And I'm going to hand it directly over to Irene and Jenny who Their bios can be found on the program page. So I'm not going to spend time doing that and let them get started They have a fantastic presentation on really working on changing the culture of midwifery care and obstetrical care in China no small task. So Irene I'm gonna Send it to you I'm going to introduce you to Meng who was going to take over the first part. We've got a lot to cover Meng's done. She's great She's a very passionate midwife and we're changing an entire system of maternity care that is led by midwives So I hope you enjoy the presentation and over to Meng Hello Hello, I'm very happy and Jenny and I'm Meng Xie from China Okay, this speech I Introduced by MMSF Okay, Chinese philosophy please are important rules in the Provision of respect for maternity care The multi-maternity service system is women, baby, and family-thinkers We want to establish happiness in maternity units the delivery wisdom and Celebrate childbirth Why I means philosophy of Chinese traditional culture is the center of the MMSF Maternity care works should keep their Condition clear, have a love and the respect for life and hold positive attitude Midwife and the obstetrician hold Comprehensive a big key to provide a whole holistic Continuity of care service multi-maternity service system philosophy is to uphold tradition to support the natural philosophy Physiological process of mother and baby the MMSF Homework is based up on reading respect for integrated maternity care Chinese traditional medicine Has been used for thousands of years Before 1929 many obstetrician War who will put this in America 19 MMSS we integrate the old System with the modern system show that care is effective reduces Intervisual witness Inter-witness and Provence many complications most importantly this Uploads to care gives the woman and the family's voice and the choice in all matters Health mother a healthy mother Healthy babies and healthy China my body my baby to provide holistic continuity and Personalized Service for mother and babies in China In China there are around 16 20 million on both for a year 30 years ago in China made wife lost its status as a professional made wife to became nurses and this country lost its middle midwifery midwifes and its midwifery education system Good clinic clinical midwife risk to girls were lost in 1980 women were equal To delivery deliver their babies in hospital because of the difference in regional cultures the country did not establish a established uni-field standard standard of midwifery There were no longer any Community midwifery services made wife and Obstetrics all worked in the hospital city the broken chain became the weapon dependent up on the president women's press or wants Press or wants there is no effective management of the maternal system in China Obstetrics team as all the lab of the hospital have different models of maternity maternal and infant services Some hospital made the wife work only in the delivery room and responsible for Observing labor and the delivery Some hospital made the wife only responsible for observing the labor and the doctor takes the skates and Deliver the baby Some hospitals do not have midwives doctors are responsible for prenatal Exemptation and the delivery in 2019 Series in China become the Highest in the world China has Read the world's healthy organization Millenium Millenium goes but in the global ranking China The old Behind Comparate with the developing Country, we still have a long way to go the midwives read Proficial was agreed if not it's well Well, yeah, it's not Optino Workloads health and the heavy this you Produce a great loss of a midwife read a midwife Midwife education is low in Lampere made the wife re-training paste and skills were lost the medical intravenous increases and The title was became a high risk increasing in 20 I in 2008 with passing and the end of my into Inter tip We is Douglas China's first the midwife Or line or links organized season China is Douglas the midwife website this became and at the academic exchange Plates former for midwives The midwives and the nurse training school They linked Advanced experience experience and physiology philosophy of developing countries North and the midwives linked to to do the used duties duties of the world's healthy organizing midwife Responsible for the management of low risk woman Obstruction is responsible for the management of the high risk woman We in Asia, we is doubly MMSF we have more than We have more than 18,000 members midwife include our midwives and Obstetrician and PDA Doctor MMS from work is They designed to be a woman baby and the family centered system MMSS supports Respect for maternity care MMSS complete Compliments China's culture and the woman's needs their pain modules or Modules share the same philosophy The seven modules cover theory Evidence is the practice and the clinical skills the remodels cover management Management skills they and pattern and Intro pattern and the post pattern services through Interprofessional team work as we Continuity and the indy Individually the pair there are there are pros and cons to all things China's Advantage comes from the Factor that obstetrician and the midwives Working in the same unit They have a great we are at the passing to work together Or longer as the Philosophy is the same. It is easy to promote the development of the MMSS module Over the last ten years we have treats Treats the maternity teams in almost 100 hospitals We opened the network teaching the school training more than 30,000 3000 people a year the government introduced a Policy to support the development of midwifery Opens a midwife in University course the lower The lower c-section rate is evident midwives coming to the forefront of again MMSS is a service module Modding that shows the world that Integrated in the we do lines and the continuity of system rates It is the basic the upon up on Chinese culture Next I will Induce introduce MMSS Let the Arring team go on. Thank you Okay, I think may need a marvelous job because two years ago She could not speak any English. So I want to say well, she's doing an amazing job But I'm going to take over from here on the next slide You're going to roughly see what the MMS module does it starts from conception and goes through the entire system. I And we also train obstetricians and with that is an interprofessional team and we also include all the holistic practices from Therapy To massage to homeopathy and Chinese traditional medicine. It's a really awesome module Sorry, it's just not So Again, we covered the first point to call is the midwife and the midwives will assess all the women and then Care will be planned with the woman and it will follow through the whole The whole system and they work together as a team because in China the obstetricians and the midwives work very closely together We're very very fortunate to be here amongst all this passion and once again We start off with the the appointment system and we go through low risk if I know risk high risk But we take the woman's psychological aspects in hand and we Have a lot of consultation Services without actually having to hand the women over We've got lots of great support here and you can see on here what we use to get high quality care Oh, once again, this is we set up the midwives clinic So again, we have the booking visit the planning of individualized care monitoring growth and well-being of baby Again individualized care putting the woman first It's quite amazing to see the communication skills of these midwives Really really been able to communicate with women and explain the woman's physiology and what's happening with her and Give her confidence to birth So again, we have we have the school we have a public class we have very strong prenatal education We teach all our health care providers to be first and foremost prenatal education and we're all sending the same message So our vision is is to create a celebrated childbirth where women are remaining control and reduce postpartum Depression Because after all it is a once or a twice in a lifetime event here, but maybe more so we have a free positioning in birth and Even though for some of you in our hospitals in New Zealand, we don't wear gowns But unfortunately we sometimes have government systems that haven't changed yet, but here we're celebrating a woman delivering Squatting on the floor, which is quite a squatting on the bed changing positions. It's a great thing to see here in China So we also have framework for breastfeeding and where we use holistic practices Chinese traditional medicine And the idea is to avoid pharmaceutical preparations We have great postpartum recovery here I think it beats the world because there's lots of respect for postpartum care Women can stay in the hospital and they're truly supported throughout the entire process So this is the hospital that main Did a few years before I came it was up to 500 beds There were 14 nurses and midwives and she put the entire system into there and reduce the C-section rates And here we are. This is a Tian Dong. We're opening the projects and cutting the ribbon We're very excited and when I stood up there and I had the whole team Together with passion to make a change and at this hospital, it's a traditional Chinese medicine hospital So it's very open to holistic practices So here we have the the group the teams We have the professional team in Tian Dong together to give confidence in the training The 10 modules there are 10 expert lectures with theoretical part training. So we're all playing we bring international experts And you can see our New Zealand doctor Emma Perry up there We have Ricardo Herbert Jones from Brazil We have obstetricians who are also homeopaths on board and very holistic So our clinical teaching is the use of ultrasound and we're teaching midwives and obstetricians And we're also trying to reduce the technical interventions Positioning of baby and you can see me they're sitting Showing the different birthing positions and demonstrating normal birth So we've got the midwives there the clinics here we use charts We created our own educational tools our charts. They're bright. They're colorful And when we first got to the hospital there was little communication at all with the women And now we've got a beautiful happy atmosphere where all the women's Questions are answered no matter where she comes whether it's in prenatal class where it's prenatal education There are health professionals teaching prenatal education supporting normal philosophy So we have yoga and meditation and we have international experts come into To teach us all how to do it So the obstacles we had was removing fear about the bag. It was Getting in there face to face Bringing back the evidence-based practice And giving being there and giving them confidence in helping make the right decisions But most of all observing birth Observing the normal process and learning from the woman and less from the machine We continuous perennial