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BEGIN:VEVENT
SUMMARY:Katharina Muller (Université Laval/Goettingen)
DTSTART;VALUE=DATE-TIME:20210924T190000Z
DTEND;VALUE=DATE-TIME:20210924T200000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/1
DESCRIPTION:Title: Iwasawa theory of class groups in the case $p=2$\nby Katharina Mulle
r (Université Laval/Goettingen) as part of Algebra and Number Theory Semi
nars at Université Laval\n\nLecture held in VCH2820.\n\nAbstract\nLet $K$
be a $CM$ number field and $K_\\infty$ be its cyclotomic $Z_p$-extension
with intermediate layers $K_n$. If $p$ is odd we get a decomposition in pl
us and minus parts of the class group and it is well known that the ideal
lift map from $K_n$ to $K_{n+1}$ is injective on the minus part of the cla
ss group. For $p=2$ this is in general not true. We will provide a differe
nt definition of the minus part and explain how inherits properties that a
re known for $p>2$. If time allows we will also present an application of
these results to compute the $2$ class group of the fields $K_n$ for certa
in base fields explicitely. Part of this is joint work with M.M. Chems-Edd
in.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Vallieres (California State University\, Chico)
DTSTART;VALUE=DATE-TIME:20211001T190000Z
DTEND;VALUE=DATE-TIME:20211001T200000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/2
DESCRIPTION:Title: An analogue of a theorem of Iwasawa in graph theory\nby Daniel Valli
eres (California State University\, Chico) as part of Algebra and Number T
heory Seminars at Université Laval\n\n\nAbstract\nIn the 1950s\, Iwasawa
proved his now celebrated theorem on the growth of the p-part of the class
number in some infinite towers of number fields. In this talk\, we will
explain our recent work in obtaining an analogous result in graph theory i
nvolving the p-part of the number of spanning trees in some infinite tower
s of graphs. Part of this work is joint with Kevin McGown.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ariel Pacetti (Universidade de Aveiro)
DTSTART;VALUE=DATE-TIME:20211105T140000Z
DTEND;VALUE=DATE-TIME:20211105T150000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/3
DESCRIPTION:Title: Modularity of some geometric objects\nby Ariel Pacetti (Universidade
de Aveiro) as part of Algebra and Number Theory Seminars at Université L
aval\n\n\nAbstract\nThe purpose of this talk is to recall different instan
ces of modularity of geometric objects. We will start recalling the case o
f rational elliptic curves (the Shimura-Taniyama conjecture)\, to move to
quadratic fields (and more general ones) and end with the case of abelian
rational surfaces (the Brumer-Kramer paramodular conjecture). We will put
special emphasis on the state of the art of the correspondence\, including
the open problems. If time allows\, we will also discuss some particular
cases of Calabi-Yau threefolds.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Filippo Nuccio (Université Jean Monnet Saint-Étienne)
DTSTART;VALUE=DATE-TIME:20211022T143000Z
DTEND;VALUE=DATE-TIME:20211022T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/4
DESCRIPTION:Title: Explaining the finiteness of the class group of a number field to a comp
uter\nby Filippo Nuccio (Université Jean Monnet Saint-Étienne) as pa
rt of Algebra and Number Theory Seminars at Université Laval\n\n\nAbstrac
t\nA proof-assistant is a computer program that can digest a mathematical
proof\, implemented as a chain of statements. If all statements follow log
ically from previously proven ones\, then the assistant is happy\, and cer
tifies the correctness of the proof\; if it is doubtful about a certain po
