3/4 Analytic aspects of Cohen-Lenstra heuristics


Formal Metadata

3/4 Analytic aspects of Cohen-Lenstra heuristics
Title of Series
Number of Parts
Fouvry, Etienne
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
Institut des Hautes Études Scientifiques (IHÉS)
Release Date

Content Metadata

Subject Area
Multiplication sign Valuation (algebra) Cohen's kappa Analytic set Power (physics) Number Prime ideal Roundness (object) Meeting/Interview Well-formed formula Square number Heuristic Area Product (category theory) Forcing (mathematics) Consistency Theory Total S.A. Bilinear form Functional (mathematics) Number Arithmetic mean Computer animation Uniform space Bounded variation
Interior (topology) Euler angles Multiplication sign Function (mathematics) Fraction (mathematics) Positional notation Meeting/Interview Square number Negative number Chromosomal crossover Amenable group Physical system Product (category theory) Moment (mathematics) Fundamental unit Price index Bilinear form Flow separation Proof theory Category of being Arithmetic mean Symmetry (physics) Equation Combinatorics Right angle Identical particles Bounded variation Fundamental theorem of algebra Resultant Directed graph Design of experiments Computer programming Sequel Divisor Civil engineering Parity (mathematics) Valuation (algebra) Inequality (mathematics) Event horizon Number Prime ideal Congruence subgroup Goodness of fit Helmholtz decomposition Well-formed formula Reduction of order Integer Friction Series (mathematics) Element (mathematics) Counting Set (mathematics) 8 (number) Cartesian coordinate system Local Group Voting Thermal radiation Bijection Family
Musical ensemble State of matter Propositional formula Parameter (computer programming) Student's t-test Food energy Rule of inference Number Power (physics) Frequency Lecture/Conference Well-formed formula Oval Square number Boundary value problem Integer Series (mathematics) Focus (optics) Product (category theory) Projective plane Exponentiation Moment (mathematics) Basis (linear algebra) Fundamental unit Infinity Set (mathematics) Price index Mathematical model Incidence algebra Functional (mathematics) Bilinear form Sequence Field extension Summation Arithmetic mean Logic Hausdorff dimension Order (biology) Universe (mathematics) Object (grammar) Bounded variation Annihilator (ring theory)
Point (geometry) Classical physics Multiplication sign Real number Sheaf (mathematics) Propositional formula Rule of inference Number Inclusion map Root Meeting/Interview Helmholtz decomposition Well-formed formula Square number Flag Integer Descriptive statistics Area Series (mathematics) Product (category theory) Exponentiation Expression Moment (mathematics) Analytic set Incidence algebra Line (geometry) Bilinear form Field extension Category of being Summation Order (biology) Game theory Resultant Sinc function
Series (mathematics) Computer programming Musical ensemble Product (category theory) Process (computing) Structural load Prisoner's dilemma Equaliser (mathematics) Basis (linear algebra) Infinity Mereology Theory Maxima and minima Causality Many-sorted logic Lecture/Conference Meeting/Interview Well-formed formula Green's function Position operator Physical system
Computer programming Group action Variety (linguistics) 1 (number) Propositional formula Food energy Theory Fraction (mathematics) Frequency Lecture/Conference Meeting/Interview Forest Square number Area Series (mathematics) Uniqueness quantification Moment (mathematics) Element (mathematics) Expression Fundamental unit Thermal expansion Field extension Arithmetic mean Shooting method Algebraic number Directed graph
as and then we will and so on and so is just
some cute but I think that this election will be the only thing is variation on the force so I recalled what we proved that we wouldn't approve the use of course so I defined the top area also the seems to the problem the company was confusing and with most of the things so this is the definition of a very important function so we proved that if I did know about the fundamental feeling and so we proves that the probability that immediately and the usage of the form of it's an oral drug rupees according to all is probably courtroom 2 Falwell minus Pesquera it that to it up payable to minors To this fall in daily quoted 0 1 2 and so on and so on was the same formula is equal to 2 miners have 1 because the need of crude yes it kind of threw it to the press 1 or 2 and this is divided by just so you to remember this when you have this 1 you replace 1 by evil and UAVs use it once OK so I want to make some of valuation this for me so it's something that I would say nothing new will it's a byproduct of work proved risky may prove recently announced for the 1st 1 windy at around the speed during Jets so we TCU's offensive production is due to the shots around ICH maybe their own so it's really says that before they make do you regret and square the offense square so I do not use a capital and later it's not maybe not the fundamental discrimination but you will be clever enough to give meaning to what I said was of CD is less of a court was surrounded by sea mine is the history court to Xerox oppressed black so that means so you know when I went to you you consider the future looks great with my did you consider is a fundamental and which is a fundamental instrument and associated so it is stand and see off but needed of imaginary and quality the 2 of us we want larger than the positive corresponding won by speedier by mere phenomenon so in that it was so it is Adger 0 and bending their around in their the condemn and the young Moses a sinful the 4th round so same statement so you up the for Arkhipov see deemed necessary .period uniforms of the many the many quarterly foreign could you persuade he became sold could Fall Rock and sedentary consistency about Woods so there is to applications so that I want to stress that there was a time in which we proved the recently announced it means that records that for for instance that's PGE mediating the government 1 that followed the total power over foreign policy toward the 1 investor number of Prime of the 10 years so you consider also the composition and the
