2/4 Analytic aspects of Cohen-Lenstra heuristics


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2/4 Analytic aspects of Cohen-Lenstra heuristics
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Fouvry, Etienne
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Institut des Hautes Études Scientifiques (IHÉS)
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this so so I I recorded what what they used to being in the lower good and insoluble so prime victims 3 in we would like to to prove that what they called C seeks to refer to original work of Clinton's long then see tend not to so what I always used so the conjecture it should be for all prime greater than 3 Paul K. equals 1 2 OK and the U.S. this is a fundamental discriminants Of said the user narrow present rules if you confuse it with all generic drugs will be it is not drum up because explains that it is African treated the same and so early in the week although this is a sensitive to KDP OK 1 and 2 in the the case press 1 PM miners and keeping people working a case In conjectural and so what is entered pleas is not the original version it's only equivalent form and KP is the continuity of this set of birds zillionaire orbital space is already you know we tossed this season the people I work with them in case of any dimensions so this is when you have at the moment and explained to you we can get the probabilities of Zachariah said it appeared to be caught to RUC officers which recorded serial for months Seoul with the 1st of all it is so what opponents of history so it is something very very important has seen in the influence of gifts around so good around this here said that the sequence ,comma we see beginning later this and 3 because there it's not a program to deal with the 2 uncles of classical of your spirit of the because Bigos history we know that it's the number of times adviser -minus so the ideal Gale said In fact we can it's a great aside heuristic to the people still but instead of continuing CDU consider square good so when does that mean that means that no Shakespeare in some command you put the and 2 see Square is a square and the same computers and a center takes wife because it it's easy to see that the bill is greater according to 3 is appear Ronca of city Square is Iraq of CDs and so what is that all wrong of city the square accused which we call a fork the forum and that the issue right to the source of foreign countries if you have a finite abelian group is the dimension of a it's too often dimensional of if people what they what it is and it used to take over for a Eleanor's awaits if you arrives at police a billion group as the product of a all prime From new cause want to introduce the product of cycling groups misses the frequency of a new zones of fork it's a Is exponent of fire to new from you if forced to go to To infinity so worried because I receive audacious extension of its associates of course we do not know the drunk is blocked by goes but we can say something about a foreign entities and he was right with the explained so firm if so of course you you you will replace it's what I did hear you take no for digital and you have in mind at the cost of 2 political to the floor OK so what was remarkable for India to the world isn't approved something which was really is the source of the problem sold in opposing team so his sister 6 correspond to the immediate here is a 1 is for the police and
I recall a formula that you already met is that the probability for instance the fall in the negative polarity fall the Getty the subject the form is according to our users to follow minus their square that it does not mention the of 2 is a product of 100 2 an escape dollar managed to from Katie cost 120 officers so we knew of his 6 EUR that's where I fall the and as a function of internationally to the product 1 minus 2 minus J of the apology and the EU From cost 1 to ingenuity and suppose that didn't this function that you unimaginative and get the structure which is which has a strong increase partition function and so on and so on so the Union factor in recent agreements and needs some time ago we we saw a cordless or contractually the cases being forced to reasons of floor as a description was a foreign tho is it he's known and exactly through
the prediction of when and how and what yeah so what did give prove so not only did the intuition of extended medical something which is very attractive so it's around I see him I can maybe on loans is still so it proved so you take a but I would say it was strange way of counting you can't indeed place outside people needed to be for the most and negative the foreign custody he's a quarter to honor concerned that the number of prime devices is according to change icons incentives Brook is the number of Prime divisive which as I can't count the prime so now In devised by the carbon energy of the it subject only to have the is equal to change plans season and nowhere to eat at it he takes a double made things take any day the inmates the number of from devise go changing BC economics 1st of 14 measure the size of the candidate and then
making Brooks the number of primary either and that the number of either goes to in which is very encouraging that he found that history made is that he's exactly 2 minutes this square attention to me was a product of 100 to industry or square so is the 1st time I think about going literacy literacy is assumed I would think crazies is very difficult function and the infighting proves that but he has not moments he has no 1 wants and what users is something we as a theory of how they mattresses for these are authorities that they this so it's something which is around 19 city so of rather ordinary and what is already so you very very vigorous so it's wrong so you consider it sold by simply vital I forgot to tell you that you have the same sinful positive that D so you the course you wanted to hear us as Virginia new considers cemeteries and which has a demand in G & M the according through the original symbolic duty over the Fiji up so I don't go into the ballot whether Bonds about 2 to sort review and very vaguely