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BEGIN:VEVENT
SUMMARY:Katharina Muller (Université Laval/Goettingen)
DTSTART:20210924T190000Z
DTEND:20210924T200000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/1
DESCRIPTION:Title: Iwasawa theory of class groups in the case $p=2$\nby Katharin
a Muller (Université Laval/Goettingen) as part of Algebra and Number Theo
ry Seminars at Université Laval\n\nLecture held in VCH2820.\n\nAbstract\n
Let $K$ be a $CM$ number field and $K_\\infty$ be its cyclotomic $Z_p$-ext
ension with intermediate layers $K_n$. If $p$ is odd we get a decompositio
n in plus 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 class group. For $p=2$ this is in general not true. We will provide a
different definition of the minus part and explain how inherits properties
that are known for $p>2$. If time allows we will also present an applicat
ion of these results to compute the $2$ class group of the fields $K_n$ fo
r certain base fields explicitely. Part of this is joint work with M.M. Ch
ems-Eddin.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Vallieres (California State University\, Chico)
DTSTART:20211001T190000Z
DTEND:20211001T200000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/2
DESCRIPTION:Title: An analogue of a theorem of Iwasawa in graph theory\nby Danie
l Vallieres (California State University\, Chico) as part of Algebra and N
umber Theory Seminars at Université Laval\n\n\nAbstract\nIn the 1950s\, I
wasawa proved his now celebrated theorem on the growth of the p-part of th
e class number in some infinite towers of number fields. In this talk\, w
e will explain our recent work in obtaining an analogous result in graph t
heory involving the p-part of the number of spanning trees in some infinit
e towers of graphs. Part of this work is joint with Kevin McGown.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ariel Pacetti (Universidade de Aveiro)
DTSTART:20211105T140000Z
DTEND:20211105T150000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/3
DESCRIPTION:Title: Modularity of some geometric objects\nby Ariel Pacetti (Unive
rsidade de Aveiro) as part of Algebra and Number Theory Seminars at Univer
sité Laval\n\n\nAbstract\nThe purpose of this talk is to recall different
instances of modularity of geometric objects. We will start recalling the
case of rational elliptic curves (the Shimura-Taniyama conjecture)\, to m
ove to quadratic fields (and more general ones) and end with the case of a
belian rational surfaces (the Brumer-Kramer paramodular conjecture). We wi
ll put special emphasis on the state of the art of the correspondence\, in
cluding the open problems. If time allows\, we will also discuss some part
icular cases of Calabi-Yau threefolds.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Filippo Nuccio (Université Jean Monnet Saint-Étienne)
DTSTART:20211022T143000Z
DTEND:20211022T153000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/4
DESCRIPTION:Title: Explaining the finiteness of the class group of a number field to
a computer\nby Filippo Nuccio (Université Jean Monnet Saint-Étienne
) as part of Algebra and Number Theory Seminars at Université Laval\n\n\n
Abstract\nA proof-assistant is a computer program that can digest a mathem
atical proof\, implemented as a chain of statements. If all statements fol
low logically from previously proven ones\, then the assistant is happy\,
and certifies the correctness of the proof\; if it is doubtful about a cer
tain point\, it will not let you continue until it gets convinced. Among o
ther proof assistants\, Lean3 is getting popular among some "classical" ma
thematicians\, who are formalising well-known proofs in order to shape a l
arger and larger mathematical library upon which subsequent work can rely.