product of fall at the commission voted so I records this season would say generated from so what companies with his 3 years ago the Olivier ended obsidian given you pull new proof of identity beyond at year-end using India as is for me using this formula and the and the violence for me it's it's impressed by this proof because it is you you you might you must as existing breeze you proved that this number will you see proof that this is an integer just foreign and then prove it is supposed to procedure to you and they look at the expression and political evolved to using only combinatorics resists not trigger home such a nightmare of of breast minus-1 insulin and now there so it is this year up securities especially coming at Oracle is Korea intellectual and the surprise of left so what what we do what we did is the following saying would remain Moscow so which is a joint probability so I'd call is this number so this 1 will be take compliance of all in just 1 day 8 to kind of I so the question is what can you say about but the priority of Sudan news but this 1 is equal to all this 1 equal to all this 1 is equal to uh all ends when to all well so I need a three-goal what probably you can't genus and capital X which has his property and you divide by the number of people the discriminants and 1st of all the exist and they did use of it so that which we prove is the following recently that Iran is the probability for any article he wrote the probability so you just a way of counting of immediately said that he was quality of the tools and for some reason the cost to all that is probably produce it and so it's easy tho not much easier these days to also the globe output beating interior of a factor which is to minors or and of course you can guess which is reportedly the when you have this year 1st won the toss off so it's a compliment which is why I'm commanders to minus all aid to and of course there is another Surin which can it that control when you got a positive ID uniformly on what of the same and the Trojans no so it's a very originally the proof is rather simple some sense because you are you reduce decomposition of success that it made I call a 2 -minus on as the problem such a lot of the event and they need to be in the form of these include all for all of the amenities His according to OK so I can't tell you have to weigh of city of counting using it TiVo was did not here I brother that you all and as ended of course you you you can this which is sold he says only to be considered when a player is the Essen-based press 1 because otherwise it 0 :colon and this will review of a raft of radiation the first one is the voice of it's easy to the heir to minors 0 0 0 is equal to victim and his because of he could just is what that is because it's yes because as the basic satisfies that is a sequel to the role another 1 which could take into account for the dignity of the decomposition of 2 around a this aid to minus the pair minus 1 equals 2 a 2 minds on this 1 also is it's only the decomposition of the bodies said so I foresee for mileage I suppose it is probably too busy city proving so we have understood type of friction when you consider that a compressor OK so but not the fact is that there are no over you have in the systemic you really have regained some of Mero property what I want to express our team that is able Yukon issued arises To many of the seats families so that the citizenry of fundamental instrument because it did not get the government to 1 would ,comma fall much at all the visible light and so now you concede that so it's you can a set of data needed to complement to what would follow the same thing you multiply it by miners fall so after cleaning the summer you arrive in the set of positive the NZZ did he want to go on mud 8th and then you might think I'm thinking by fall by useful and you arrive at the indignities the program to 1 watchful certainly
get mural property and for instance you these properties that base to minus all as you want to add to players this I have been known as a seat is taken care of all the quarters please it took us all on Monday according to a press so you of property you see on creation and U.S. engineered systems and you find a solution because you know there's something about you so it's very easy to find the the value of his property In the end the crossover proved you cross over the probably so is the 1st application of the variation on the 1st and the associated with their boats Frere often restate my last session would be about a free or almost free rocking the synergies that exist since we do not know that we have only 1 moment for those who have no we cannot speak of probability such so don't know if they want to speak about the negatives that equation of the form was here know that but are so your question is that is a very