suturing you can see as they're demonstrating that where the woman's pain-free And helping the woman feel her own baby's position and we get the partners and everyone to be involved too So we allow the endorphins to work the husband supporting and we're training the husband how to accompany it so we You know, we do have donors in remote areas of china where we don't have enough midwives But we believe in teaching the the husband the family and the support to do the do the work And the midwife is the observant one. They're just stepping if something goes wrong And get it back on track, of course with optimal positioning So how to accompany the maternal and infant skin contact in c-section? We started that and of course Involving husbands too. They're now allowing husbands in We're helping the team to improve management so that we're learning but one thing about china There's no lack of white and I love it here because it's not over-regulated We've got a lot of common sense and teamwork actually making the decisions Assessment time and communicating this asking and doing and ways of exploring and having happy mothers and listening to the woman And the midwives master of training so this is the midwives actually holding their own midwife clinics and Showing the woman optimal positioning or working with flip charts. It's just amazing to see this turnaround in such a Short time. I think we were about four or five months into yandong weren't we Jenny? So the confidence and empowerment is telling women I can birth and women choose where they're birth celebrating the outcomes Well the positions for birth. We've got beautiful Tian Dong is just getting its new hospital ready and Jenny and I went around picking the colors for the the hospital So we're actually going to sleep with colors on our mind We did an ecv giving the obstetricians Confidence to do an ecv with ultrasound there and this woman went on and had a normal baby He was called the opposite to turn breach babies and here we are the hospital is not quite ready It's just about getting ready. This is the postpartum area at the end is the entrance to the delivery unit and we're trying to pick nice soft home like colors This will be the reception area downstairs in the hospital We picked the red desk because it reflects culture and we've designed colors to go with that The women's health education class will have bureaus and a plenty of room to use the pilates balls yoga balls And access to bring family and we want to bring the grandmothers and everyone else from the community in here This is the chinese characteristic postpartum ward. This was designed by mms all the hospital designs we've done So we're very proud that mms is designed by midwives For women from design from content of education All the way through it. So we're very proud. We're too amazing. I'm so lucky to be here with with jenny doing this Here we have a VIP room For women who do want up to pay privately award-winning area Postpartum recovery here We've got a roma therapy and massage for women after they've had their babies And it's something that the chinese women love because I know myself since being in china I've had more massage and a roma therapy and traditional practices Then I did in new zealand Which I was looking in new zealand too because my doctor is a chinese traditional medicine practitioner So we have infants water care and exercise and massage therapy rooms for children from birth all the way through Get water massage and touch therapy And this is our team with their choosing the new uniforms. So they chose that they wanted the The the the the purpley color the the lighter one to blend in and we let them choose what they felt happy being And here we are after barbeque the team full of strength joy and happiness to everyone So even contact is in the following way. We're looking for exceptional midwives or Healthcare providers who want to join us. We have an international We have a large conference coming up in october if you want to come over and support us You can contact us at cnzcs8163.com And you can just tab it to jenny or myself So, thank you Thank you. We're looking forward to your questions That was fantastic I know that we probably have a great deal of questions and Maybe some input. We have a lot of comments for Irene and Meng on the comment side just With lots of exclamations of how amazing it is So it would be great if people have questions. You can just type them into the box on the right Absolutely We're more than happy to help and support Because of course with with jenny and myself Midwifery is our first passion and only well, I've got many because I've got a whole me offered to And the traditional therapies and I believe the midwives of the future won't need to prescribe farmer college Farmer ecology we will be able to prevent complications by the use of holistic practitioner holistic care Yes, yes Can I ask um, you guys to give a little bit of a background of how you interfaced with the chinese medical system to make such amazing changes in such a short period of time Well, it hasn't been easy You know, we we actually went out into the provinces because sometimes the cities are still quite western medicine, you know, so We we found the experts across china and globally that supported us and we found hospitals like tiandong hospital Supporting us. So tiandong is the history of this. It's we we're putting in an entire system a software system management system And we're launching it in june. We're actually going to develop the holistic midwifery research center there Can you tell us one of the Questions we have was um, how did you get started? What was your first step? How you two came together maybe Well, I actually wrote the book my body my baby and I was on linkedin and meng found me out of the blue and she says hey, can you come and lecture in china and You know, I would have my own Practice my a beautiful practice in new zealand But when she when I saw her passion and I came to china and the passion of the people That was it. She after uh, a few months. She says right come on over here and join me So she did she invited me here That's how we met it was kind of it's meant to be You know, it's like all my surnames chain and it's an amogram of china. So I think I must be meant to be here Lovely and there's a question also about whether the midwives do deliveries Yes, the midwives do all Do all the deliveries? Yes in our hospitals the midwives are in charge on delivery units They call the obstetrician if they have a problem Well china is a very big place, you know, it's about 10 countries in one So we're getting it across china and I think jenny Would like to answer this in To us. Uh, yes, we have the the conference there Uh, we've traveled up and down china and across it, but there are many many more A lot of work still to go Yeah, that question was um, has the change been recognized in all the areas of china? Well, that's what i'm saying. No, it's not in all areas of china because china is such a vast place and it's run by Lots of different provinces. So if you're in shandong province, you've got about 11 Uh million people and you've got one governor and then there's another governor in another province So it's not easy to uh get directly into the into the major league of the government To you know to voice what we're doing, but we've done lots of media Uh, the hospitals have been supporting us. So it's growing What types of midwifery training problems do we have for midwives? Midwives Every year Yes, well At the moment we don't have enough midwives So the the types of Yeah, we with mms We kind of up skill and train the midwives Because they're all trained as nurses first The midwives that we train midwife. Yes And we are upskilling the nurses to become midwives Because that's the only route we have at the moment Yes Now midwife can Do midwifery a walk So the time is the government uh, a me midwife Midwives with developing So China is open a country Uh developing quickly Now the midwife Is important to you Support the family support the family ume and bb so, um next So we are We are growing and uh tinsy Yes, so it's hoping that you know, jenny works very closely with the china Union the confederation of midwives.  That's helping it grow Yes The time is the environment is better Uh for midwife training for midwives for walking so the The obtutry sin obtutry sin live or working with the midwife Midwife together And we're hoping to develop this training based in based in a hospital setting a hospital that has uh The philosophies of normal supporting normal birth both in high and low risk. So the training will be practical Yes Yes, uh the government want to lose the C-sexy risk So, uh, the midwife and the obtutry sin together work hard To load the C-sexy And as we're saying before we hope to have direct entry midwifery But we're not ready in china yet for this We're you're very lucky in new zealand and other countries where you have this we don't have it here in china Uh, so we're hoping that's going to happen But we yes, we we do train the nurses in midwifery at the moment Yes If you would just read the question Yeah, i'm just going through them now because i've just answered the direct entry midwifery and now there's one about the chinese people and the Women's movement towards medicalization of birth and seeking midwifery care Well, as i said that the midwives off, you know, we're just building up midwifery here So, yes, we've got a lot of changes to do with the public just as we had in new zealand when we were Obstetrically dominated before midwifery had its profession So, uh, a lot of we so we are promoting midwifery care in the provinces and the hospitals that we're going to So it takes time I told meng that i came two years ago and it would take five years to see good changes They're saying that when some medical unit will start a four-year direct entry. Yes midwifery program. Yes This is this this autumn You know this one jenny ven saia Can you answer the university? Direct entry midwifery education program Uh, yeah. Yes, ten is ten. I have Eight Eight universities for Midwifery Yes, yes Yes, so they have eight universities for midwifery, but the training is not holistic Yes, and that's why we believe mms will be together in the future will change it Uh, not only ring to medical university. Yes, so it's not only when venzao medical university There are there are talk of eight universities according to jenny, but she's in touch with uh The unit some of the universities that teach the midwifery. Yes Eight universities eight universities. Yes. I'm the time that and uh, uh, many Cities, that's right. Yeah, Guangzhou uh, Beijing, Shanghai uh, Suzhou eight Uh, universe midwifery universe. That's right, which is nowhere near enough To cover what china needs so we have to rely on mms to upskill the nurses And up skill the midwives and the obstetricians and the entire team So, uh, mms is special Put the nurse become uh midwifery with the obstetrician together working for uh, so the woman and the baby uh, think the