Why is the Eisenstein ideal paper so great?












59












$begingroup$


I am currently trying to decipher Mazur's Eisenstein ideal paper (not a comment about his clarity, rather about my current abilities). One of the reasons I am doing that is that many people told me that the paper was somehow revolutionary and introduced a new method into number theory.



Could somebody informed about these matters explain exactly what subsequent developments did the paper bring, what ideas in the paper were considered more-or-less original (at the time it was published), and exactly what difficulties did these ideas resolve that people failed to resolve before the paper was published (if any)?










share|cite|improve this question









$endgroup$



















    59












    $begingroup$


    I am currently trying to decipher Mazur's Eisenstein ideal paper (not a comment about his clarity, rather about my current abilities). One of the reasons I am doing that is that many people told me that the paper was somehow revolutionary and introduced a new method into number theory.



    Could somebody informed about these matters explain exactly what subsequent developments did the paper bring, what ideas in the paper were considered more-or-less original (at the time it was published), and exactly what difficulties did these ideas resolve that people failed to resolve before the paper was published (if any)?










    share|cite|improve this question









    $endgroup$

















      59












      59








      59


      22



      $begingroup$


      I am currently trying to decipher Mazur's Eisenstein ideal paper (not a comment about his clarity, rather about my current abilities). One of the reasons I am doing that is that many people told me that the paper was somehow revolutionary and introduced a new method into number theory.



      Could somebody informed about these matters explain exactly what subsequent developments did the paper bring, what ideas in the paper were considered more-or-less original (at the time it was published), and exactly what difficulties did these ideas resolve that people failed to resolve before the paper was published (if any)?










      share|cite|improve this question









      $endgroup$




      I am currently trying to decipher Mazur's Eisenstein ideal paper (not a comment about his clarity, rather about my current abilities). One of the reasons I am doing that is that many people told me that the paper was somehow revolutionary and introduced a new method into number theory.



      Could somebody informed about these matters explain exactly what subsequent developments did the paper bring, what ideas in the paper were considered more-or-less original (at the time it was published), and exactly what difficulties did these ideas resolve that people failed to resolve before the paper was published (if any)?







      nt.number-theory ho.history-overview






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked May 20 at 14:36







      user140761
































          1 Answer
          1






          active

          oldest

          votes


















          93












          $begingroup$

          First, Mazur's paper is arguably the first paper where the new ideas (and language) of the Grothendieck revolution in algebraic geometry were fully embraced and crucially used in pure number theory. Here are several notable examples: Mazur makes crucial use of the theory of finite flat group schemes to understand the behavior of the $p$-adic Tate modules of Jacobians at the prime $p$. He studies modular forms of level one over finite rings (which need not lift to characteristic zero when the residue characteristic is $2$ or $3$). He proves theorems about mod-$p$ modular forms using what are essentially comparison theorems between etale cohomology and de Rham cohomology, and many more examples. The proof of the main theorem ($S5$, starting at page 156) is itself a very modern proof which fundamentally uses the viewpoint of $X_0(N)$ as a scheme.



          Second, there are many beautiful ideas which have their original in this paper: it contains many of the first innovative ideas for studying $2$-dimensional (and beyond) Galois representations, including the link between geometric properties (multiplicity one) and arithmetic properties, geometric conceptions for studying congruences between Galois representations, understanding the importance of the finite-flat property of group schemes, and the identification of the Gorenstein property. There is a theoretical $p$-descent on the Eisenstein quotient when previously descents were almost all explicit $2$-descents with specific equations. It introduces the winding quotient, and so on.



          Third, while it is a dense paper, it is dense in the best possible way: many of the small diversions could have made interesting papers on their own. Indeed, even close readers of the paper today can find connections between Mazur's asides and cutting edge mathematics. When Mazur raises a question in the text, it is almost invariably very interesting. One particular (great) habit that Mazur has is thinking about various isomorphisms and by pinning down various canonical choices identifies refined invariants. To take a random example, consider his exploration of the Shimura subgroup at the end of section 11. He finishes with a question which to a casual reader may as well be a throw-away remark. But this question was first solved by Merel, and more recently generalized in some very nice work of Emmanuel Lecouturier. Lecouturier's ideas then played an important role in the work of Michael Harris and Akshay Venkatesh. Again, one could give many more such examples of this. Very few papers have the richness of footnotes and asides that this paper does. Never forget that one of the hardest things in mathematics is coming up with interesting questions and observations, and this paper contains many great ones - it is bursting with the ideas of a truly creative mathematician.