sticking with the so that they Attilio of today matrices we know is that the all since the business I don't know if it is to address 1 of the miners the wrong other indignities peacefully over you eulogy that face which is not exactly symmetry should because of the use of cancers after With the procedural vote at the meeting with no vote is uncertain but that's uh yes so it was a very positive see receipts he said his former not but we can say that it is a strange way of counting missing is like continue to remember even Nixon's talk about 8 should do you have a natural way which is according to the size of the and and Yukon comes according to the the events of June year in the counts about some cynicism it is the size of the mattress all brought to and then the goal to infinite sold the it's so what I want to go to tell you is that about the use of Surin of myself and cleanup which shows that in fact is that 6 seats in the the true and the probability that the forming of COD please equal to all the inquiries and coincided the victory of wedges and management that at greenside is going insult predictions so we can say that it's in my mind it's impossible to move from a guest 0 and to hear the way of counting so I did use some in salute of the pool yeah so in fact I recorded so that I shall concentrate of the proof of the season means another theorem which is same I wrote it so I ran so that means I should more concentrated of negative the committee is the equivalent independent K entered to cancel and so you face to infinity and I make some comments are the 1st of 4 is so what in fact we really prove that Zeus C-minus these things so the see it a delirious with you if the seniors around something like some consistency noted X and if you go without saying that direction we prove the Everett these capital of all off DEC's very probable winner a mix of so tiny follow minus to minus K press it's time for using Medicaid costs and so Euratom is very smart and something which is very surprising if I write it here you will not be surprised but is appropriate we are surprised that this is true these are some of the if you are the on both sides so to me it's the same since fall the positive but of course to replace by press 1 to 2 minutes and came to be
divided by the 2 publications OK and so would you very surprising that year but in the quizzical the Post said that I can it so it is so is the form 6 type profited from them attendees from produce immediately and then as a power of 2 which would be of no power of 2 as a tool to the borrower to all totals 3 so you can come into the rules when it sold I say it's not surprising that is proving surprising for me it's a miracle I don't understand what happens when you're doing community service the yes sir so OK so now what is the what is the show we say it I must say that we invented the world because implicitly order that I'm going to say during a 10 minute eased it is implicit in many of the papers like Day of course although died and we were always aware chance to put that in the context of McDonald's that an unedited combustion consistent so so I'm going to make some underage numbers you arrange fishing absolutely no revolution so I write it like 1st of all I ride the on the phone P 1 P and may be here if it is to 0 you may not see exponent of unions exponent of 1 to all of us we can and what I consider know is the sole Atkinson the Primedia which of them so that means the 1 key allies the U.S. and we all can be I'm OK and no I considered as a set of currently being which is at the center of the 1 you want puberty at with anyone and was at the time between 0 and 1 so 1st of 4 something which is going Ingalls NGOs that means that if should take a CD of Indian divided by but at the square so you consider this group cushion group it would it's just I write at each class was exactly representative but In this case and so you see our menu of 2 apology dividing tool and you arrive at what they want somewhere above would go somewhere with it but this is seen by some as a way to 2 states so no the Irish I so something which I have no plans to go to there so the amount waterways a fundamental and so the first one the carbon energy units to polish officers there is no is regarded other square square off in class said Jack before his principal and counting amounts NO something which is also Jessica is at a quarter to 1 of the so-called amenities offered the incorrectly said that a newcomer right this is the of the product of an idea and the other file it belongs to the brink of integer and is an idea of what so he it's only 2 to discuss policies is phenomenal Bento state sources he's absolutely no new ways it so I shall introduce a system by which we which is something like the invests in To date a in year danger not the integer rational numbers and you say that it is equal to 1 of 0 used so is the user credited from thick square minors was wondering solutions and 0 was what so which is which is
awarded the so we see when we area appear Satanic "quotation mark lose although Surin some in some menus that will be the case and so forth our characters edition is a forum 1 whole proposition so I say use which did to put in it is expected that away so users of so the total power the according to 1 of the continuity of the positive the square free the advisedly indeed this is symbolic of the mind is the prime is equal to 1 way the prime square free square free In they said b and this is the court the prime see square .period fossil the secret court to 1 so it is very difficult to understand not not very difficult if so what what about all this stuff from the start his own way Due to work on this problem here is that with you to the prior to I remember local and sold only in the context in that context remember 2 is not a prime fully surprised exactly where they so if you made it so there is you another somewhere that may not to Anderson of course but it means that you must have united in this going to wonder what for it is the prime time they did indeed negative selection approved for dignity and decorum to 1 what folks in fact the segment is equivalent