Recall recall the dental graph And uh, good education Education and uh support the family Yes, other universities are supposed to admit their first classes. Yes. So that's what we're saying We are we're aware of what is starting to happen But the current situation is like I said, we don't have enough so mms is focusing on on Upskilling the midwives the obstetricians To be prepared for when these midwives comes through Yes Any more questions? Fantastic If anybody has another question for Irene and Meng Sure This question is I see in many countries pregnant women are going first to obstetrician Or obi doctors and we see the rate of C section has been increased But in your case it is the opposite Yes, we're turning it down. We're removing the We're integrating it. So we have the medical knowledge but the But because we're so open to the holistic care and preventative Prenatal education We are getting it. Uh, we are reducing the C section by You know, the obstetricians are learning normal birth. I was teaching yesterday in uh, in Qingchong And I was teaching the obstetricians the hospital leaders The pediatricians the midwives and the nurses and we're all moving our pelvis and learning about birth So yes, we we we we're having the difference with mms is that we have the entire philosophy With the whole system and the entire maternity service care providers, which is unique because it doesn't happen In many other countries we've got obstetricians doing one thing pediatricians doing another thing midwives doing another thing nurses doing another thing and Yet the the knowledge of the training should be that universal philosophy of supporting normal birth and preventative care And working together its teamwork and I think the future is into professional teams Thank you so much. We do have a comment that I do want to share with you It says not a question but sending congratulations. I think that this this comment speaks for us all Um, and best wishes to keep the great progress in women and holistic care moving forward across the whole of china So I really thank you too for for for the work you've done Well, you know, thank you all too. I've just come back from new york and I've just recorded an online Midwifery led homeopathic course for midwives with the new york academy, which is part of our training at mms So but what we want is the midwives from around the world to join us So any professional any special project you have anything you want us to share with china You know, we can invite some of you to come to our conference come to tiandong and see what we do You'll be very very welcome You know, we want to see midwives lead this and midwifery philosophy held everywhere and You know, men and I have such a passion for all you midwives. That's why we're here You know, my family is all over the world and I got to get chance to see them So, you know, I want to thank you all for listening today and send you wonderful messages for From men and I for this special day for independent day of the midwives. So I'm going to say Midwives rock so on on and we look forward to hearing from you Fantastic


thanks to the organizers for the invitation to give this talk. I was first in contact with them, I think it was sometime last year and I kind of let it slide for a very long time. So I'm glad when I finally got back to them they were still agreeable to be giving this talk. So three notes before I leave this slide. So this is a drama in five acts, which means there will be brief intermissions between the acts. Those will be some opportunities to ask questions. Second is that the slides are available both, well the link has been posted in chat for my slides. That's just a link to my webpage. There's also a handout version if you prefer not to see the overlays. And third, normally I would start by acknowledging that my campus sits on unceded and such a land of the Kumeyaay Nation. But as Mike pointed out today, I'm in Washington D.C., which turns out if you check nativeland.ca, which you should, it's the ancestral land of the Piscataway Nakhachetank people who Europeans call Anacostian, because that's easier to pronounce. Great. Okay. So act one is where we meet the characters. So the characters in this drama are abelian varieties and they're associated vape polynomials. So A is always going to be an abelian variety over a finite field FQ. And I'm going to refer frequently to the order of an abelian variety, which is a slightly nonstandard terminology. When I say the order of A, I always mean the order of the group of rational points of that. So I'm not forgetting that A is an object of algebraic geometry. I'm just keeping track of how many points it has over its field of definition. And of course, that's a group. But we're not going to be using the group structure very much in this talk. Mostly we're going to be using a vase interpretation of this order as the value of a polynomial. So there's a polynomial P of t, which has integer coefficients, such that this order of P of one, and this polynomial comes from, well, it's okay, it's the characteristic polynomial for Benius acting on the Latic tape module of A for any prime L not dividing Q, any prime L not equal to the characteristic. But what you really need to know for this talk is that it's a polynomial, which is monic of degree two times the dimension of the abelian variety. And as a polynomial viewed over the complex numbers, all of its roots have absolute value Q to the one half. And such things are called Q-v polynomials. And this talk is mostly not about abelian varieties per se, it's mostly about the abelian varieties. So the abelian varieties are part of the motivation, but they recede into the background of the action fairly quickly. They may they will appear briefly again at the end. So I'll just mention that a lot of what I'm doing depends on the fact that there is built-in code in Sage to enumerate v polynomials. This is based on an algorithm that I started working out more than 10 years ago by now, but relatively recently, this code got incorporated into Sage. So you can run experiments like this where you ask Sage for all the v polynomials for a given Q and a given D. D is the degree of the polynomial. So it's two times G. And as long as you don't make these parameters too large, you get the answer back fairly efficiently. So the main thing that makes the abelian varieties not so critical for this talk is the fact that this is very tight correspondence due to undertake between isogenic classes of simple abelian varieties over FQ and irreducible Qv polynomials. So if A is a simple abelian variety over FQ, it's associated polynomial would be a power of some irreducible where the integer can be read off from the polynomial, but most of the time in practice, it's equal to 1. So in particular, if Q is prime, which will be most of the time in this talk, it's more or less always the case that the characteristic polynomial of A will actually be irreducible. So sort of the abelian variety being simple, which is kind of like irreducible, it doesn't factor as a product to isogenic, that corresponds to the polynomial being irreducible in the usual sense. As a reminder, when I say isogenic, I mean, it's a morphism, which is rejected with finite kernel. So we're not going to be talking so much about isomorphism classes of abelian varieties. That's a little bit more subtle. We're going to be talking about isogenic classes because these correspond exactly to vapor polynomials. And the order of an abelian variety will be invariant under isogenic because it only depends on the vapor. So it's natural when we're talking about orders to just work up to isogenic. And yeah, so as I made this point earlier on a previous slide, the upshot of honetate is that problems about existence of abelian varieties of a given order or over a given FQ are really problems about existence of vapor polynomials with certain properties. What that means is that for most of the action, we will not be seeing abelian varieties explicitly, we will be just manipulating polynomials and deducing consequences about abelian varieties from that. So most of the work in this talk is happening at the level of constructing or ruling out the existence of vapor polynomials under certain conditions. So really, this is a number theory talk and not an arithmetic geometry talk except at the end, maybe some curves in their Jacobians will sneak back in. I mentioned on the previous slide that the order of an abelian variety is an isogenic variant, but the isomorphism class is not. So for example, you can have one abelian variety whose group of rational points is cyclic of order two times cyclic of order two being isogenous to another one whose group is cyclic of order four. The calculations that I'm going to be doing since they're only at the level of the vapor polynomial, they will only see the order and not the isomorphism class. Questions about existence of abelian varieties with a given group structure are also tractable, but they use a different set of ideas in combination with these. So there's a recent preprint of Marcellia and Springer that explains the relationship between these kind of results up to isogyny which prescribe a given group order and results at the level of isomorphism that prescribe a given isomorphism class of the group of rational points. That's the, sorry, that's almost the end of act one. Now it is the end of act one. So let me pause for a moment to see if there are any questions. I don't see any in chat. Seeing none and hearing none. Let me continue. So in act two, we look at orders over F2. So now for the moment, let me not specify F2, but on this slide, let a be an abelian variety of some dimension G over some finite field FQ. Remember what I said about the order being P of one where P is a vapor polynomial. Well, if you just use the information that's implicit in that statement about the roots on the complex roots all being on the circle, absolute value of T is equal to square root of Q that implies a lower bound and an upper bound on this order. So the upper the upper bound is square root of Q plus one power two G. And the lower bound is square root of Q minus one power two G. So the leading order term is Q to the G. The correction is Q to the G minus a half. So if Q is large compared, if G is large, sorry, when Q is large compared to G, there are gaps between these intervals for consecutive values of G. So this number will be close to QG. And if you go to G plus one, that number will be close to Q to the G plus one. If Q is large, then there will be a gap between the intervals that you get. But when Q equals two, there isn't going to be any gap because this number is actually less than one. So the lower bound for Q equals two is meaningless. It tells you nothing because of course the order is at least one because it's a group. It has an identity element. It can't be zero. So when Q equals two, you don't have any intervals that any integers that are obviously excluded from occurring. And so you might ask, well, is it the case that every integer