In this talk\, I will show how to discuss with Lean3\, I will show some e
xamples and I will report on a recent work\, joint with A. Baanen\, S. Daa
men and Ashvni N.\, where we formalised the finiteness of the class group
of a number field in Lean3.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiacheng Xia (Laval)
DTSTART:20211015T190000Z
DTEND:20211015T200000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/5
DESCRIPTION:Title: Modularity of generating functions of special cycles on unitary S
himura varieties\nby Jiacheng Xia (Laval) as part of Algebra and Numbe
r Theory Seminars at Université Laval\n\nLecture held in VCH2820.\n\nAbst
ract\nSpecial cycles on orthogonal and unitary Shimura varieties are analo
gues of Heegner points on modular curves in higher dimensions. Following w
ork of Hirzebruch--Zagier\, Gross--Zagier\, Gross--Keating\, and Kudla--Mi
llson\, Kudla predicted the modularity of generating functions of these sp
ecial cycles in the 1990s. \n\nI will review some historic development of
this conjecture\, and summarize recent results built upon earlier work of
Borcherds and Zhang. I will also talk about arithmetic applications\, espe
cially the recent work of Li--Liu on arithmetic inner product formula. Tim
e permitting\, I will sketch the method of Bruinier--Raum and discuss its
scope.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Siddarth Sankaran (University of Manitoba)
DTSTART:20211126T200000Z
DTEND:20211126T210000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/6
DESCRIPTION:Title: Green forms\, special cycles and modular forms\nby Siddarth S
ankaran (University of Manitoba) as part of Algebra and Number Theory Semi
nars at Université Laval\n\n\nAbstract\nShimura varieties attached to ort
hogonal groups (of which modular curves are examples) are interesting obje
cts of study for many reasons\, not least of which is the fact that they p
ossess an abundance of “special” cycles. These cycles are at the centr
e of a conjectural program proposed by Kudla\; roughly speaking\, Kudla’
s conjectures suggest that upon passing to an (arithmetic) Chow group\, th
e special cycles behave like the Fourier coefficients of automorphic forms
. These conjectures also include more precise identities\; for example\, t
he arithmetic Siegel-Weil formula relates arithmetic heights of special cy
cles to derivatives of Eisenstein series. In this talk\, I’ll describe a
construction (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 Kudla’s conjectures.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rylan Gajek-Leonard (UMass Amherst)
DTSTART:20211112T200000Z
DTEND:20211112T210000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/7
DESCRIPTION:Title: Iwasawa Invariants of Modular Forms with $a_p=0$\nby Rylan Ga
jek-Leonard (UMass Amherst) as part of Algebra and Number Theory Seminars
at Université Laval\n\n\nAbstract\nMazur-Tate elements provide a convenie
nt method to study the analytic Iwasawa theory of $p$-nonordinary modular
forms\, where the associated $p$-adic $L$-functions have unbounded coeffic
ients. The Iwasawa invariants of Mazur-Tate elements are well-understood i
n the case of weight 2 modular forms\, where they can be related to the gr
owth of $p$-Selmer groups and decompositions of the $p$-adic $L$-function.
At higher weights\, less is known. By constructing certain lifts to the f
ull Iwasawa algebra\, we compute the Iwasawa invariants of Mazur-Tate elem
ents for higher weight modular forms with $a_p=0$ in terms of the plus/min
us invariants of the $p$-adic $L$-function. Combined with results of Polla
ck-Weston\, this forces a relation between the plus/minus invariants at we
ights 2 and $p+1$.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yongxiong Li (Tsinghua University\, Beijing)
DTSTART:20220422T143000Z
DTEND:20220422T153000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/8
DESCRIPTION:Title: On some arithmetic of Satge curves\nby Yongxiong Li (Tsinghua
University\, Beijing) as part of Algebra and Number Theory Seminars at Un
iversité Laval\n\n\nAbstract\nLet n>2 be a cube free integer\, we conside
r the elliptic 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.K
ezuka.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:John Bergdall (Bryn Mawr College)
DTSTART:20220318T143000Z
DTEND:20220318T153000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/9
DESCRIPTION:Title: Recent investigations of L-invariants of modular forms.\nby J
ohn Bergdall (Bryn Mawr College) as part of Algebra and Number Theory Semi
nars at Université Laval\n\n\nAbstract\nIn this talk I will explain new r
esearch on L-invariants of modular forms\, including ongoing joint work wi
th Robert Pollack. L-invariants\, which are p-adic invariants of modular f
orms\, were discovered in the 1980's\, by Mazur\, Tate\, and Teitelbaum\,
who were formulating a p-adic analogue of Birch and Swinnerton-Dyer's conj
ecture 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\,Qp)\, p-adic families of modular forms\, Coleman integration on the
p-adic upper half-plane\, and Fontaine's p-adic Hodge theory for Galois re
presentations. In this talk I will focus on recent numerical and statistic
al investigations of these L-invariants\, which touch on at least four of
the theories just mentioned. I will try to put everything into the overall
context of practical questions in the theory of automorphic forms and Gal
ois representations\, keeping everything as concrete as possible\, and exp
lain what the future holds.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chris Williams (University of Warwick)