clear about the people of home exhibit the most serious Estonia Sechselaeuten ripe upon the problem but the 1st to win easily you know executing but not so need not written so and what is the use of arms is the result more than he is today so I got on so I know that I embraced people wouldn't to what is the reason they're at the center of the inequalities bubbles appear on but so that's why we don't always agree on valuation is maybe more strictly because of rain border on the committee or if you want that means that put we all know that please police and suddenly he was reminded of that means the fundamental unit has no minus 1 so we can make it to the drawing of what happened so what's been the waterways and we positive and from then on endit but we know it is a concern the property ever at 7 in the quarter to 1 so it's not in event a new record you redefined the style called to me all the people this is equivalent to the square mile so it's certainly don't you have some time to take a cue to because here there may be considered a diminutive to also fallen soldiers you can't manage minus 1 and so this is equivalent to to "quotation mark discussed ways to expand it into continued fraction and this is the moment wasn't going all continued fraction is all sources not very all that and you know ,comma at the moment so so what so the audience and so is this implies and so did the basically implies the program to cut the number of American 1 framework and it be carefully people sometimes made 0 that he sees equivalent which is absolutely not to lose a case the end the 1st example so you will not find example for G prime but they series the 1st example is that the number he 205 which is a 5 2 and the 41 and so when you expenditure continued fraction annually 14 sleep on for anyone who seeks create a clean exit character caught this notation so to me and it evolving that means the severity and base is equal to the rental so this is a good attitude number filter of D 1 mostly people 2 the and with their votes yeah this 1 hasn't even been here and nevertheless always Frankfurt talk one-to-one 1 watchful so easy to ours is condo example fickle so eyes are so you can see that I would say united States people usual about a parity of 4 burial of quality call during more than 2 bright membership and the usual property a record to the properties that this thing is above all of that and here you have always had the symmetry it so now our continued so I can continue to say that this thing is that we were wondering if the president walk credible is the same as ordinary credit and we know that this is just group the ordinary just rupees de-facto group of the narrow present group we index when all 2 so that means this secret means that for border trade it was a to year-round called city the CIA others say OK so after we know will focus on characters and 1 NO yes we a sprint to use that in a way that is fair cookbook when these 2 groups are not the same as it necessary yes it is a Toronto always the same when or primary residence of India congruent 1 with for and so we can find the right or generator could too 2 rocket user to talk of civility and it's
easy to see seized since this the sequence of integer decreasing if you are in the form of a is equal to 0 this implies the end you up or phenomenal usage so it's what will because we have to the full of so now we have contained so I shall write index S 2 record fishermen went to occur especially discrimination and the special when did I did people grounds that not congruent to stream it's exactly the same thing as to say that it is the sum of 2 square and we have we produce joke accounting function which on the basis of banks which sees is the set of the the lists and X the OK then the best of banks which as the state of the art to minors d of the press of X suggests the normal residence on the not did to 1 so the question is whether we have you have to settle for for different in the case of faith in you is that the miners s affects at least the number it is the other man was on so we can use it many exceptions so if the question is all what is ideologues it's got competitors were so if you are it's a very nice here said or it's a question of party I would say it's 50 50 because it's huge tuition fees means an awful but this summer than of use not very interesting as thing is is very naive and work we consider recent what's a very great respect to conjecture but to know what to recourse something that seemed to work he said the best of banks in the center of the 2 fuel tanks will go projects this season the size most evil the boat and functions or the prime Levis a onto 1 waterfall sold by dimension when I've said that you must know also that this is set at this and next week 1 1 1 4 it including thing the best of the best the best minus 1 is very easy for fork all in sticky so that means he should be congruent to 1 fall because just a few Jews screwed the the fundamental unit has not minus 1 exist :colon energy something like eggs of obligates came and know what I want to say used to say that there is yes so is a contigent Stephen again and what is the to close of the case so he did so it defines the under-fire to be the product one-liners the the money 1 forging greater than 1 core 80 years since the value of this seems about so if is a very sensitive cleared a probabilistic model based on what happens in Itaborai . 