occurs? And this is the content of a theorem of mine with Everett Howe. We prove that, yes, every positive integer is the order of at least one abelian variety over F2. More precisely, we show that in this range, every positive integer going from, say, this about four-thirds to the D minus one to about four-thirds to the D occurs for a polynomial of degree 2D. For some reason, the D here is what I was calling G before. So these intervals, you'll notice, they line up perfectly. So you don't miss anything. You pick up every integer because you've sort of tiled the positive integers with these intervals. We can also ensure that the polynomial is ordinary, which in this case means that the middle coefficient is odd. Geometrically, this means that the resulting abelian variety has two torsions subgroup over F2 bar as large as possible, or rank is D. I see Sarnak has asked, do you use the distribution of roots? I mean, this is a statement about just finding one abelian variety. So we more or less give an explicit formula for a vague polynomial. So this is not a question that involves analyzing the distribution. This is a question about really producing a concrete polynomial for every choice of the order. And I'll show you in a moment how we do that. My question was, can you hear me? Yes. Yeah, thanks. My question was, whether in that example, okay, we'll see how the roots are distributed. Okay. Yeah, maybe when I get to the end of this act, that might be a good time to ask this again. So let me go through briefly how this is proved. Prove is actually only a couple of pages, so I can give you most of it. The key lemma is an older result of de Pippo and how. So this is a sufficient condition for a polynomial hat to be of a polynomial, at least if it has integer coefficients. But this is really a statement about polynomials with real number coefficients. So I take a sequence of real numbers, which are not too large, they satisfy some kind of weight condition. So a1 through an are bounded in some way. And if they're small like this, then the complex roots of this polynomial that I get by using a1 through an as the first coefficients, and then making it reciprocal to finish, the complex roots of this polynomial are pairwise distinct and lie in the circle absolute value of z equals square root of q. So this is essentially sort of a winding number argument. We'll see another one like it later. A very short way to say this, which your impudence suggested to us is that if you look at the function f of z over z to the end on this circle, it's real valued, and it has sign changes at the roots of unity. If you start with if you start with if you start with the ai's all being zero, then you have a polynomial that manifestly has roots on the circle, and the sign changes sort of survive when you perturb this much. There's a question in chat which I'm going to answer on a later slide, so I'll hold on to it for Yeah, so now you have to show that every positive integer actually occurs as f of one for some polynomial like this. Now when we originally were working on this, right, to Pippo and how wrote this limit down in the late 1990s, and they in their paper, they point out that it gives you some intervals. But for a long time, I think everybody including me and including Everett thought that these intervals weren't going to be wide enough to actually cover all integers. And so we thought we were going to have to upgrade this criterion in order to actually prove this. And then we somehow discovered initially by doing numerical experiments and then observing what was going on that actually this is enough. And the proof of this is a tweak of a very classic fact from the early history of computer science that every positive integer admits a unique non-adjacent binary representation. So you can write it as a sum of powers of two with signs where you don't use any two consecutive powers of two. So this might be familiar to those of you who are used to implementing elliptic curve arithmetic for crypto purposes, for example. So this is a tweak of that because it's not quite the problem we're solving, right? We essentially have to represent something. We have to put two to the i plus one here instead of two to the i. And then it's not exactly the same proof, but it's very similar. And then once you've done that, then you get a v polynomial that does what you want. So there's the there's the concrete construction. And this, I think, will answer a question that was raised in the chat. What about larger q? So yeah, q equals three. You don't get much mileage out of the v bounds either. But it is at least the case that so for every prime power q, every sufficiently large positive integer is the order of at least one abelian variety of rafq. This is a result of fanbamo, costa, we, punin, and smith. For q equals three, I believe they do get every order. For q equals five, I forget which where I think for q equals seven, you don't get the order two. But up to that, you get all orders. And you can, you can ask for more conditions like ordinary, simple, or principally polarizable, but that will change what sufficiently large means. So for example, for q equals four, you get all orders. But if you insist on ordinary, then you miss one of the small numbers. I forget which one. Yeah. And again, the point is that the v bounds do start to overlap once g is large enough. It's not the case that every interval integer in this interval occurs, but you can make this interval a little bit shorter and get an interval in which you can find everything by a kind of more sophisticated version of what we did with Everett. This is also refinement of results of Uri, Alouille, Lechot, and Kadyitz. The result of Kadyitz is in some sense a motivation for a lot of what I'm talking about. So I want to give a shout out to that. I think that's the end of act two. So this is a good point. Maybe Peter can circle back to his question and I can see if there are any other questions. So where are the roots of this polynomial? Are they running around? As you know, you can make the roots avoid intervals. I mean, Sarah has studied that. So in this polynomial that you constructed, you know where the roots are. We didn't make any efforts to sort of constrain the roots. I mean, in some sense, they're relatively close to the roots of the, you know, they're relatively close to q to the one half times roots of unity because we're deforming that polynomial. So that will give you some condition, but we didn't check very carefully. Yeah, thanks. We're not sort of probing arbitrarily far into the space of a problem. You can't do this with curves. Well, I'll talk about curves at the end a little bit. That's act five. When you have your polynomial constructed in your way, let me say polynomial satisfying all your conditions, are able to give us an example of a billion variety that for that polynomial, we did not attempt to go, we did not attempt to explicitly write down the abelian varieties that come up. A hint how it is looking. Well, it's just what you are doing your opera in five parts, right? Yes. So, yeah, so this I mean zeros on the unit. Yeah, so of course, you in principle, you could invert, you could run through the proof of candidate, right?  There's also a more algebraic approach by maybe try and or. But neither of those is extremely explicit. So it would be, I have not tried to make explicit examples, except in a few cases that I'll talk about an act five.  Yeah. Yeah, I don't know how much that would cover of the whole range. I mean, of course, the problem with taking a product of elliptic curves is that the order will be a product as well. So you won't get all possibilities, right? That way. Thank you. All right. I don't see more questions. So if I don't hear any more, maybe we can move to act three. So in act three, each order happens many, many times. So the motivation for this act is this statement, which is also a refinement of a theorem of Aubrey-Alioula show. This is the version by, well, this, maybe you can deduce this already from Aubrey-Alioula show, but there's a stronger version of this theorem in the paper by Cadets. For every prime power q greater than two, you actually get an exponential lower bound if you insist on simple abelian varieties. So this is another answer to the question about q equals three that came up in chat. So even for q equals three for symbol abelian varieties and q equals four, I should mention, for symbol abelian varieties, you do get an exponential lower bound with one exception. Namely, there are elliptic curves of order one over q three and q four. They fall on the Haase interval. And right, it's due to, right, it's an old theorem of doiring that everything in the Haase interval, at least for prime order shows up, but also for q equals four in this case. So, okay, so for q is at least seven, this is immediate from the bay bound. For smaller q, this is a kind of classic strategy in algebraic number theory. This goes back to Castles, this kind of approach to identifying algebraic integers of small trace and norm. This kind of question was also studied in detail by Raphael Robinson.  This kind of threshold is what Robinson studied very carefully. And you actually hit the threshold for q equals two and something very different happens, which I'll mention on the next slide. So, this is all kind of related to things like the sure as eagle Smith trace problem. I wanted to give a shout out to Alex Smith. He's been working on this. He has a not yet public, I think, preprint making some progress on that question. But that will not directly intervene in this story because That's on the archive now, Kiran. Oh, it's great. It's on the archive. Congratulations, Alex. It's a really nice result. Yes. But I'm not talking about that because I'm focusing on q equals two here. And in q equals two, as I mentioned, the situation is kind of different. So, there's a good reason why you can't improve the way bound even for simple ability varieties for two equals two. And it's because this old theorem of modern, this combines a theorem of modern Paul with a theorem of Robinson. If you take a simple ability and variety over fq of order one, then first of all, this can only exist when q is at most for zero. The zero ability and variety is not simple. So that doesn't that's not a counter example. Right? Just like one isn't prime. If q is three or four, then the only thing you get is the elliptic curve that I alluded to earlier. But for q equals two, you get infinitely many simple ability and varieties of order one. And you can describe their Frobenius eigenvalues rather explicitly. They are things you get by taking a root of unity and then solving this quadratic equation. Most of the time, this gives you a quadratic extension. There are a couple of exceptional cases, n equals seven and n equals 30, where a root of unity of order n gives you a split quadratic equation. And so, in those cases, you get two different isogenic classes. But most of the time, you get a unique isogenic class corresponding to each root of unity. And