DTSTART:20220121T153000Z
DTEND:20220121T163000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/10
DESCRIPTION:Title: p-adic L-fucntions for GL(3)\nby Chris Williams (University
of Warwick) as part of Algebra and Number Theory Seminars at Université L
aval\n\n\nAbstract\nLet $\\pi$ be a p-ordinary cohomological cuspidal auto
morphic representation of $GL(n\,A_Q)$. A conjecture of Coates--Perrin-Rio
u predicts that the (twisted) critical values of its L-function $L(\\pi x\
\chi\,s)$\, for Dirichlet characters $\\chi$ of p-power conductor\, satisf
y systematic congruence properties modulo powers of p\, captured in the ex
istence of a p-adic L-function. For n = 1\,2 this conjecture has been know
n for decades\, but for n > 2 it is known only in special cases\, e.g. sym
metric squares of modular forms\; and in all previously known cases\, \\pi
is a functorial transfer via a proper subgroup of GL(n). In this talk\, I
will explain what a p-adic L-function is\, state the conjecture more prec
isely\, and then describe recent joint work with David Loeffler\, in which
we prove this conjecture for n=3 (without any transfer or self-duality as
sumptions).\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Longo (Universita di Padova)
DTSTART:20220128T153000Z
DTEND:20220128T163000Z
DTSTAMP:20260404T111243Z
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 pa
rt of Algebra and Number Theory Seminars at Université Laval\n\n\nAbstrac
t\nI plan to present a work in progress\, in collaboration with Stefano Vi
gni\, in which we study the equivariant Tamagawa number conjecture\, formu
lated by Bloch-Kato\, in the case of motives attached to cuspforms. This c
onjecture can be seen as a generalisation to (pre)motives of the (full) Bi
rch and Swinnerton-Dyer conjecture for elliptic curves\, and is still wide
open. The case of motives of modular forms can be studied using methods a
nalogous to those exploited in the case of elliptic curves. After an intro
duction in which I will recall the main results in the case of elliptic cu
rves\, I will discuss our results in the case of motives of modular forms.
\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cameron Franc (McMaster University)
DTSTART:20220204T153000Z
DTEND:20220204T163000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/12
DESCRIPTION:Title: Characters of VOAs and modular forms\nby Cameron Franc (McMa
ster University) as part of Algebra and Number Theory Seminars at Universi
té Laval\n\n\nAbstract\nModular forms have appeared throughout the repres
entation theory of vertex operator algebras (VOAs) from the very beginning
of the subject\, for example\, via the study of VOAs modeled on represent
ations of infinite dimensional lie algebras\, as well as spectacular examp
les such as the monster module. In this talk we will explain how the theor
y of modular forms can be used to study representations of VOAs\, in a sim
ilar way to how character tables can aid the study of representation theor
y of finite groups. No prior knowledge of VOA theory will be assumed.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Somnath Jha (IIT Kanapur)
DTSTART:20220401T143000Z
DTEND:20220401T153000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/13
DESCRIPTION:Title: Fine Selmer group of elliptic curves\nby Somnath Jha (IIT Ka
napur) as part of Algebra and Number Theory Seminars at Université Laval\
n\n\nAbstract\nThe (p-infinity) fine Selmer group (also called the 0-Selme
r 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 Iwa
sawa cohomology groups. Coates-Sujatha observed that the structure of the
fine Selmer group over the cyclotomic Z_p extension of a number field K is
intricately related to Iwasawa's \\mu-invariant vanishing conjecture on t
he growth of p-part of the ideal class group of K in the cyclotomic tower.
In this talk\, we will discuss the structure and properties of the fine S
elmer group over certain p-adic Lie extensions of global fields. This talk
is based on joint work with Sohan Ghosh and Sudhanshu Shekhar.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cornelius Greither (Universität der Bundeswehr München)
DTSTART:20220225T153000Z
DTEND:20220225T163000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/14
DESCRIPTION:Title: An equivalence relation for modules\, and Fitting ideals of clas
s groups\nby Cornelius Greither (Universität der Bundeswehr München)
as part of Algebra and Number Theory Seminars at Université Laval\n\n\nA
bstract\nIt is well known that analytic sources\, like zeta and\nL-functi
ons\, provide information on class groups.\nNot only the order of a class
group but also its\nstructure as a module over a suitable group ring has b
een\nstudied in this way. The strongest imaginable result\nwould be determ
ining class groups\nup to module isomorphism\, but this seems extremely\nd
ifficult. A popular ``best approximation'' consists in\ndetermining the Fi
tting ideal. The prototypical result (we omit all\nhypotheses\, restrictio
ns and embellishments) predicts the Fitting ideal\nof a class group as the
product of a certain ideal $J$ and a so-called\nequivariant L-value $\\om
ega$ in a group ring. The element $\\omega$\ngenerates a principal ideal\,
but its description is analytic\nand complicated. On the other hand\, the
ideal $J$ is usually far from \nprincipal but has a much more elementary
description. -- In this talk we intend to\ndescribe a few recent results o
f this kind\, and we explain\na new concept of ``equivalence'' of modules.