0 euros this question in the sense that there is a contradiction also the base of a minus is the sum of the 2 one-liners on fire the base of thinks so that means that we 2 conjecture that means that about 50 some both special of and it certainly was minus 1 pixel 58 this so this number has to do with my data functional case it Darchinyan complement those of us who you sober what we proved so what was the situation it that the situation was that always a 2003 will filter knows goes something like that so what I say to you is that the case is that there an excellent objects using my remarks about this style and Brohman the decrees exponent of substantive details that except with logic it's 1 of the most summed it up but it can be used and that was the encompassed by what I see that no 1 was no who don't one-liners that so intelligently 3 1 so this was a member and crew what we pooled resources at the 1st of foolish of the issues in this yes so we think criminals have induced from informal hearings this year and he has to say is that the company to the so is greater than the unfair miners smaller pool of 1 the best but takes the place of we prove that at least 42 persons the 2 candidates I in the fundamental unitary snow minus 1 focus square on decreases since we have a fight when my father and after Osasuna because Europe 1 in Surin too we really proved to be less than do so through the small 1 yes so it's absolutely clear on we still again student argument congestion OK so approves approval would say 3 he's not and do
you already always a tourist because that is the is especially the city so I some formula here a series of clues Rolison bone and there's a computer is supposed to know and in the case of the From the I suppose that this is going to to wonder what followed my for taste to follow their fault the president would want to to the diesel is the same formula assist BJ so you composed in him and what happens if indeed is not special you of some U.S. some extract their reliance this 1 it but seems that if you suppose that the special it especially the role arose out as always Frankfurt the ,comma 1 what fall below so this 1 is equal to 1 0 you some could this product to reverse products here you use them as a reciprocity will in you see that is or with 1 so in that case the former is much simpler than in it's on me the year rules for the year of the 2 were so it is much more simple and I sprained you would take out to raise that to some K power using the cold cold at nite and there's of basic due to post to carry ended with 3 arriving with the euro as a song about which is absolutely it was Cimmino about approved with another proposition so when you consider the case also means sustained ended the fundamental discriminants shall obtain this is as undertake to take so we're so you're constantly and here you have it's not there is a dimension of the number of auditors subspecies of 2 cases most important these 1 prior to J from India equals 2 0 0 implement 1 cancel you of this proposition which says that we know awesome amounts NO explained to you in my furniture out to go from moments probability Andrea Richard B. because this fact all does not go too quickly to infinity asking created and I probably 2 years into which we know is that it's the probability the register with probability of what it is that especially discrimination the subject area foreign on the courtroom what I caught on fire incident Of all were on fire engines of holidays according to our father Stephen LeGrand constant approved to Jerry minus 1 from quarter 1 cool off the means to give particpated some variation about what to expect to be supporting the newly next to 1 of the same and Europe Saturday and after you have you know so as I would say it's an extension of women's inside going I not to announce the proposal conjecture .period because it was inevitability visiting our case and the OK from now In it's finished because the probability of but as as a probability of that unflinching of the World Music Award cases that means I always letter write I just your OK so it should use appropriate it because 1 OK so now receives is done around so that means we proves that the parties to 41 per cent of animal of the end so now what can you say know about the about so far as the upper bound reuse and as today yeah if so you see when they use this fight I say nothing special about this all the narrative so it's the order not just ordinary corporate draws rule is mall dedicated on for me dozens now requires so to give another boundary implications that means that therefore since the course to the fall of Saddam but the question is always too the this is a quality guy always able to do that and gave her question the probability of unreadable so again really use a very classy code the In the committee subsidiary classical I must absolutely right to the name of or justifiably the crossed the above all I I've shown in 4 and even if went back to the university for some special so is peopled with green around 1968 1948 and the as you remember when you were my criterium was doing we got it for "quotation mark from ended in fact read decomposing the number 9 in the the former GE wanted to and we wanted a compliment each square reminded the one-way where the capacity of the Reynoso signed orders a nontrivial solution Sodium if today the spoke you knew that of course then you spoke about the composition of the 1st and now with the composer netminders with the composer and do the Argenbright number Suri so fashion reasons because I was think so the fact that the flood that's