this more or less the proof is that if you look at alpha minus one over alpha bar minus one, right, the order of the ability and variety is essentially the norm of alpha minus one. And so, since that's one, alpha minus one is a unit in the ring of algebraic integers, as is alpha bar minus one. So the ratio is an algebraic integer whose conjugate all have absolute value one. So, Kronecker's theorem tells you it's a root of unity. So that implies this equation. And then the contribution of Robinson is to use the strategy of Castles to show that you only get a splitting when n equals seven or 30. So you compute the discriminance of this equation and ask whether it's a square. This is a problem about writing something as a sum number as a sum of very few roots of unity. So it's kind of classic algebraic number theory to solve that problem. Yeah. So there are these infinite family of ability and varieties of order one. And so the main result of this act is this theorem that says that every positive integer, not just one, but every positive integer occurs as the order of infinitely many simple abelian varieties over F2. Now, for m equals one, we have an explicit recipe for doing this. For m greater than one, it's, we don't give, we give a somewhat explicit recipe in that I can, we construct infinite sequences of vay polynomials realizing a particular order. So the sequence of vay polynomials, you know, giving you a given order is explicit. It's a kind of tweak of the polynomials that come out of the modern Paul construction. The catch is they don't a priori come to you as irreducible polynomials and we're looking for symbol abelian varieties in this case. So you have to massage things in order to actually be able to prove that they're simple. And essentially what you do is you rig everything up so that you can prove irreducibility over q2 either directly using the Eisenstein criterion, or which is due to Schoenemann originally, or, you know, a sort of generalized either side criterion where you say, I cook things up so that the Newton polygon, the two added Newton polygon has only a single slope with no interior vertices. So maybe it has rise three and run co-prime to three. So that's not the vanilla Schoenemann Eisenstein criterion, but it does still imply irreducibility over q2 and hence over q. Now, we don't quite do this for the original polynomial. We show that there is a factorization with a cofactor of bounded degree. And then the rest I claim is what I want most of the time, because these polynomials will satisfy some second order linear occurrences of Chebyshev type. And you can show that any factor that occurs more than once actually gives you order one. So pulling those out doesn't affect the order. So that's the strategy. Now, let me show you how to write down these polynomials. So I'm going to start with this, you know, the number theorists normalization of the Chebyshev polynomial, which is the one that makes it monic. So Tn evaluated 2 cosine theta equals 2 cosine n theta, right? It's also the number theorists normalization because these are supposed to be algebraic integers when you plug in a rational multiple of pi. So I'm going to sort of tweak this polynomial in a particular way. This factor x plus x inverse minus four, I don't think I included a graph in here. So I'll just say this Laurent polynomial has the property, well, maybe I'll come back to why this Laurent polynomial is a special one in a moment. So this basically gives you the construct, the polynomials that come from the modern Paul construct. That's in some sense why I'm doing this, but I can say more. And then I kind of rig up some weird linear combination of these things and then divide out a big factor of x minus one. So this is some recipe that you can follow. And this is partially rigged up by, I did some experiments and it worked, but there's a better reason why it worked, which is on the next slide, which is that there's something kind of like the De Pippo-Hau lemma that applies here that says the following. If I take a sequence of real numbers, ending in one, such that if I use these as the coefficient of a polynomial, I get a polynomial whose complex roots have absolute values at most square root of two. So again, this is a condition that says that these numbers are not too large. If I use this, then if I make a coefficient, if I use these as the coefficient of a linear combination of the polynomials g and k from this slide, which again are kind of modified tweaked from Chebyshev polynomials in some way. Because I have a moving index n here, I get an infinite sequence of polynomials and all of these polynomials give you roots, which are real pairwise distinct. And in this interval, three minus two root two, three plus two root two. And again, this is some kind of winding number calculation that proves this. And the point is the following. So on one hand, this is an interval whose end points multiplied one, right? Because nine minus eight. On the other hand, if I translate, then I get minus two root two plus two root two, which is where you end up if you take a complex number of absolute value root two and add it to its complex conjugate. So this polynomial shifted will then give me the what I call the trace polynomial associated to a two-v polynomial with the right order. Whose order is whatever this thing is, the trick is that g and i, did I say this? Oh, I don't think I said this. g and i has value two to the i. So whatever binary representation, whatever this binary representation of sorts fits out, that number is represented as the order of infinitely many vapor elements. Yeah, so now, right, so the polynomial, the mysterious Laurent polynomial x plus x inverse minus four is the one that carries this interval to plus two root two down to minus two root two at the value one, and then back up to plus two root two. So it's rigged up to have to sort of shift the usual graph of a Chebyshev polynomial over to this interval that's relevant for constructing Bay numbers. Okay, so that's the end of this act. So if there are questions about this, I can take those here. Okay, I don't see anything. So let's talk about act four. So act four is another look at Abelian varieties of order one. So in this act, I want to distinguish between simple and geometrically simple. An Abelian variety is geometrically simple. If it stays simple when you base extend it to any finite extension of the base field or equivalently to an algebraic closure. Now again, this is really a condition on Bay polynomials, at least if the characteristic polynomial of A is irreducible, which more or less always is in my situation. So this condition says that no two of the the Frobenius eigenvalues so no two roots of the Bay polynomial have a ratio which is a non trivial root of unity. So this is a for any given polynomial this is a relatively easy thing to check computationally. If you want to check this for infinite families, this is also tractable as we'll see. For any given m greater than one, I expect there are infinitely many geometrically simple Abelian varieties of order m, but I have no idea how to prove this. So I'm not going to focus on that question because that's my approach on the previous in the previous act doesn't seem to give this. At least I don't know how to how to control that. So instead, let me look at m equals one where we don't have any choice of the Bay polynomial. We just have a list. We know what all of them are. And we just look at that construction. We see what happens. So here's what we do. So the main result in this act is a theorem with torrent and L a warady who was a summer research intern with me. So they are Berkeley undergrad who's applying to grad schools right now. So you should admit them. Great. So this theorem says the following. If I pick a positive integer in that gives me one or possibly two in a few cases, I saw any classes of simple Abelian varieties of order one over F2. And I ask, well, what happens when I decompose that over F2 bar? Well, it's always going to be isogenous to some power of a simple Abelian variety over F2 bar. And so the real content in this question is what is the power? And it turns out that you do get geometrically simple examples infinitely often when n is a power of two, you do get something geometrically simple. Most of the rest of the time you get F equals two. The exceptional cases give you exceptional answers. But yeah, so most of the time you get two factors and there's a good reason for this. And this but this reason doesn't apply to powers of two. And in that case, you usually only get one factor. So here's an interesting corollary of this, which I don't know how to prove directly. It's just a corollary of this classification. You can also compute the Newton polygons of these things without too much trouble. And you show that A is ordinary if and only if n is not a power of two. And so what that means is that there was no simple Abelian variety over F2, which has order one, which is ordinary, and which is geometrically simple. So those conditions together run into each other. And it's a little surprising you can prove this given that there are infinitely many simple Abelian varieties of order one. But it's a consequence of the fact that we have an explicit classification. So we can do this. Okay, so how do we do this? Well, this is a type of problem that may be familiar to some people in this audience. I have, you know, say I take two for many as eigenvalues. For this, I don't have to assume they occur in the same simple Abelian variety. So this actually gives you some information that I'll use later when you do this with different Abelian varieties.  