int\, it will not let you continue until it gets convinced. Among other pr
oof assistants\, Lean3 is getting popular among some "classical" mathemati
cians\, who are formalising well-known proofs in order to shape a larger a
nd larger mathematical library upon which subsequent work can rely. In thi
s talk\, I will show how to discuss with Lean3\, I will show some examples
and I will report on a recent work\, joint with A. Baanen\, S. Daamen and
Ashvni N.\, where we formalised the finiteness of the class group of a nu
mber field in Lean3.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiacheng Xia (Laval)
DTSTART;VALUE=DATE-TIME:20211015T190000Z
DTEND;VALUE=DATE-TIME:20211015T200000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/5
DESCRIPTION:Title: Modularity of generating functions of special cycles on unitary Shimura
varieties\nby Jiacheng Xia (Laval) as part of Algebra and Number Theor
y Seminars at Université Laval\n\nLecture held in VCH2820.\n\nAbstract\nS
pecial cycles on orthogonal and unitary Shimura varieties are analogues of
Heegner points on modular curves in higher dimensions. Following work of
Hirzebruch--Zagier\, Gross--Zagier\, Gross--Keating\, and Kudla--Millson\,
Kudla predicted the modularity of generating functions of these special c
ycles in the 1990s. \n\nI will review some historic development of this co
njecture\, and summarize recent results built upon earlier work of Borcher
ds and Zhang. I will also talk about arithmetic applications\, especially
the recent work of Li--Liu on arithmetic inner product formula. Time permi
tting\, I will sketch the method of Bruinier--Raum and discuss its scope.\
n
LOCATION:https://researchseminars.org/talk/ANTULaval/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Siddarth Sankaran (University of Manitoba)
DTSTART;VALUE=DATE-TIME:20211126T200000Z
DTEND;VALUE=DATE-TIME:20211126T210000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/6
DESCRIPTION:Title: Green forms\, special cycles and modular forms\nby Siddarth Sankaran
(University of Manitoba) as part of Algebra and Number Theory Seminars at
Université Laval\n\n\nAbstract\nShimura varieties attached to orthogonal
groups (of which modular curves are examples) are interesting objects of
study for many reasons\, not least of which is the fact that they possess
an abundance of “special” cycles. These cycles are at the centre of a
conjectural program proposed by Kudla\; roughly speaking\, Kudla’s conje
ctures suggest that upon passing to an (arithmetic) Chow group\, the speci
al cycles behave like the Fourier coefficients of automorphic forms. These
conjectures also include more precise identities\; for example\, the arit
hmetic Siegel-Weil formula relates arithmetic heights of special cycles to
derivatives of Eisenstein series. In this talk\, I’ll describe a constr
uction (in joint work with Luis Garcia) of Green currents for these cycles
\, which are an essential ingredient in the “Archimedean” part of the
story\; I’ll also sketch a few applications of this construction to Kudl
a’s conjectures.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rylan Gajek-Leonard (UMass Amherst)
DTSTART;VALUE=DATE-TIME:20211112T200000Z
DTEND;VALUE=DATE-TIME:20211112T210000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/7
DESCRIPTION:Title: Iwasawa Invariants of Modular Forms with $a_p=0$\nby Rylan Gajek-Leo
nard (UMass Amherst) as part of Algebra and Number Theory Seminars at Univ
ersité Laval\n\n\nAbstract\nMazur-Tate elements provide a convenient meth
od to study the analytic Iwasawa theory of $p$-nonordinary modular forms\,
where the associated $p$-adic $L$-functions have unbounded coefficients.