This leads\, ideally\, to\na finer description of the class groups a prio
ri 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://stable.researchseminars.org/talk/ANTULaval/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marc Masdeu (Universitat Autònoma de Barcelona)
DTSTART:20220408T143000Z
DTEND:20220408T153000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/15
DESCRIPTION:Title: Numerical experiments with plectic Darmon points\nby Marc Ma
sdeu (Universitat Autònoma de Barcelona) as part of Algebra and Number Th
eory Seminars at Université Laval\n\n\nAbstract\nLet E/F be an elliptic c
urve defined 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 (o
r the more general Darmon points) to produce (at least conjecturally) a no
ntorsion generator of E(K). If the analytic rank of E(K) is larger than on
e\, the problem of constructing algebraic points is still very open. In re
cent work\, Michele Fornea and Lennart Gehrmann have introduced certain p-
adic quantities that may be conjecturally related to the existence of thes
e points. In this talk I will explain their construction\, and illustrate
with some numerical experiments some support for their conjecture. This is
joint work with Michele Fornea and Xevi Guitart.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ari Shnidman (Hebrew University of Jerusalem)
DTSTART:20220211T153000Z
DTEND:20220211T163000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/16
DESCRIPTION:Title: Manin-Drinfeld cycles and L-functions\nby Ari Shnidman (Hebr
ew University of Jerusalem) as part of Algebra and Number Theory Seminars
at Université 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 repr
esentation for PGL_2 over a function field to an intersection pairing of t
wo special algebraic cycles in a moduli space of shtukas. The proof\, whi
ch I will try to sketch\, is via the geometric relative trace formula of J
acquet-Yun-Zhang. The formula leads to interesting questions about Manin-D
rinfeld cycles\, which are generalizations of the cusps on modular curves\
, as I will explain.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christian Wuthrich (University of Nottingham)
DTSTART:20220325T143000Z
DTEND:20220325T153000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/17
DESCRIPTION:Title: The denominator of twisted $L$-values of elliptic curves\nby
Christian Wuthrich (University of Nottingham) as part of Algebra and Numb
er Theory Seminars at Université Laval\n\n\nAbstract\nIn the context of t
he generalised Birch and Swinnerton-Dyer conjecture\, one considers the va
lue at $s=1$ of the L-function of an elliptic curve $E/\\mathbb{Q}$ twiste
d by a Dirichlet character $\\chi$. When normalised with a period\, one ob
tains an algebraic number $\\mathscr{L}(E\,\\chi)$. I will discuss the que
stion under what conditions $\\mathscr{L}(E\,\\chi)$ is an algebraic integ
er and what the possible denominators could be.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mohamed Mahmoud Chems-Eddin (Sidi Mohamed Ben Abdellah University\
, Fez)
DTSTART:20220304T153000Z
DTEND:20220304T163000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/18
DESCRIPTION:Title: Unit Groups and 2-Class Field Towers\; Techniques and Computatio
ns\nby Mohamed Mahmoud Chems-Eddin (Sidi Mohamed Ben Abdellah Universi
ty\, Fez) as part of Algebra and Number Theory Seminars at Université Lav
al\n\n\nAbstract\nDuring this talk we are going to expose some techniques
for computing the unit groups of multiquadratic number fields. Furthermor
e\, 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
$q$ are two prime numbers. In the second part this talk\, we shall use un
its to study the $2$-class field tower of some imaginary biquadratic numb
er fields.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Samuel (Université Laval)
DTSTART:20220315T203000Z
DTEND:20220315T213000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/19
DESCRIPTION:Title: An investigation of local zeta functions of self-similar fractal
strings\nby David Samuel (Université Laval) as part of Algebra and N
umber Theory Seminars at Université Laval\n\n\nAbstract\nWe give an overv
iew of fractal strings and examine the relationship between their Minkowsk
i dimension/content to their complex dimensions and their geometric zeta f
unctions with the aim of demonstrating the geometric information made avai
lable by studying these entities. Building on this knowledge\, we propose
a way of searching for locally defined geometric zeta functions by looking