          Finally, the result itself is amazing, and (pretty much) remains the only method available for proving the main theorem (the second proof due to Mazur is very related to this one). To give a sense of how great the theorem is, note that if $E$ is a semistable elliptic curve, then either $E$ is isogenous to a curve with a $p$-torsion point, or $E[p]$ is absolutely irreducible. This result (added for clarity: explicitly, Mazur's Theorem that $E/mathbf{Q}$ doesn't have a $p$-torsion point for $p > 7$) is crucially used in Wiles' proof of Fermat. One could certainly argue that without this paper (and how it transformed algebraic number theory) we would not have had Wiles' proof of Fermat, but it's even literally true that Mazur's theorem was (and remains so today, over 40 years later) an essential step in any proof of Fermat.






          share|cite|improve this answer











          $endgroup$











          • 26




            $begingroup$
            welcome to Mathoverflow, Lycergus cup (Lycurgus?) Here's hoping for many more highly informative answers such as this one!
            $endgroup$
            – Carlo Beenakker
            May 20 at 19:35








          • 12




            $begingroup$
            Hmmm, that's a lot of votes for what is a reasonable but fairly anodyne description of Mazur's work that is well known (to those who know it). I can only conclude that you guys must really really miss Matthew Emerton.
            $endgroup$
            – Lycurgus cup
            May 21 at 21:37






          • 10




            $begingroup$
            I miss BCnrd, too.
            $endgroup$
            – Victor Protsak
            May 21 at 22:01














          Your Answer








          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "504"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f332022%2fwhy-is-the-eisenstein-ideal-paper-so-great%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown
























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          93












          $begingroup$

          First, Mazur's paper is arguably the first paper where the new ideas (and language) of the Grothendieck revolution in algebraic geometry were fully embraced and crucially used in pure number theory. Here are several notable examples: Mazur makes crucial use of the theory of finite flat group schemes to understand the behavior of the $p$-adic Tate modules of Jacobians at the prime $p$. He studies modular forms of level one over finite rings (which need not lift to characteristic zero when the residue characteristic is $2$ or $3$). He proves theorems about mod-$p$ modular forms using what are essentially comparison theorems between etale cohomology and de Rham cohomology, and many more examples. The proof of the main theorem ($S5$, starting at page 156) is itself a very modern proof which fundamentally uses the viewpoint of $X_0(N)$ as a scheme.



          Second, there are many beautiful ideas which have their original in this paper: it contains many of the first innovative ideas for studying $2$-dimensional (and beyond) Galois representations, including the link between geometric properties (multiplicity one) and arithmetic properties, geometric conceptions for studying congruences between Galois representations, understanding the importance of the finite-flat property of group schemes, and the identification of the Gorenstein property. There is a theoretical $p$-descent on the Eisenstein quotient when previously descents were almost all explicit $2$-descents with specific equations. It introduces the winding quotient, and so on.



          Third, while it is a dense paper, it is dense in the best possible way: many of the small diversions could have made interesting papers on their own. Indeed, even close readers of the paper today can find connections between Mazur's asides and cutting edge mathematics. When Mazur raises a question in the text, it is almost invariably very interesting. One particular (great) habit that Mazur has is thinking about various isomorphisms and by pinning down various canonical choices identifies refined invariants. To take a random example, consider his exploration of the Shimura subgroup at the end of section 11. He finishes with a question which to a casual reader may as well be a throw-away remark. But this question was first solved by Merel, and more recently generalized in some very nice work of Emmanuel Lecouturier. Lecouturier's ideas then played an important role in the work of Michael Harris and Akshay Venkatesh. Again, one could give many more such examples of this. Very few papers have the richness of footnotes and asides that this paper does. Never forget that one of the hardest things in mathematics is coming up with interesting questions and observations, and this paper contains many great ones - it is bursting with the ideas of a truly creative mathematician.



          Finally, the result itself is amazing, and (pretty much) remains the only method available for proving the main theorem (the second proof due to Mazur is very related to this one). To give a sense of how great the theorem is, note that if $E$ is a semistable elliptic curve, then either $E$ is isogenous to a curve with a $p$-torsion point, or $E[p]$ is absolutely irreducible. This result (added for clarity: explicitly, Mazur's Theorem that $E/mathbf{Q}$ doesn't have a $p$-torsion point for $p > 7$) is crucially used in Wiles' proof of Fermat. One could certainly argue that without this paper (and how it transformed algebraic number theory) we would not have had Wiles' proof of Fermat, but it's even literally true that Mazur's theorem was (and remains so today, over 40 years later) an essential step in any proof of Fermat.