The Iwasawa invariants of Mazur-Tate elements are well-understood in the c
ase of weight 2 modular forms\, where they can be related to the growth of
$p$-Selmer groups and decompositions of the $p$-adic $L$-function. At hig
her weights\, less is known. By constructing certain lifts to the full Iwa
sawa algebra\, we compute the Iwasawa invariants of Mazur-Tate elements fo
r higher weight modular forms with $a_p=0$ in terms of the plus/minus inva
riants of the $p$-adic $L$-function. Combined with results of Pollack-West
on\, this forces a relation between the plus/minus invariants at weights 2
and $p+1$.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yongxiong Li (Tsinghua University\, Beijing)
DTSTART;VALUE=DATE-TIME:20220422T143000Z
DTEND;VALUE=DATE-TIME:20220422T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/8
DESCRIPTION:Title: On some arithmetic of Satge curves\nby Yongxiong Li (Tsinghua Univer
sity\, Beijing) as part of Algebra and Number Theory Seminars at Universit
é Laval\n\n\nAbstract\nLet n>2 be a cube free integer\, we consider the e
lliptic curves of the form C_n: x^3+y^3=n. \nIn this talk\, we will prove
that the 3-part of BSD conjecture for C_2p (resp. C_2p^2)\, where p ≡ 2
(resp. 5) mod 9 is an odd prime. The 2-part of the Tate-Shafarevich group
of those curves will also be discussed. This is joint work with Y.Kezuka.\
n
LOCATION:https://researchseminars.org/talk/ANTULaval/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Bergdall (Bryn Mawr College)
DTSTART;VALUE=DATE-TIME:20220318T143000Z
DTEND;VALUE=DATE-TIME:20220318T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/9
DESCRIPTION:Title: Recent investigations of L-invariants of modular forms.\nby John Ber
gdall (Bryn Mawr College) as part of Algebra and Number Theory Seminars at
Université Laval\n\n\nAbstract\nIn this talk I will explain new research
on L-invariants of modular forms\, including ongoing joint work with Robe
rt Pollack. L-invariants\, which are p-adic invariants of modular forms\,
were discovered in the 1980's\, by Mazur\, Tate\, and Teitelbaum\, who wer
e formulating a p-adic analogue of Birch and Swinnerton-Dyer's conjecture
on elliptic curves. In the decades since\, L-invariants have shown up in a
ton of places: p-adic L-series for higher weight modular forms or higher
rank automorphic forms\, the Banach space representation theory of GL(2\,Q
p)\, p-adic families of modular forms\, Coleman integration on the p-adic
upper half-plane\, and Fontaine's p-adic Hodge theory for Galois represent
ations. In this talk I will focus on recent numerical and statistical inve
stigations of these L-invariants\, which touch on at least four of the the
ories just mentioned. I will try to put everything into the overall contex
t of practical questions in the theory of automorphic forms and Galois rep
resentations\, keeping everything as concrete as possible\, and explain wh
at the future holds.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Williams (University of Warwick)
DTSTART;VALUE=DATE-TIME:20220121T153000Z
DTEND;VALUE=DATE-TIME:20220121T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/10
DESCRIPTION:Title: p-adic L-fucntions for GL(3)\nby Chris Williams (University of Warw
ick) as part of Algebra and Number Theory Seminars at Université Laval\n\
n\nAbstract\nLet $\\pi$ be a p-ordinary cohomological cuspidal automorphic
representation of $GL(n\,A_Q)$. A conjecture of Coates--Perrin-Riou predi
cts that the (twisted) critical values of its L-function $L(\\pi x\\chi\,s
)$\, for Dirichlet characters $\\chi$ of p-power conductor\, satisfy syste
matic congruence properties modulo powers of p\, captured in the existence
of a p-adic L-function. For n = 1\,2 this conjecture has been known for d
ecades\, but for n > 2 it is known only in special cases\, e.g. symmetric
squares of modular forms\; and in all previously known cases\, \\pi is a f
unctorial transfer via a proper subgroup of GL(n). In this talk\, I will e
xplain what a p-adic L-function is\, state the conjecture more precisely\,
and then describe recent joint work with David Loeffler\, in which we pro
ve this conjecture for n=3 (without any transfer or self-duality assumptio
ns).\n
LOCATION:https://researchseminars.org/talk/ANTULaval/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Longo (Universita di Padova)
DTSTART;VALUE=DATE-TIME:20220128T153000Z
DTEND;VALUE=DATE-TIME:20220128T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/11
DESCRIPTION:Title: On the p-part of the equivariant Tamagawa number conjecture for motives
of modular forms\nby Matteo Longo (Universita di Padova) as part of A
lgebra and Number Theory Seminars at Université Laval\n\n\nAbstract\nI pl
an to present a work in progress\, in collaboration with Stefano Vigni\, i
n which we study the equivariant Tamagawa number conjecture\, formulated b
y Bloch-Kato\, in the case of motives attached to cuspforms. This conjectu
re can be seen as a generalisation to (pre)motives of the (full) Birch and
Swinnerton-Dyer conjecture for elliptic curves\, and is still wide open.