to the following scene equal to 1 off the commonality of the said that but the positive the device the and the symbol the -minus over key issues according to 1 yes gives you taken music division the staff of the square is only 4 it is a powerless to so the Israeli just came to know all too to prove through the it's so you must see that there or 1st to organize isn't always easy always in almost all oppose that idea which which I've known we are dealing with the security OK 0 1 off come from and I say this is 1 on which is here NO you played so the To using that we want all Casals approved reduces to a square according to I thought this what I would here defendant if the land due on the the court to 1 so as to direction can be done I taken 1 so I shall do in that direction for so and you start from the ceiling of a lot of easy relations because we also have using those it is edition is shown with which is the symbol of the this is the kind multiplied by you squarely until that has a lot of it he don't of use property and when you do that in that sense so you know please stop from this is equal to 1 so by using tonsil you see that it is ready to be as you know the big question minus 8 miners did was scared miners Jesus square is cerebral in In that means that in fact so that means that in is enormous in Kewaskum whatever the so by jeering than minute those we know that there there and fun said John and fighting in of said that normal 4 out of 5 easy quarter to resist the W square so Sonoma and W belongs to the candidate so after we would this and then you can buy you you can suppose the NAP I thought he doesn't belong to to OK he survived carrying demeanor tells and then use it as a boring to explain and after you finish by using the audience for this user finds ideas we Peace Square I was generated by PC all the squared off an idea of known as Pete so use resident withstood born boring but you have resident your eyes so I suppose that if you arrive at this equality here and use it to fight that the you when he's DU of order representative in coded beat is the opposite direction still another 1 if you want to do is use of direction so you suppose at a square is according to have fun b so we the same reserve said so that means that a normal and far is equal to b known all of this is that I don't even about the squares so that means that so so the known we know is no wonder familiar months so if you take square miners Devora squared minus 1 of these is that squarely solid and so you won't want across the called the money prime so it's so it's rather the classical has arrived the series so with a insist about the we do not invented inventor will but we're energy to portions of erection of quality and that no but very is a consequence of that so so it a lengthy existed because there is cure so that this is a fact of is the primary vote somewhere stores is used XIX exposition to seize square it's always to do with the ball off to so that the schools wanted for the ingenuity of the conglomerate and 1 would fall so busy with up when I say I am counting is isn't the square Friday which are devised the and the prime here is universal but that means that means that that means that is questioner minus you will be the little something so here we renewal a quick carry on with the speaker of the US-appointed look recounted body is OK he referred to was wrong and wrong says that this city is survived by Newfoundland usually will checks assignment and you see as a lookout position so that you it is the same as to be a square with resisting synthesis
Soviets square call so we retain so yes so that the full amount .period discrimination what we have is to assume the people who won all 4 of and off you're right with Canada and Europe using eco-friendly features so you work for I think this is going because Disney getting used to write it and that's all is a square when you indeed the square when you all to music for Use of Ronald to and have a varied histories and severity to define its insurers which which is not written explicitly and there's still a lot although it is at the 2006 dives formula 6 type of screaming and and this 1 is especially the most simplest most important 1 for instance that supposedly this positive ending hunger run to rumor that 8 so we have we have this will so we have a problem with the number 8 so continuity so you ride to the on the form and the equalizer saved a B so you right you use you asked to minus 2 to be the square would you be and B the square you also throws 1 of community so we were the same minus the square Monday and to be In the end and to leave the square when you so you see is what I said to you is right for me to be in the Senate miracle at a very surprised that if I were pooled with know the same formula when you in Seltzer over for the condition when I arrived in the condition and you see with such a mess always agree much stronger as those of the simmered pool repeated Zia if exactly matches to Quentin it's hard guest so now we go more 2 and that is this so so you know so I will every concentrate for the to the government to 1 that follows and so it's very easy and always series you are happy with the same going because he sees that loser on Simba Jacoby rewrite we guess that we really use consolation about a character Solomon so-and-so which recording France and French rule horses as the so what did the work you do years of men mine is so often the case this look we ride so I uses this is the creator all these square when will be and I suppose that a school you need daisies vise is formula which is a product so you can be devised that would be just 1 person a of P so I suppose to be correct that be this was in the military on under and be square free to be in city is equal to 1 was always it is equal to 0 however you this civil case and always use a senior if a user square module please so the day right and must we should look to a program it is equal to 1 so you see when you factorize the from Monterrey discriminants particulate futures order you are sure to have co-prime component because it just square foot soldiers hiss it was created so that what