So I take two for many as eigenvalues, both of which have this form, which is say they both are roots of quadratic equations involving roots of unity 801 and 802. And I also want to assume that their ratio is a root of unity. So that's a third eta three that I put in here. So now I have three equations in five variables. What do I do? I eliminate alpha one and alpha two. That gives me one polynomial in three variables. And I'd like to find solutions of that equation in roots of unity. And there's a sort of standard techniques for doing this. We use the method of Conway Jones, which involves writing down all the additive relations among short sums of root of unity. You can classify those. I think at the moment, the record is that those are classified up to 24 summands. There's a preprint by Christie and Klepp, I think, that goes out to 24. We actually use some non-trivial subset of that result. We also use some sage math code that I wrote for this earlier paper, paper from last year with Kolpokalpon and Rubinstein, where we classified tetrahedral with rational dihedral angles. That was actually a much more serious instance of this problem that was a polynomial in six variables because it involves six variables corresponding to six dihedral angles of a tetrahedron. So I had to write a whole bunch of code to solve intermediate problems involving three variables. So that code off the shelf handles this problem relatively easily. So that did a lot of the heavy lifting for us. Then we have to interpret what the answers are. You do get some parametric solutions. So one of the parametric solutions is the reason why you get a two here and not a one. So there's a systematic collision that happens between two of these roots of unity. It's a fun exercise to find it or you can just look in our pre-print. Then there are sporadic cases that you use to explain the other entries at the table. So that's the end of Act 4. So let me pause once more to see if there are questions about this. I have a question to Act 3 actually. Oh yeah, we can go back to you said that all numbers are represented well under the conditions. Yeah, so right, so I'm sorry. complexity, right? And do you have any idea how not necessarily uniformly but although of magnitude other certain values special that appear very rarely or some other I mean just lower at the bound. I did not look into this question about the distribution but much easier and but still you know whether there's certain special values in this. Yeah, we I didn't look I didn't try to make any effort to look at lower estimates. So in Act 2 the the work of fanbomal etc. does maybe give you some information like this. For this problem right it's it's I found it very difficult to to do anything quantitative because I mean right I'm I mean if you look at the space of the polynomials I'm in some very small corner and you know the number of points in this corner is a much lower you know is a is a smaller power than the main term. So I have to I have to use an optic where I really sort of look from this point of view and make it more quantitative and I haven't entirely figured out how to do that. So it would it will take more ideas I think to really get quantitative estimates about you know lower bounds for how many how many times you can represent a given integer. I mean I had a question I am really not familiar with the subject but if you I mean you don't you don't discuss that the examples of this are being the right is having having the very point again you would have to go through haunted tape to find you know it's up to the isogenic classes but the question is is and if you don't have these examples how do you how do you know that that somewhat equal you know in size the multiplicities I mean your construction in your mind behind the back you are not giving it's just only shows that these numbers do are represented indeed. Yeah but statistics would probably require interference into the construction you know. Yeah I mean I think that's a question for future investigation I think is to try to either by constructing example on the baleen variety side or more likely doing a better analysis on the vape polynomial side to get more quantitative information. Okay that's okay that's all thank you. All right if there are no other questions about parts one through acts one through four let me talk a little bit about act five so act five is of course where I close up some loose ends but also tease a sequel so so here's where we actually get we'll start to see some abelian varieties back in the picture and so in particular I want to start talking about Jacobians of curves we talked a little bit earlier about to what extent you can do things with Jacobians of curves so this theorem of various authors says that well there aren't very many curves over finite fields at least of positive genus if you exclude p1 so there are not very many curves of positive genus over finite fields for which the class number of function field is equal to one which is to say the order of the Jacobian variety is equal to one so there are one uh it turns out that not only over f3 and f4 there are one isogeny class each but there's actually one isomorphism class each and so all other cases are going to be over f2 so uh in the in the 1970s lights of modon and queen found five examples in general one two and three uh and then proved uh that this list was complete and this proof in air quotes stood until the mid 2010s when stirpe noticed that they were missing a case stirpe found an example in genus four so that's why seven has to be crossed out here this theorem was published with the word with the value seven but then it had to be corrected to the value eight about 40 years later so mercury stirpe gave a corrected proof and Shen and she gave an independent corrected proof so now it really is eight but this is an interesting cautionary tale oh you should uh trust but verify when you use old results uh but uh anyway so here's uh here's a relative version of this problem so uh so in the in the rest of this and the rest of these slides i want to talk about the relative version of this so the relative problem is if you have an extension a finite extension between two function fields when can it be the case that the relative class number is equal to one i.e the the two function fields have the same class number so the when you make an extension of function fields the class number um of the bottom function field is always a divisor of the the class number of the top function field the ratio is is the order of the generalized prim variety if you like and so you can ask well when is that ratio equal to one well one version of this question is when you take a curve and you take a base field extension so that's in the in the language of function fields that's like taking a function field and then you know taking a constant extension taking the compositing with it with a constant extension of fq and so um you can prove uh that there are relatively few cases except um i should have said g is not equal to zero on this slide so i'll say that out loud um because if g equals zero of course there's all you always get one on both sides but if g is bigger than zero and the order of the jacobian does not change going from fq to fq to the d then you only have um a specific list of cases for these values of q d and g and i have an explicit table there's a whole bunch of values a whole bunch of cases over f2 and a few cases over f3 and f4 again genus one in both of those cases but over f2 there is an example of genus three which i believe was found by lights of modern and queen and back in the 70s so the way you analyze this is to say well i mentioned this that there's an abelian variety a that measures the ratio between these two things in this case it's it's you take j of c you take its base extension then you take the v restriction of that and you compare that to j of c um the co-kernel of the map between the two is an abelian viad of order one so when d is bigger than two this thing has firmanius eigenvalues that involve d throats of unity uh in some way and so you have to have pairs of eigenvalues whose ratio is the d throats of unity and so act four comes in especially the part of act four where i allowed you to compare different simplest simple classes so that that that's secretly why i needed to do that in act four that takes care of all the cases where d is bigger than two now when d equals two that doesn't help because when d equals two a is just a quadratic twist of j of c so within a there's no anomaly it's j of c has one set of eigenvalues a has another set of eigenvalues i can't sort of do any internal comparison of eigenvalues to observe anything anomalous but what i can do is use the fact that occur because there's a curve and a curve over fq squared has at least as many points if a curve is over fq then it has at least as many points over q squared as it does over fq and this puts a restriction on a that turns out to be enough to to to reduce this to a finite computation so it's quite critical that we're talking about curves here this is where we use the fact that it's actually a curve uh the Jacobian of a curve and not just something generic abelian driving okay so this is this is a of the relative function field the class number one problem for extensions of function field that only differ by a constant field extension the opposite extreme is if you actually have an extension which comes from a a morphism of curves with the same base field this is this case is much harder but um i can tell you what happens when q is bigger than two this should also be in preparation i should point out um um so if i had so geometrically i have a morphism of curves of degree greater than one um and um i guess i'm also assuming that uh i also have to rule out the case where both curves are genus zero and birth curves are genus one um so ignore those exceptional cases um so if if c and c prime have the same class number and q is not equal to two then there is a complete list of possibilities so you can find examples of you know genus three to genus five over q equals three i think this might have also been an example in light somata and queen and over q equals four the best you can do is genus two goes to genus four um so i believe over i didn't write this on the slide but for genus two to genus four you might be wondering whether that's a degree two map or a degree three map i think both possibilities can occur so you can have a ramified map of degree two or an un-ram or in a tall map of degree three and there's a complete list of possibilities um so the way you you approach this is again there's an abelian variety that whose order measures this ratio it's the it's the prime variety for this covering and again you use information about numbers of points so specifically the upstairs curve has to have a non-negative number of fq points between uh the structure of a which is limited by the theorem that i stated earlier about abelian variety's order one in this case since q is greater than two a is just isogenous to a power of a certain elliptic curve so once you know the order of c you know the order of c prime and vice versa so c prime having at least zero points implies that c has quite a few points and uh then you compare that to uh sort of classical by now um lower bounds on the number of points these are the ones that come from the the so-called linear programming method um of ihara and especially ustelae and what which you find in uh the notes of sarah on uh rational curves rational points on curves that were finite fields from his harvard lectures in the 1980s which are now published uh by the smf so you use a lower bound on number of points uh on a curve for its genus and that runs sorry you use an upper bound on that you compare it to the lower bound that comes from the constraints of this problem and that tension creates um a finite list of possibilities which you then investigate so so that sort of solves the geometric version of the relative class number one problem for function fields over f three or larger what about f two um well you can do something similar over f two but you get a worse answer you get g less than or equal to seven g prime less than or equal to 13 this is sharp by the way uh there actually is an example of a degree of a of a genus seven curve with a double cover an unramified double cover therefore a genus 13 um where j of c and j of the double cover have the