his Serie a and we spend a lot of time to put forward the idea is in the context of the use immunity and combustion that made to arrive at the end with character workers so the 1st thing is to say results so we of the combination of of consumed on you and and we could not so the idea is that 1st of all we really need to remember the more flexible former Real Mode deep inside is a formula for 2 so is is the 3rd is this number counting findings is 1 that is equivalent to continue contained the city for extension all K equals to scrutinize and ecologists extension and I most of the property which along the enemy lines so he following the this shows that result from human OK so you will want to get a particular group of and then you have chief fall in here it's a C for extension you what we could keep to the extension of the 3 2 we want this to be the seafloor extension under me fight so I we used to report and which is at the center achieve similar billion much humor but the extension the from the fight at final and what of exponent some people so it was a result the property when you but you have asserted that around with his property of folks you know is that went for you will find this too few new from in the compositions the 2 the Commission returned ruled yes so yeah when you have to be careful about the capacity you with the composition of the D & G 1 and G 2 and you have is that there are like by composition and what happened is that the G 1 when you have the key for the 1 is a square would you object to and the curious square maturity want so I forgot to tell you because I know I want to insist that to be sure that the special sums from a section of so if you refinance you find again just as is decomposition incident dive reciprocally specifically was could we do so when you when you start from a lot of instability of the decomposition of the can create Vijay fall the explicitly that means you you will take careful to be tailored to the square root of fossil this is not a Stephen Mangan and Fox were on fire is defined by a Express was promoted to the 1 in bakes and wipe out precise solution of the question it's square miles 1 worst score mind the 2 was that square costly misstated his the game is when you are out when you do when you have achieved following Loverdi went to when you have a G 1 2 2 2 coalition of that the 2nd time you can be for so you can or should me so what the troubled what are you using your time since he's more simple To do it we square the former stated last so it which is really Dera seeing that no if you want to come all generic classical Yukon Abkhazia which up to 10 years case you you you pass from ordinary interest to know narrow present ordinary guzzled by imposing too careful to be subsidiary can and since we have an exact description of this shit and it's a property that we can follow and he is it's at the point that we will them going to order paper produce and so what is the problem of the whole the Ukraine so on and so on so it since we can follow what we are doing we have a very nice the properties which is quite good for analytic number series so I defined anew the new I you symbolizes phone number so that I so I define a vessel for prior to called tools 1 KPK equal to 1 of .period much of the minus 1 the a Peterson who us still there is a to 1 and a please not to fall the rule was a so if it is going to square OK and I would .period straight for people still studios at the finish and its chooses to give
it its who was boring encumbrances and would you want would 16 of songs and you defined after us hope for the riddles and 1 you defined paid before be the product of a piece I knew I had time he should be is about to befall uh committed to and so so we follow the flags which is key for us to be deployed areas in that uses this proposition so you can't could say that fall especially we know that to follow their fall asleep the this quarter 1 off the collegiality of taking the and into hands of the so you decompose like that and you remember we were asking for it to be a square on would be used to be a square more year we read the mall the extensive ABC equal to it is equal to 1 all a fault would be in the 4th quarter to mind 1 so we have the earlier Peter finds out we have a way to detect the it's a fact which is the value of foreign calls at all the Americas the Samoa epistles assessment because of to detect the false lowered to produced some characters about the reading of the inclusion in the integer of something like that of member some character which have for which an achievement that can work .period we have some of the Nunavut fish stew involved character and we have also what they called Dolores additional uses the situation is very similar but the way to know the question is we would like to the consensus is that we want to do together the probability that is a form of the disease called all then usually when we got to city and the need to be accord to best so if we were working with them but we have to work increases the Dublin moments to something that 2 OK 3 of the crew following the fall of the evidence but it would be a nightmare the enactment of a coup but since I know that the best thing for and to conceal their faces to all all extras equal to all minus 1 in fact we have to come on top of me Miss able also what is the real chance when no no and no
minimum on to compute