The case of motives of modular forms can be studied using methods analogou
s to those exploited in the case of elliptic curves. After an introduction
in which I will recall the main results in the case of elliptic curves\,
I will discuss our results in the case of motives of modular forms.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cameron Franc (McMaster University)
DTSTART;VALUE=DATE-TIME:20220204T153000Z
DTEND;VALUE=DATE-TIME:20220204T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/12
DESCRIPTION:Title: Characters of VOAs and modular forms\nby Cameron Franc (McMaster Un
iversity) as part of Algebra and Number Theory Seminars at Université Lav
al\n\n\nAbstract\nModular forms have appeared throughout the representatio
n theory of vertex operator algebras (VOAs) from the very beginning of the
subject\, for example\, via the study of VOAs modeled on representations
of infinite dimensional lie algebras\, as well as spectacular examples suc
h as the monster module. In this talk we will explain how the theory of mo
dular forms can be used to study representations of VOAs\, in a similar wa
y to how character tables can aid the study of representation theory of fi
nite groups. No prior knowledge of VOA theory will be assumed.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Somnath Jha (IIT Kanapur)
DTSTART;VALUE=DATE-TIME:20220401T143000Z
DTEND;VALUE=DATE-TIME:20220401T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/13
DESCRIPTION:Title: Fine Selmer group of elliptic curves\nby Somnath Jha (IIT Kanapur)
as part of Algebra and Number Theory Seminars at Université Laval\n\n\nAb
stract\nThe (p-infinity) fine Selmer group (also called the 0-Selmer group
) of an elliptic curve is a subgroup of the usual p-infinity Selmer group
of an elliptic curve and is related to the first and the second Iwasawa co
homology groups. Coates-Sujatha observed that the structure of the fine Se
lmer group over the cyclotomic Z_p extension of a number field K is intric
ately related to Iwasawa's \\mu-invariant vanishing conjecture on the grow
th of p-part of the ideal class group of K in the cyclotomic tower. In thi
s talk\, we will discuss the structure and properties of the fine Selmer g
roup over certain p-adic Lie extensions of global fields. This talk is bas
ed on joint work with Sohan Ghosh and Sudhanshu Shekhar.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cornelius Greither (Universität der Bundeswehr München)
DTSTART;VALUE=DATE-TIME:20220225T153000Z
DTEND;VALUE=DATE-TIME:20220225T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/14
DESCRIPTION:Title: An equivalence relation for modules\, and Fitting ideals of class group
s\nby Cornelius Greither (Universität der Bundeswehr München) as par
t of Algebra and Number Theory Seminars at Université Laval\n\n\nAbstract
\nIt is well known that analytic sources\, like zeta and\nL-functions\, p
rovide information on class groups.\nNot only the order of a class group b
ut also its\nstructure as a module over a suitable group ring has been\nst
udied in this way. The strongest imaginable result\nwould be determining c
lass groups\nup to module isomorphism\, but this seems extremely\ndifficul
t. A popular ``best approximation'' consists in\ndetermining the Fitting i
deal. The prototypical result (we omit all\nhypotheses\, restrictions and
embellishments) predicts the Fitting ideal\nof a class group as the produc
t of a certain ideal $J$ and a so-called\nequivariant L-value $\\omega$ in
a group ring. The element $\\omega$\ngenerates a principal ideal\, but it
s description is analytic\nand complicated. On the other hand\, the ideal
$J$ is usually far from \nprincipal but has a much more elementary descrip
tion. -- In this talk we intend to\ndescribe a few recent results of this
kind\, and we explain\na new concept of ``equivalence'' of modules. This l
eads\, ideally\, to\na finer description of the class groups a priori than
just determining\nits Fitting ideal\; in other words\, we look for a way