          share|cite|improve this answer











          $endgroup$











          • 26




            $begingroup$
            welcome to Mathoverflow, Lycergus cup (Lycurgus?) Here's hoping for many more highly informative answers such as this one!
            $endgroup$
            – Carlo Beenakker
            May 20 at 19:35








          • 12




            $begingroup$
            Hmmm, that's a lot of votes for what is a reasonable but fairly anodyne description of Mazur's work that is well known (to those who know it). I can only conclude that you guys must really really miss Matthew Emerton.
            $endgroup$
            – Lycurgus cup
            May 21 at 21:37






          • 10




            $begingroup$
            I miss BCnrd, too.
            $endgroup$
            – Victor Protsak
            May 21 at 22:01
















          93












          $begingroup$

          First, Mazur's paper is arguably the first paper where the new ideas (and language) of the Grothendieck revolution in algebraic geometry were fully embraced and crucially used in pure number theory. Here are several notable examples: Mazur makes crucial use of the theory of finite flat group schemes to understand the behavior of the $p$-adic Tate modules of Jacobians at the prime $p$. He studies modular forms of level one over finite rings (which need not lift to characteristic zero when the residue characteristic is $2$ or $3$). He proves theorems about mod-$p$ modular forms using what are essentially comparison theorems between etale cohomology and de Rham cohomology, and many more examples. The proof of the main theorem ($S5$, starting at page 156) is itself a very modern proof which fundamentally uses the viewpoint of $X_0(N)$ as a scheme.



          Second, there are many beautiful ideas which have their original in this paper: it contains many of the first innovative ideas for studying $2$-dimensional (and beyond) Galois representations, including the link between geometric properties (multiplicity one) and arithmetic properties, geometric conceptions for studying congruences between Galois representations, understanding the importance of the finite-flat property of group schemes, and the identification of the Gorenstein property. There is a theoretical $p$-descent on the Eisenstein quotient when previously descents were almost all explicit $2$-descents with specific equations. It introduces the winding quotient, and so on.



          Third, while it is a dense paper, it is dense in the best possible way: many of the small diversions could have made interesting papers on their own. Indeed, even close readers of the paper today can find connections between Mazur's asides and cutting edge mathematics. When Mazur raises a question in the text, it is almost invariably very interesting. One particular (great) habit that Mazur has is thinking about various isomorphisms and by pinning down various canonical choices identifies refined invariants. To take a random example, consider his exploration of the Shimura subgroup at the end of section 11. He finishes with a question which to a casual reader may as well be a throw-away remark. But this question was first solved by Merel, and more recently generalized in some very nice work of Emmanuel Lecouturier. Lecouturier's ideas then played an important role in the work of Michael Harris and Akshay Venkatesh. Again, one could give many more such examples of this. Very few papers have the richness of footnotes and asides that this paper does. Never forget that one of the hardest things in mathematics is coming up with interesting questions and observations, and this paper contains many great ones - it is bursting with the ideas of a truly creative mathematician.



          Finally, the result itself is amazing, and (pretty much) remains the only method available for proving the main theorem (the second proof due to Mazur is very related to this one). To give a sense of how great the theorem is, note that if $E$ is a semistable elliptic curve, then either $E$ is isogenous to a curve with a $p$-torsion point, or $E[p]$ is absolutely irreducible. This result (added for clarity: explicitly, Mazur's Theorem that $E/mathbf{Q}$ doesn't have a $p$-torsion point for $p > 7$) is crucially used in Wiles' proof of Fermat. One could certainly argue that without this paper (and how it transformed algebraic number theory) we would not have had Wiles' proof of Fermat, but it's even literally true that Mazur's theorem was (and remains so today, over 40 years later) an essential step in any proof of Fermat.






          share|cite|improve this answer











          $endgroup$











          • 26




            $begingroup$
            welcome to Mathoverflow, Lycergus cup (Lycurgus?) Here's hoping for many more highly informative answers such as this one!
            $endgroup$
            – Carlo Beenakker
            May 20 at 19:35








          • 12




            $begingroup$
            Hmmm, that's a lot of votes for what is a reasonable but fairly anodyne description of Mazur's work that is well known (to those who know it). I can only conclude that you guys must really really miss Matthew Emerton.
            $endgroup$
            – Lycurgus cup
            May 21 at 21:37






          • 10




            $begingroup$
            I miss BCnrd, too.
            $endgroup$
            – Victor Protsak
            May 21 at 22:01