is so so I continues the analysis and I'm concentrate always poses case endit but what I want to say is that with can we read this exact form relied solely on the 1 to follow often made up of the and then we have the line is the costs everybody but above the sun the revised want predicts it will be the end of the product so you see you the compositional of mind is the is positive government program and so it is a square free so you didn't give a rope and you and it's rather simple that you are 2 Omega of diesel throughput put absolute value this and that you're right multiplied line so you say you you expand this product and you grew you remade as the denominator of any devices of being in a device of a you put that together and you obtained that it is something like the cause a 1 8 2 B 1 and B 2 so you read of it wanted it to be divided by B 1 we want to really it once so this is it reviewed the rise of the and so on the complementarities 82 or something and so of course security because some of the Bush would rather put minus 9 minus the this and so you fertilize it on but to be 1 it used to anyone airline of the 1 could be whenever they want so we so you you may say yes and you know that it is reciprocity law but we don't you think for the moment to the end their right on Lake is a fundamental 1st at the equality which is that after some of some well which is that so the moment of although 1 1 of the older 1 which is the sum of 2 polar bears can see all the children take -minus delays and cakes and I hardly going to want what follows is a courtroom 1 one-half so we of a a trip son so stands at 8 1 8 2 B-1 B-2 is is an excellent so fundamentally discriminated OK and here you put the 8 to what would be 1 so the 2 . it
was the 1 the 1 over in India must not forget is a gracious shunt which is 1 of the 2 women of England appeared to be 1 of the castle is a summary state 9 is the full complement and then follows a decade in liquid is the syndicate which is musical crucial the coronary disease .period out to the days of form so we have associated fears of of where few the Commission voted Emily over and always was and that's the numeric unless they meet with 1 of the awards 1 so you may be sure that when Jesus squarely does not exceed it yes I know but square is not free condoms in the Topeka injuries so what happened here from receiver to issue I want to this what is mentor which is the main town isn't it demonstrates is that you undertaking Wednesday product of forcing do does not assume so we're take a 1 equals 1 they do it cause 1 so he knew the first one in 1 equals 1 it too is equal to 1 I found the 2nd 1 which is everyone is equal to what want is equal to 1 the 1 2 1 so this is 1 of very easy to to guests streaming towns another 1 is more than a check to the correspond to take to equals 1 and then the 2 in the quarter was so in that case so do quarter the 2 acquire land so you recognize not only this like but it is a symbol it's the the new maritime unit mutiny from from 1 but already it is it does not assume it's because the when do we know we know that we have in mind is the and so that means minus anyone paid to be 1 of the 2 it's common to 1 what fault so a two-week course 1 and the 2 equals 1 so that means that they won the 1 ,comma on 2 3 months ends this right yes a similar to is a quarter to 1 because the product is going to Fremont fall and the city this in case is symbol is equal to 1 so we all fall received from Maine terminate Reese my fears of look at NO what went to Norway's is maintained Disney down we do like that so for instance you see anyone into also you broke this doesn't associate it 1 off is a sum of 1 2 or made up of a decent ,comma on cities and use some of his engineer to follow the media in the so you know that you are forming tales and it's an exercise to see you have no you check with the plans you obtain the maintenance is some to take forward to it undertake to twofold to the sum of minds is the present the comment 1 the fall flat in which is for over 2 cities to locate and executed a number of always he is also the dollar's spaces in a tour ineffective so there is as you all and to do so this is a very simple and Risingsun no water at all and Shigenori dead because what about Euratom sold out to is now I want to do presented is always to raise to In inequality so we we cross the figure today that on other foreign when the when computer the size of the Tosanoumi who offered to take care the which of government programs and issues is using his favor the October in the 1st 1 in the instruments we make way may the 1st moment severed boss and then you make or is it the orders all he expends so you see so we have an exact form along and the U.S. and know what about so I have to raise Aziz formula which is raises formula to Delaware the so it's a nightmare Yukon 0 4 it is a square 16 pence and then went through what you want and we're we followed these I date is to use a Estill it's to square to cold this is because the combination that means that instead of a wider doing so your ID so miners in the eyes of the 0 0 at the 0 1 in the 0 0 1 and you consider this is this is a senior nearly 2 x 2 square this in at no you are ready to go to the case Farwell it however it's not easy absolutely not so you will see that summer c of the women something like I'd I'm very very ignorant already so you take in this season in F to follow to care and then there's the To can hear you report did you although GE which is assembled office could be symbolic here you put you off new the and then sold the it the next problem around here you put the product of deal it is a product of the next "quotation mark banks in which it is you really don't know what happened when you it when you whistleblower K which which is a voice as you see that voice also the the pop up on a 1 0 0 8 0 does not appear to be and so what is the future of Humvee the 2 of you really he is kinetic form no matter where he is you want Brazil yuan per mile you want pressing to trust you to give breasts -minus 1 president I took a minus 1 the it's yes it's obscene that Chris gravity cases that is important it's quite he said either it's being module in any reason you will in a stone so easy exponentially quarter was 0 is Zeus aeration Jacoby Sinbad appear he said differences equal to 1 using and it is this way of coding you see you are sure that it usually went in the right results just so so you can say that if you're going to be exact support to 1 defendant on 1 of seemed do you overdid with all the publicity you but so we have an excellent way of coding