so for on-site supposedly belonging to the S X band we are be able to we find that this is a center the dollar minus 1 plus 1 independent product the request was to so it's a surety he has been extended until then there is no ordinary uses the music theory of the infinite system then we can reduce the value of all is the probability is that Ford had been auditing systems especially in such of for of CDC he is equal to "quotation mark too they put in the position of the Toronto green silence in the courtroom but to minus probability so what they call the not so much as a probability that there is a need them to In this sense a since people who can so we knew out so we have to win the question afforded a costly basis so the to Robert approved for OMX Article 2 and so this is probability and to minus so admitted Wednesday the increase there it uses to fall around but frequently equal OK so we knew what we have an exact formula falls out of course and then it's only I will teach you so you rides out parts you have to deal with this summer From the aid of course to a 0 0 2 so the foreign ghostly the court Tuesday for Uncle Sam case according to a canceled you some predict needs to with that same miracle you find at the end to set pieces that you sue the book for so this 1 can be this 1 is on the obtained by considering the forum of CD is equal to the role and this 1 on that is to equality of the 2 4 the you which is in the job so you may be surprised at this year when you single this fall OK this is equivalent to the is the quantity of the 2 K wrongful Borges and was only 4 K cause of the 2 and we are we are absolutely not ridiculous theory 42 per cent and sister she says because 1 because this series agency is very quickly going there to win you take to Wallstreet down you on more sort this was so I wasn't here so the that .period question and after so here we have a program that will provide its original suspicions of sound so we wouldn't wouldn't dream all out to all too understands that a strong by Sunday we do not know precisely the uh what would be what could prison chief subject we controllers a trunk of studio even in the only in the case of full indiscriminately and specialists remain on which the most important so no the yes so what tests what devoted to try with load story of very those
1st of all we would like to have created on wasn't as nice as he swung we understand this is history variety of this is an edited mystery in reality unity of people adjusting the government-controlled element so we are only a proposition if door so I supposedly special and so then I suppose and said that is for all of the important 1 because they imposed be the case then the fall the aid from custody it is equal to 1 also needs be it is very near from that expression the cardinality so as usual you decomposed the In the phone G 1 2 and you do it that what the what it equal to need 21 for the quarter to 1 Greece and only that's great through and some sense it is a minute off of this it should be he is expected to be the of so a coffee member we must be careful ask but none of these costs to so will knew of his artistic or yes we can do rule maybe because using the character to detect Commission and on means that cares for and they do you that is the implication so always so moments and probably clearly since then what action you that in mind is that the government took the special discriminants subject to is a fundamental unit us no 1 so see I come back here and there but if I use air fall see he told 1 area it is equal to that for the policy and the so for ongoing side enters the stage shrugged of society is peopled with you if you are in this situation is annual show that you will and then you produce new example of decent that is ceremonies according to you know minus 1 cancel is so so I write for the dead this is going to be the role of the US government and the so of course isn't it it's the energy so we control we knows is probably the and this 1 we can incorporated by the cafeteria I Iowa he those seeds and sent revenue at the moment we are not working research at the moment of all those 3 straight down and we have some of the the users to be so what is of the story is story is that exists here and that the U.S. minus state please with that 5 out of the miners went off 1 of its effects OK so you remember I had onslaught which is being in the "quotation mark inserting the discussion about the state-run I 25 per cent so what is very important for me is that it isn't just number unique or 2 something like you and identify sources :colon elegant called to the company is also case in the office so this is under fire is is 1 of the shooting she's not advocacy 1 minus far year's fight on far so Forest with tenacity .period "quotation mark for people were following where we have more than 1
also needs the receipts forced to see seeing this evidence to periodic of it's a period Of the square of the expansion continued fraction of screen whatever the reason the special off 50 she she faces of and Stevena agonies I believe each school and of course we see you are losing seniors but not to this difficult but you really you may be noted 2 of new improved slightly as a constant with a lot of problems but it's because of the series quickly b we still have this is not it that some really you the government rocking about that assistants vicinity No 16 and over what he said that I want I like is that the Conference of algebraic number theory and an energy and reality because can to see if you say that some extension easier billion that so come if program becomes all in all of which thank was


  605 ms - page object


AV-Portal 3.9.2 (c7d7a940c57b22d0bc6d7f70d6f13fde2ef2d4b8)