same class number i found exactly one example so far i'm starting to think it's unique but i haven't managed to prove that yet um here it's a it's a similar argument in that you start with a prime variety that has order one but now there are lots of possibilities because there are infinitely many simple ones and i have to allow combinations of the simple ones too so this seems like a mess uh the linear programming method uh over f two is not enough um because there's an example with there's there's an abelian variety of order one with trace zero but the linear programming method conveniently gives you more than advertised it actually gives you a lower bound an upper bound not just on the number of points on a curve over f two but it gives you some correction terms that involve the number of points over f four over f eight and so on with smaller weights and we actually use uh those error terms to our advantage to to to tighten up the argument that i sketched on the previous slide over f two and that actually ends up being enough to give you a bound um that doesn't quite give you this bound you then have to um i think it gives you g equals nine and then you have to do some searching over vapor polynomials to say that uh you have to spend like two hours searching vapor polynomials to show that you can't get g equals nine or g equals eight so uh this is something you can do um and now you can ask well okay i have a bound can i find the complete list well uh not yet but i can report one more partial result to where it's finding the complete list which is that um i can prove that all of these all of these cases where you get the um you know where where if g is bigger than one so i'm not messing with genus zero curves or genus one curves because you know i can those cases i can kind of make lots of i can you know insert isogenes and things so if i have hyperbolic curves uh and i have a not an isomorphism of course again to avoid trivialities and i have this and both curves in the cover both the curve and its cover have the same class number then i actually have to have a cyclic galois covering uh it doesn't have to be at all so galois here is in the sense of allowing ramification but it is a sick the extension of function fields has cyclic galois so this involves looking at all possible pairs of vapor polynomials for c and c prime so you have to do some more searching of the of the sort that i described on the very first slide and then of course you can ignore the cases where d equals two because quadratic covers are automatically galois so you just worry about covers where d is greater than two um there are about 70 of them uh then you you kind of go through this is a kind of argument that's frequently used to uh refine the upper bounds and the number of points on a curve over a finite field you say well i have some some vapor polynomials i look at how close points of c split in the covering and then because i have a non-trivial galois closure there are other curves that appear as quotients of the galois closure and so i sort of transfer information to get points counts about those curves and then they have isogenic factors in their Jacobian so i get some more vapor polynomials and i kind of mash all that together and uh to complete the classification uh based on this you need to find all the cyclic covers i'll just mention there actually is a cover of degree seven that occurs so there there are some interesting examples this would be straightforward in bagma just computing class filter function fields except that i need a i need a table of curves of genus up to seven over f2 and this does not currently exist there's a table up to genus four in the lm fdb and there's work on genus five that's ongoing genus six and seven should be doable because mukai has these beautiful descriptions of canonical curves in terms of homogeneous certain homogeneous spaces but there's a computational problem that is not yet done to to finish this classification and i think except for my references that's the last slide so i will end there


Good morning.  You're not going to have me come up like that. Hold on. A few house rules. When speakers come to the stage, we do standing ovations before and standing ovations after.  You can get to practice. Sit down. Good morning. Welcome to KubeCon 2017. Looks like all of the Kubernetes users are actually here globally. One thing that amazes me is that this still seems to be the community conference that has started.  They're here when they started this conference. They focused on community and it's to remain that way since. So let's give them a big round of applause. So I'm going to get right into your first speaker. It's me. And I'm going to allow me to reintroduce myself. Is that okay? Please welcome the stage staff developer advocate at Google Kelsey Hightower. So I was supposed to give a project. I love you too probably. Don't quite know you yet. So we're not quite there yet. So Twitter friends are not your real friends. So I was supposed to give a project update and going through the change log, I see that we hit a major milestone. The changes were so boring. I have no updates for you. And this was the goal the whole time to get Kubernetes to a place where you can actually build things on top of it, extend it and grow the community and the ecosystem from there and keep the core boring.  Big round of applause for getting Kubernetes close to boring. If you're new to Kubernetes, it's going to set you free.  While it's boring to many of us, there's still a lot of challenges for it. But one thing I want you to understand is that Kubernetes is not the end game. The goal as I see people do this all the time, they get Kubernetes installed everywhere and they go out and giving everyone kubectl. Who is that? Is that you in the crowd? I'm just going to give everyone our back roles and everyone just use kubectl. Kubectl is the new SSH. Do you remember virtualization? If you had to SSH into the server to do deployments or to do the majority of your job, you knew something was wrong. I think kubectl has now reached that point. If you're using kubectl to deploy from your laptop to production, you're missing a few steps. We're going to go through now as talk about how you should be thinking about all of these tools. Throughout the day, you're going to see lots of tracks.  But the thing is, they're at their best when they're hidden from everyone in the organization. If you're doing it right, no one should probably even know that you're using Kubernetes. So we're going to try to walk through that today. That's my only slide, by the way. So we're going to be doing live demos. All right, let's do this. I'm going to switch to the slides. So we all know what Kubernetes is. If not, you'll learn. We have a nice 101 track today that will teach you a lot of Kubernetes from the ground up. What I'm going to do is try to talk about Kubernetes and the way you should be thinking about it. So I wrote this guy called Kubernetes the hard way so that everyone could learn how to do Kubernetes. There is a big difference from installing Kubernetes and using Kubernetes. Chances are, you're not on both sides, and we've come a long way. Raise your hand if you believe installing Kubernetes is easy. That's like seven hands. Here's my first demo. I'm hoping it works. I think it's gotten to the point where it's so easy that you can ask for a cluster. So we're going to try to see if we can just ask for a cluster and see if it will appear. Fingers crossed. Say one for the demo gods. So we're going to try this. No one sent me a DM or anything to interrupt the flow. Let's try this. Okay, Google, talk to Kubernetes engine. Sure. Here's the test version of Kubernetes engine. Good day. Create a Kubernetes cluster. How many nodes shall I provision? Creating a eight node Kubernetes cluster in the swest one region. That's what I call Kubernetes the easy way. What do you mean? All right. So after today, it's not hard anymore. What's hard is that collection of hardware that you have on-prem. That's not our fault. You've got to figure that one out. So the next thing we want to talk about is, once you have a Kubernetes cluster, again, our goal is not to use kubectl for everything that we want to do. So what I like to see people do and people ask me all the time, Kelsey, what is the best practice for running Kubernetes in production? The truth is, there are some known patterns, but I'm not quite sure everything is super mature and set in stone where we can go off and just have a complete list of best practices. You do this and you're good. So what I like to think about is separate clusters based on your organization structure. If you work in the enterprise and you have to open a ticket to get a deployment, you probably need to have separate clusters for each of your environments. Trust me, it will be easier that way. Do not go down the R back rabbit hole if you don't have to. But as a developer, how many people are developers in the crowd? Are you looking forward to getting your nice kubectl installed on your laptop? No. You see that ops people? No one cares. As a developer, you have probably one flow in mind. Everything else is noise. So I have this registry, or they have this app, it's called Hello World, complex, I know. And as a developer, this is how you envision the world. And it's all of its perfection. You want to clone a repo, not install Kubernetes, it's not your first step that comes to mind. I'm going to clone the repo that I'll be developing against. We'll do this here. We'll go to my laptop. And I'm going to do get clone. Amazing. Internet works. All right?  Ideally, if I make a code change, all I want is a URL to tell me where it's running. You get bonus points if you give me metrics to tell me how well it's running. So the next thing we want to do here is create a branch. So we'll say get check out. We'll create a branch called change message. So we're going to change it from Hello World to something else. So I'm going to come here and we're just going to edit this app. Go is the best programming language in the world. I'm not biased at all. Yes, you can clap for that. Hell, yeah. All the PHP people like, no, it's not. It's okay. All right. So we're going to change the message. And we'll look at the diff here. All right? Looks good to me. Now, the contract between the developer is how to build this application and what the dependencies are. That's what the developer owns. And that's okay. Contract to provide. And that's where the Docker file comes in. Sweet. Those are the only things I care about. My source code, my Docker file, everything else is implementation detail or should be. So once we have this, we'll commit it. So get, commit, always use helpful messages like change message. Oh, yeah. Who does this one?  It's like, what did you add? So you got your message and now we're going to push it. So remember, we have this branch. Get, push, origin, and then we're going to push this branch. Okay? So at this point, what do you expect to happen? You know, switch to Coup CTO, write some YAML. Lots of people like that. Sweet YAML. Nobody wants to write YAML. What you expect is when this gets checked into a branch that it should be somewhere ready for me to test in a staging environment.  