and yet stew and what person I spoke to you all food make spaces so I would say I would wouldn't say it appears that number that depend on them so when you have the scope of directive the electoral subspecies completed when it gets to the bottom of the community in the fall full-featured so I was quite happy it's a nightmare for me to study whether the trauma over it to me you know what is in there so we have a quality control some October manager and then his is a little so now there is so EU yes is so you don't you do the job ends then disease is for me amazing computer opponents of mayors and and then again just a few what's going on inside the guests and particularly when you look into the issue 6 type of this ground support by putting all of the miners buried somewhere will watch what we what would be effect the end he does no effect Everything is book so so no air what about uh yes so I want to see you combinatorics although 15 or so I would like to do to make to remarks he said no when you're doing computational number Suri people want to check by by using the computers reported and to exit the costs involved politician you compute with stories to tell which you can use its story get score of 1 because it is it seems that you want to computer the is moment the series of meetings and the legs not looking so you go to my interpretation is that when I did the 1st on Monday there already fall valuable so he is there will be when you expanded 16 which of the Diocese of DU was you always you want so you have a product of the user Jacoby singled and so when you want to other association you want to be sure that the president is in the eyes of the as of month decomposition so that means that the at least 16 brand-new consistency which is an integer which has at doing premieres it's rather large and if you want the art that means that whenever editors knows at home in office and is something that club again so for instance if you want to see you but you should know this so you have to do with Chris banks whose child died he to power the to the of 16 so we seem not seen and these phenomena that year on quoted authorities as saying the paper became so Segovia utterances these can be done so know what about underneath the 2 were false and said there are other I would say the classical particularly I put himself to remain indoors as they that are all took the first one is but with the coming from area functions during that means around which the museum but type you would recording Monday officials in something that someone with issuance your inferences about any of need these fits it's it and then here what 3 obvious not like that is something like some of them money pair of then what is the context there will be some soon which should be at Conte and Anderson takes the is something that nobody accessible and this is this function of 1 or 2 McGovern is as difficult those in some point of view that deal either from Seoul to know that you up this award dignity authority but you opt for instance the data the square root of resists function taking the square was that means that you have to control of 0 free region OK so it's not same is not true that 1st ingredients so let me that I will result is an effective and the 2nd 1 is what a good number of sedation the latest addition Surin so this never as it is now so I give you an example ice age is when so I sings August 1 is due to iron Braun and forces you to not approved by the stronger no 1 is due to his brand which the famous paper adjusted the codes outlasted inequality for Kentucky character also is very simple so you take is that were some so you think out far and not data and then to sure I write is to be sure that they are all dead square free and I sub and around Caputo them and their own computers In the yes and I multiplied by Jacoby symbolic em over and so what do we know is that we have the feeling that end over end does that factorize as a product of 2 function soul is its approval in this way so if I suppose that I thought was theirs and want to make up and is equal to 1 this saying is less than mn which is as a treaty on on and then you can gain you go game that provided that mn to and logo mn I will so this result is whether it is easy to pool it's only a using you use caution 12 also on you until its emission and either you use you have the no-huddle for all you use of Pfizer a character when you summit of repeated you'll so his his your all of this so this is the 1st 1 2 3 1 1 in the the world of the absurd to milk the strength of 1 the you in book so thank you the few would what is the source of the smaller than year before me what is the source of the smaller of the 2 decisions was U.S. the the souls of Iowa also it's gone from here yes it's gone from him was Uighur read attended by always come and lost so and I'll consider that the room of Irish and how viable is very small and it seems that it is very easy so the toward our very classic a simple implement to arrive and that to save the take seasonal and you got some spaces and the was the it's in the world the other thing this he was the 1st the of it was yes throughout U.S. you square you have to take care of menacing iconic straining once again when the autumn easily so small the the case and the nominal anyone on the
1 hand is it is to poetry is like you see is that you have to vote to this image has to do with this case economics they knew what was often a lot was to explain so you see we have to do when it's all repairman and will be some part of the of for all of this you would
use a jobs and either new digs out all these operas I would of course must keep


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