of\nimproving the above-mentioned ``best approximation''.\nThis is recent
joint work with Takenori Kataoka.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marc Masdeu (Universitat Autònoma de Barcelona)
DTSTART;VALUE=DATE-TIME:20220408T143000Z
DTEND;VALUE=DATE-TIME:20220408T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/15
DESCRIPTION:Title: Numerical experiments with plectic Darmon points\nby Marc Masdeu (U
niversitat Autònoma de Barcelona) as part of Algebra and Number Theory Se
minars at Université Laval\n\n\nAbstract\nLet E/F be an elliptic curve de
fined over a number field F\, and let K/F be a quadratic extension. If the
analytic rank of E(K) is one\, one can often use Heegner points (or the m
ore general Darmon points) to produce (at least conjecturally) a nontorsio
n generator of E(K). If the analytic rank of E(K) is larger than one\, the
problem of constructing algebraic points is still very open. In recent wo
rk\, Michele Fornea and Lennart Gehrmann have introduced certain p-adic qu
antities that may be conjecturally related to the existence of these point
s. In this talk I will explain their construction\, and illustrate with so
me numerical experiments some support for their conjecture. This is joint
work with Michele Fornea and Xevi Guitart.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ari Shnidman (Hebrew University of Jerusalem)
DTSTART;VALUE=DATE-TIME:20220211T153000Z
DTEND;VALUE=DATE-TIME:20220211T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/16
DESCRIPTION:Title: Manin-Drinfeld cycles and L-functions\nby Ari Shnidman (Hebrew Univ
ersity of Jerusalem) as part of Algebra and Number Theory Seminars at Univ
ersité Laval\n\n\nAbstract\nI'll describe a formula I proved a few years
ago relating the derivative of an L-function of an automorphic representat
ion for PGL_2 over a function field to an intersection pairing of two spec
ial algebraic cycles in a moduli space of shtukas. The proof\, which I wi
ll try to sketch\, is via the geometric relative trace formula of Jacquet-
Yun-Zhang. The formula leads to interesting questions about Manin-Drinfeld
cycles\, which are generalizations of the cusps on modular curves\, as I
will explain.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christian Wuthrich (University of Nottingham)
DTSTART;VALUE=DATE-TIME:20220325T143000Z
DTEND;VALUE=DATE-TIME:20220325T153000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/17
DESCRIPTION:Title: The denominator of twisted $L$-values of elliptic curves\nby Christ
ian Wuthrich (University of Nottingham) as part of Algebra and Number Theo
ry Seminars at Université Laval\n\n\nAbstract\nIn the context of the gene
ralised Birch and Swinnerton-Dyer conjecture\, one considers the value at
$s=1$ of the L-function of an elliptic curve $E/\\mathbb{Q}$ twisted by a
Dirichlet character $\\chi$. When normalised with a period\, one obtains a
n algebraic number $\\mathscr{L}(E\,\\chi)$. I will discuss the question u
nder what conditions $\\mathscr{L}(E\,\\chi)$ is an algebraic integer and
what the possible denominators could be.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mohamed Mahmoud Chems-Eddin (Sidi Mohamed Ben Abdellah University\
, Fez)
DTSTART;VALUE=DATE-TIME:20220304T153000Z
DTEND;VALUE=DATE-TIME:20220304T163000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/18
DESCRIPTION:Title: Unit Groups and 2-Class Field Towers\; Techniques and Computations\
nby Mohamed Mahmoud Chems-Eddin (Sidi Mohamed Ben Abdellah University\, Fe
z) as part of Algebra and Number Theory Seminars at Université Laval\n\n\
nAbstract\nDuring this talk we are going to expose some techniques for com
puting the unit groups of multiquadratic number fields. Furthermore\, we
shall present a new simple method to deal with the $2$-class field towers
of some number fields whose $2$-class groups are of type $(2\,2)$. More
precisely\, we shall compute the unit group of the number field $\\mathbb{
Q}( \\sqrt{p}\, \\sqrt{q}\,\\sqrt{2} \,\\sqrt{-1})$\, where $p$ and $q$ ar