If I hit, let's look at our bill pipeline. How many people have end-to-end continuous delivery bill pipelines? Damn. We should have had a continuous delivery track. Because I think once you get to Kubernetes, this is the next Holy Grail that you have to do. At some point, this is going to be table stakes. I think it already is. But this is where you really should be focusing your time, big time. So here's this bill pipeline. I have a few rules based on a developer's intentions. One is if I push to any branch that's set for master, based on this regular expression, I want to deploy to the staging environment. If I tag it, then we want to go to QA and we'll build a container image based on that tag and propagate it to production. That is the goal. Now some of you are like, where's the YAML files? Where's the deployment descriptor? Implementation detail, folks. No one cares or shouldn't. So now we go here and we look at our bill history. So I push to this branch and we see just now a bill completed and succeeded. It built an image with this particular commit. So this is the commit of the repository so the developer can chase it back to see how it was built. And then if we go to our pods and I hit refresh, we see we have our app in the staging environment. Now you don't actually have to show this to anyone. Ideally, you should have a URL that maps to this running application. There's no need to use QCT to figure out what the IP is. Anyone ever heard of this thing called DNS? Oh, it's sweet if you've never used it. People reinvent DNS all the time. What are you doing? That's DNS. No, it's not. It's a distributed service discovery. DNS is what you invented. Again, this week. So we see our changes life. So what happened here? Let's not worry about it quite yet. So once it's in staging, the next thing I want to do is maybe it looks right to me, I'm going to merge this, I'm going to tag a release. Hopefully you're tagging releases and not building Docker containers on your laptop. If you are, be in shame and then change your behavior. All right. So the next thing to do as a developer is I'm going to check out master and then I'm going to merge in the change message branch. Everything looks good here. And now I'm going to push it or gin master. This is the experience when you adopt all these tools that you should be after. Do not leak your implementation details into the developer workflow. You don't have to. Once I push this change, nothing should happen because I haven't done the thing to trigger. Everyone at once, what do I need to do to trigger the next step? Tag a release. So we're going to tag this release. Get tag 1.0. Get push origin tags. So now that I've tagged this release, my workflow, I assume at some point if the bill passes and the test pass that it should end up wherever you tell me QA is. I don't care where it is. Give me a URL. If you give me a URL, we'll come here. And we know we need to go through our bill pipeline. So we'll take a peek at that. So if we go to images really quickly, and we'll see down here in the hello world, we see that right now our bill is still probably going. And when it's done, what we should see is 1.0 tagged here. And if that's successful, we should see that deployed to the next environment. I'm impatient. So I'm going to hit refresh as if it's going to go faster. It probably won't. Let's look at the bill history. So let's see what it's doing. We'll take a peek into the bill pipeline. So what I'm doing now is checking out the repository, grabbing the tag, and then I'm actually using kubectl to actually check out my YAML file from a separate Git repository that actually holds my Kubernetes deployment structure for that entire environment. Now, the nice thing about this is that you can actually have multiple teams work on those Kubernetes configs independently of the pipeline. And what I'm doing in this case is I'm checking out the infrastructure repository, and I'm using the kubectl patch command to patch the one container image that I'm concerned with, and I'll issue a commit to the infrastructure repository. I want a history for all of these things. So if this works out well, I can go to my QA infrastructure repository and see now we have a new commit. If I look at that new commit, what we'll see here is the change that I expect. And again, no one needs to see this repository, but it's a good place to actually track all of the changes that you do, whether it's a human doing the change or a CICD system doing the change. So once this is in place, where do you now expect the deployment to be? Let's try with just the URL without looking at anything else. So we come here, and we'll go back to DNS, and then we'll just hit QA. So it looks like it's working, and we'll verify the deployment here, hit refresh, and it's there. Now there's a missing piece here. If you don't want people asking for kubectl again, then you've got to give them visibility. They can't just be flying blind here. That's when people start asking for access again. So you've got to give them more tools. Now, anyone notice anything hidden in my deployments? I'm going to show you something here. Kubectl, Git pods. Actually, I'm going to show you the functionality that I get first. So I'm going to go here, and I have Grafana running over a proxy, and I'm going to show you what value looks like. We get caught up so much in the tools that sometimes we forget about the actual value that they bring. What you want to do is give people visibility. One way of doing that is to give them a curated dashboard that gives them insight into what's actually happening. So this is my QA dashboard, and ideally, if someone were to hit the QA URL, maybe they're doing some testing, you've got to put those sleeves. Oh, we need curl, actually. Okay. So now we're hitting the actual application. Maybe you're running some integration tests. So ideally, if you give people visibility, they will stop asking for tools like Kubectl to do their job, because now they can actually just kind of observe what's actually happening in the cluster. So in addition to metrics, you probably want logs, HTTP traces. The more visibility or tools that you give to people, the less they're going to probably want to touch the infrastructure, right? And that frees you up to actually upgrade it, move it around without people getting attached to where it's actually running. So once everything looks good in production, what do we expect to happen now? Not everyone can deploy straight to production. This is why I don't like to show these end-to-end pipelines where you go straight to production. Anyone ever had an outage in production? What does the developer say? It was a small change. That everything is down, not my fault, man. So you might want to put that manual check in there just to say what's actually happening. So let's take a look and see what's going on. So in my case, I need to trust but verify. So what I'm doing in the last stage of my pipeline, I'm just going to issue a pull request from the build system. The build system will also propose changes to the infrastructure so that the team can understand what's actually happening until we get a little bit more comfortable with the system. So in this case, I'm proposing a commit message by this system that I want to change this particular container image to this tag, the same one that's used in QA. A lot of people get confused about this one. Do not rebuild the container image between environments that defeats the whole point of testing in QA. But maybe between stage and as you're kind of prototyping and figuring things out. Review the change, looks good, 1.0. And then now what I want to do here is review it, looks good to me, rebase and merge, confirm, and you guessed it. If I do that, what do you expect it to be? You expect it to be in production. So we'll look at the last part of the build pipeline. So we'll come back here. And as soon as I merge into the master branch, whether I was doing it as a human, maybe I want to change the entire structure of the deployment config, maybe I want to update the service type that's underneath there. Again, all of that is subtracted away from the developer because what I'm doing now is I'm taking all of my infrastructure stuff and I put it in this one directory structure. And what you can do is kubectl apply recursively and everything underneath the Kubernetes directory, it will just apply again, including the changes that you made from the CICD system that are just concerning the app. So that means your services, ingress, all your deployments, if they're all combined, and that way you can actually stop messing with this thing once you start to get to a stable point where you're not changing your configs as much anymore. Now, what tool should you use to manage your configs?  There's tons of them. There's Helm, there's case on it. There are many of these tools that will make it much easier than kubectl patch to manage these things. So once you all have this up and running, the last thing that you think about is how do I actually do other changes without going to the pipeline? You need an easy way to break the glass. You can't have everyone, there may be emergencies that you need to sidestep the process at some point. You'll notice one thing here as I wrap up is that in our infrastructure directory, when you look at my deployment config, you'll notice one thing missing. I don't include the replica count. If I don't hard code the replica count here, I'm free to use an auto-scaler, okay, side by side, or you can use some other system within the infrastructure to do this. So we're going to see how we can actually integrate a last system to see what this looks like once we get everything working. So what we're going to do is look at the number of pods in production and see what it looks like when another system comes up on top. So here we'll say, while true, get the pods, context, production. So if you're new to Kubernetes, you can actually use kubectl when you have to. And if you do, it actually does support multiple clusters. Go away, Siri. All right, we see we have one pod running. And since the replica count is not included in my config, we'll do our last bit to see if we can actually have multiple systems interact with the cluster because Kubernetes maintains the cluster state. So we have multiple actors interacting with these configs at the same time. So here we go. We're going to try this last thing. Demo guys have been on my side so far. Okay, Google, talk to Kubernetes engine. Okay, let's get the test version of Kubernetes engine. Greetings. Scale the Hello World deployment. How many replicas would you like? You crazy. You're going to break everything. 10. Scaling the Hello World deployment to 10 replicas. Thank you. I gotta admit, that was pretty dope. And with that, I would like to end the presentation. Thank you.