at simple examples of self-similar fractal strings.\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felix Baril Boudreau (postdoc at U. of Lethbridge)
DTSTART:20230127T210000Z
DTEND:20230127T220000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/20
DESCRIPTION:Title: Fonctions L de courbes elliptiques en caractéristique positive
(Partie I: Rationalité et algorithme de Schoof))\nby Felix Baril Boud
reau (postdoc at U. of Lethbridge) as part of Algebra and Number Theory Se
minars at Université Laval\n\n\nAbstract\nL’hypothèse de Riemann et la
conjecture de Birch et Swinnerton-Dyer (BSD) sont de célèbres problème
s non résolus en théorie des nombres dans le contexte des corps de nombr
es (extensions finies de Q). Du côté des corps de fonctions (extensions
finies de Fq(t))\, Weil (1949) formula\, et démontra dans certain cas\, d
es conjectures portant sur les fonctions zêta de variétés projectives l
isses définies sur F_q . Ces conjectures portaient entre autres choses su
r la rationalité des fonctions zêta et sur une propriété analogue à l
’hypothèse de Riemann qu’elles vérifiaient. Les conjectures de Weil
furent généralisées à certaines fonctions L (dont les fonctions zêtas
en sont un exemple)\, et démontrées entre 1960 et 1980\, principalement
par Dwork\, Grothendieck\, Artin et Deligne.\n\nMalgré cet énorme succ
ès\, ces fonctions L ne sont pas encore complètement bien comprises. Par
exemple\, il est difficile de les calculer en pratique. De plus\, l’ana
logue de la conjecture de BSD pour une courbe elliptique définie sur un c
orps de fonctions n’est pas résolu en général.\n\nDans ce premier exp
osé de deux\, nous esquisserons une preuve de la rationalité de la fonct
ion zêta 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 da
ns F_q vérifiant une équation de la forme y^2 = x^3 + Ax + B sur F_q . E
ssayer un à un les points (x\, y) vérifiant cette équation est peu effi
cace lorsque F_q est grand. Comme a_q est un entier\, nous pouvons tenter
de calculer directement sa réduction modulo un nombre suffisamment de pet
its premiers et ensuite reconstruire aq grâce au théorème chinois. Cett
e idée est la base de l’algorithme développé par Schoof (1985)\, dont
nous parlerons brièvement. Enfin\, nous conclurons cette présentation p
ar une esquisse de preuve de la rationalité de la fonction L d’une cour
be elliptique E/K définie au-dessus d’un corps de fonctions K. Ce premi
er exposé ne contient aucun résultat nouveau. Il prépare cependant le t
errain pour le second exposé. Ce dernier portera sur des contributions or
iginales du conférencier à l’étude analogue mais plus complexe de la
réduction (du numérateur) de la fonction L de E/K modulo des entiers pre
miers à q.\n\n*the talk will be in French\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felix Baril Boudreau (U. of Lethbridge)
DTSTART:20230411T193000Z
DTEND:20230411T203000Z
DTSTAMP:20260404T111243Z
UID:ANTULaval/21
DESCRIPTION:Title: L-Functions of Elliptic Curves in Positive Characteristic (Part
II : Studying L-Functions of Elliptic Curves over Function Fields via thei
r Reduction Modulo Integers)\nby Felix Baril Boudreau (U. of Lethbridg
e) as part of Algebra and Number Theory Seminars at Université Laval\n\n\
nAbstract\nElliptic curves are a central object of study in number theory.
In this talk\, we focus on those defined over function fields and with no
nconstant j-invariant. The L-function of such an elliptic curve E/K is pol
ynomial with integer coefficients.\n\nInspired by Schoof's algorithm\, we
study the reduction modulo integers of the L-function. More precisely\, wh
en E(K) has nontrivial N-torsion\, we give formulas for the reductions mod
ulo 2 and N for any quadratic twist of E/K. This generalizes a formula obt
ained by Chris Hall for E/K. We give examples where we can compute the glo
bal root number of the quadratic twists\, the order of vanishing of the L-
function at a special value and even the whole L-function from these reduc
tions. However\, the group E(K) is finitely generated and in particular ha
s finite torsion. Time permiting\, we discuss some of our work in progress
in this situation. More precisely\, given a prime ell different from char
(K)\, we provide\, in absence of nontrivial ell-torsion and in a quite gen
eral context\, expressions for the reduction modulo ell of the L-function.
\n\nThe talk will be given in English\n
LOCATION:https://stable.researchseminars.org/talk/ANTULaval/21/
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