e two prime numbers. In the second part this talk\, we shall use units to
study the $2$-class field tower of some imaginary biquadratic number fiel
ds.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Samuel (Université Laval)
DTSTART;VALUE=DATE-TIME:20220315T203000Z
DTEND;VALUE=DATE-TIME:20220315T213000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/19
DESCRIPTION:Title: An investigation of local zeta functions of self-similar fractal string
s\nby David Samuel (Université Laval) as part of Algebra and Number T
heory Seminars at Université Laval\n\n\nAbstract\nWe give an overview of
fractal strings and examine the relationship between their Minkowski dimen
sion/content to their complex dimensions and their geometric zeta function
s with the aim of demonstrating the geometric information made available b
y studying these entities. Building on this knowledge\, we propose a way o
f searching for locally defined geometric zeta functions by looking at sim
ple examples of self-similar fractal strings.\n
LOCATION:https://researchseminars.org/talk/ANTULaval/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felix Baril Boudreau (postdoc at U. of Lethbridge)
DTSTART;VALUE=DATE-TIME:20230127T210000Z
DTEND;VALUE=DATE-TIME:20230127T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T071959Z
UID:ANTULaval/20
DESCRIPTION:Title: Fonctions L de courbes elliptiques en caractéristique positive (Partie
I: Rationalité et algorithme de Schoof))\nby Felix Baril Boudreau (p
ostdoc at U. of Lethbridge) as part of Algebra and Number Theory Seminars
at Université Laval\n\n\nAbstract\nL’hypothèse de Riemann et la conjec
ture de Birch et Swinnerton-Dyer (BSD) sont de célèbres problèmes non r
ésolus en théorie des nombres dans le contexte des corps de nombres (ext
ensions finies de Q). Du côté des corps de fonctions (extensions finies
de Fq(t))\, Weil (1949) formula\, et démontra dans certain cas\, des conj
ectures portant sur les fonctions zêta de variétés projectives lisses d
éfinies sur F_q . Ces conjectures portaient entre autres choses sur la ra
tionalité des fonctions zêta et sur une propriété analogue à l’hypo
thèse de Riemann qu’elles vérifiaient. Les conjectures de Weil furent
généralisées à certaines fonctions L (dont les fonctions zêtas en son
t un exemple)\, et démontrées entre 1960 et 1980\, principalement par Dw
ork\, Grothendieck\, Artin et Deligne.\n\nMalgré cet énorme succès\, ce
s fonctions L ne sont pas encore complètement bien comprises. Par exemple
\, il est difficile de les calculer en pratique. De plus\, l’analogue de
la conjecture de BSD pour une courbe elliptique définie sur un corps de
fonctions n’est pas résolu en général.\n\nDans ce premier exposé de
deux\, nous esquisserons une preuve de la rationalité de la fonction zêt
a d’une courbe elliptique définie au-dessus d’un corps fini F_q . Son
numérateur est un polynôme quadratique à coefficients entiers dont le
terme linéaire a_q dépend du nombre de points à coordonnées dans F_q v
érifiant une équation de la forme y^2 = x^3 + Ax + B sur F_q . Essayer u
n à un les points (x\, y) vérifiant cette équation est peu efficace lor
sque F_q est grand. Comme a_q est un entier\, nous pouvons tenter de calcu
ler directement sa réduction modulo un nombre suffisamment de petits prem
iers et ensuite reconstruire aq grâce au théorème chinois. Cette idée
est la base de l’algorithme développé par Schoof (1985)\, dont nous pa
rlerons brièvement. Enfin\, nous conclurons cette présentation par une e
squisse de preuve de la rationalité de la fonction L d’une courbe ellip
tique E/K définie au-dessus d’un corps de fonctions K. Ce premier expos
é ne contient aucun résultat nouveau. Il prépare cependant le terrain p
our le second exposé. Ce dernier portera sur des contributions originales
du conférencier à l’étude analogue mais plus complexe de la réducti
on (du numérateur) de la fonction L de E/K modulo des entiers premiers à
q.\n\n*the talk will be in French\n
LOCATION:https://researchseminars.org/talk/ANTULaval/20/
END:VEVENT
END:VCALENDAR