=DATE-TIME:20211028T140000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/20
DESCRIPTION:Title:
Congruences for the partition function\nby Scott Ahlgren (UIUC) as par
t of Dublin Algebra and Number Theory Seminar\n\n\nAbstract\nThe partition
function p(n) counts the number of ways to break a natural number n into
parts. The arithmetic properties of this function have been the topic of i
ntensive study for many years. Much of the interest (and the difficulty)
in this problem arises from the fact that values of the partition function
are given by the coefficients of a weakly holomorphic modular form of hal
f-integral weight. I’ll describe some recent work with Olivia Beckwith
and Martin Raum\, and with Patrick Allen and Shiang Tang which goes a long
way towards explaining exactly when congruences for the partition functio
n can occur. The main tools are techniques from the theory of modular for
ms\, Galois representations\, and analytic number theory.\n
LOCATION:https://researchseminars.org/talk/UCDANT/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mahesh Kakde (IISc)
DTSTART;VALUE=DATE-TIME:20211125T140000Z
DTEND;VALUE=DATE-TIME:20211125T150000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/21
DESCRIPTION:Title:
On the Brumer-Stark conjecture\nby Mahesh Kakde (IISc) as part of Dubl
in Algebra and Number Theory Seminar\n\n\nAbstract\nThe talk will start wi
th an introduction to Stark’s conjectures. We will then specialise to th
e situation of Brumer-Stark conjecture and its various refinements. I will
then sketch a proof of the conjecture. This is joint work with Samit Dasg
upta.\n
LOCATION:https://researchseminars.org/talk/UCDANT/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Johannes Sprang (Universität Duisburg-Essen)
DTSTART;VALUE=DATE-TIME:20211007T130000Z
DTEND;VALUE=DATE-TIME:20211007T140000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/22
DESCRIPTION:Title:
Algebraicity of critical Hecke L-values\nby Johannes Sprang (Universit
ät Duisburg-Essen) as part of Dublin Algebra and Number Theory Seminar\n\
n\nAbstract\nIn 1735\, Euler discovered his well-known formula for the val
ues of the Riemann zeta function at the positive even integers. In particu
lar\, Euler's result shows that all these values are rational up to multip
lication with a particular period\, here the period is a power of 2πi. Co
njecturally this is expected to hold for all critical L-values of motives.
In this talk\, we will focus on L-functions of number fields. In the firs
t part of the talk\, we will discuss the 'critical' and 'non-critical' L-v
alues exemplary for the Riemann zeta function. Afterwards\, we will head o
n to more general number fields and explain a joint result with Guido Kin
gs on the algebraicity of critical Hecke L-values for totally imaginary fi
elds up to explicit periods.\n
LOCATION:https://researchseminars.org/talk/UCDANT/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Medvedovsky (Boston University)
DTSTART;VALUE=DATE-TIME:20211013T143000Z
DTEND;VALUE=DATE-TIME:20211013T153000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/23
DESCRIPTION:Title:
Counting modular forms with fixed mod-$p$ Galois representation and Atkin-
Lehner-at-$p$ eigenvalue\nby Anna Medvedovsky (Boston University) as p
art of Dublin Algebra and Number Theory Seminar\n\n\nAbstract\nWork in pro
gress joint with Samuele Anni and Alexandru Ghitza. For $N$ prime to $p$\,
we count the number of classical modular forms of level $Np$ and weight $
k$ with fixed mod-$p$ Galois representation and Atkin-Lehner-at-$p$ sign\,
generalizing both recent results of Martin generalizing work of Wakatsuki
and Yamauchi (no residual representation constraint) and the $\\overline{
\\rho}$-dimension-counting formulas of Bergdall-Pollack and Jochnowitz. Wo
rking with the Atkin-Lehner involution typically requires inverting $p$\,
which naturally complicates investigations modulo $p$. To resolve this ten
sion\, we use the trace formula to establish up-to-semisimplifcation isomo
rphisms between certain mod-$p$ Hecke modules (namely\, refinements of the
weight-filtration graded pieces $W_k$) by exhibiting ever-deeper congruen
ces between traces of prime-power Hecke operators acting on characteristic
-zero Hecke modules. This last technique\, relying on our refinement of a
special case of Brauer-Nesbitt\, is new\, purely algebraic\, and may well
be of independent interest. We will begin with this algebra theorem\, then
discuss the classical Atkin-Lehner dimension split\, and only then move o
n to our refined dimension-counting results.\n
LOCATION:https://researchseminars.org/talk/UCDANT/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kazuto Ota (Osaka University)
DTSTART;VALUE=DATE-TIME:20220127T140000Z
DTEND;VALUE=DATE-TIME:20220127T150000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/24
DESCRIPTION:Title:
On Iwasawa theory for CM elliptic curves at inert primes\nby Kazuto Ot
a (Osaka University) as part of Dublin Algebra and Number Theory Seminar\n
\nInteractive livestream: https://ucd-ie.zoom.us/j/5963025149\n\nAbstract\
nI will report on joint work with Ashay Burungale and Shinichi Kobayashi o
n anticyclotomic Iwasawa theory for CM elliptic curves at inert primes. A
key result is a structure theorem of local units predicted by Rubin\, whic
h leads to new developments in supersingular Iwasawa theory such as a Bert
olini-Darmon-Prasanna style formula for Rubin’s anticyclotomic p-adic L-
function. I this talk\, I will explain the conjecture of Rubin and such a
development.\n
LOCATION:https://researchseminars.org/talk/UCDANT/24/
URL:https://ucd-ie.zoom.us/j/5963025149
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ellen Eischen (University of Oregon)
DTSTART;VALUE=DATE-TIME:20220407T160000Z
DTEND;VALUE=DATE-TIME:20220407T170000Z
DTSTAMP;VALUE=DATE-TIME:20220124T073458Z
UID:UCDANT/25
DESCRIPTION:by Ellen Eischen (University of Oregon) as part of Dublin Alge
bra and Number Theory Seminar\n\nInteractive livestream: https://ucd-ie.zo
om.us/j/5963025149\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/UCDANT/25/
URL:https://ucd-ie.zoom.us/j/5963025149
END:VEVENT
END:VCALENDAR