          93












          93








          93





          $begingroup$

          First, Mazur's paper is arguably the first paper where the new ideas (and language) of the Grothendieck revolution in algebraic geometry were fully embraced and crucially used in pure number theory. Here are several notable examples: Mazur makes crucial use of the theory of finite flat group schemes to understand the behavior of the $p$-adic Tate modules of Jacobians at the prime $p$. He studies modular forms of level one over finite rings (which need not lift to characteristic zero when the residue characteristic is $2$ or $3$). He proves theorems about mod-$p$ modular forms using what are essentially comparison theorems between etale cohomology and de Rham cohomology, and many more examples. The proof of the main theorem ($S5$, starting at page 156) is itself a very modern proof which fundamentally uses the viewpoint of $X_0(N)$ as a scheme.



          Second, there are many beautiful ideas which have their original in this paper: it contains many of the first innovative ideas for studying $2$-dimensional (and beyond) Galois representations, including the link between geometric properties (multiplicity one) and arithmetic properties, geometric conceptions for studying congruences between Galois representations, understanding the importance of the finite-flat property of group schemes, and the identification of the Gorenstein property. There is a theoretical $p$-descent on the Eisenstein quotient when previously descents were almost all explicit $2$-descents with specific equations. It introduces the winding quotient, and so on.



          Third, while it is a dense paper, it is dense in the best possible way: many of the small diversions could have made interesting papers on their own. Indeed, even close readers of the paper today can find connections between Mazur's asides and cutting edge mathematics. When Mazur raises a question in the text, it is almost invariably very interesting. One particular (great) habit that Mazur has is thinking about various isomorphisms and by pinning down various canonical choices identifies refined invariants. To take a random example, consider his exploration of the Shimura subgroup at the end of section 11. He finishes with a question which to a casual reader may as well be a throw-away remark. But this question was first solved by Merel, and more recently generalized in some very nice work of Emmanuel Lecouturier. Lecouturier's ideas then played an important role in the work of Michael Harris and Akshay Venkatesh. Again, one could give many more such examples of this. Very few papers have the richness of footnotes and asides that this paper does. Never forget that one of the hardest things in mathematics is coming up with interesting questions and observations, and this paper contains many great ones - it is bursting with the ideas of a truly creative mathematician.



          Finally, the result itself is amazing, and (pretty much) remains the only method available for proving the main theorem (the second proof due to Mazur is very related to this one). To give a sense of how great the theorem is, note that if $E$ is a semistable elliptic curve, then either $E$ is isogenous to a curve with a $p$-torsion point, or $E[p]$ is absolutely irreducible. This result (added for clarity: explicitly, Mazur's Theorem that $E/mathbf{Q}$ doesn't have a $p$-torsion point for $p > 7$) is crucially used in Wiles' proof of Fermat. One could certainly argue that without this paper (and how it transformed algebraic number theory) we would not have had Wiles' proof of Fermat, but it's even literally true that Mazur's theorem was (and remains so today, over 40 years later) an essential step in any proof of Fermat.






          share|cite|improve this answer











          $endgroup$



          First, Mazur's paper is arguably the first paper where the new ideas (and language) of the Grothendieck revolution in algebraic geometry were fully embraced and crucially used in pure number theory. Here are several notable examples: Mazur makes crucial use of the theory of finite flat group schemes to understand the behavior of the $p$-adic Tate modules of Jacobians at the prime $p$. He studies modular forms of level one over finite rings (which need not lift to characteristic zero when the residue characteristic is $2$ or $3$). He proves theorems about mod-$p$ modular forms using what are essentially comparison theorems between etale cohomology and de Rham cohomology, and many more examples. The proof of the main theorem ($S5$, starting at page 156) is itself a very modern proof which fundamentally uses the viewpoint of $X_0(N)$ as a scheme.



          Second, there are many beautiful ideas which have their original in this paper: it contains many of the first innovative ideas for studying $2$-dimensional (and beyond) Galois representations, including the link between geometric properties (multiplicity one) and arithmetic properties, geometric conceptions for studying congruences between Galois representations, understanding the importance of the finite-flat property of group schemes, and the identification of the Gorenstein property. There is a theoretical $p$-descent on the Eisenstein quotient when previously descents were almost all explicit $2$-descents with specific equations. It introduces the winding quotient, and so on.



          Third, while it is a dense paper, it is dense in the best possible way: many of the small diversions could have made interesting papers on their own. Indeed, even close readers of the paper today can find connections between Mazur's asides and cutting edge mathematics. When Mazur raises a question in the text, it is almost invariably very interesting. One particular (great) habit that Mazur has is thinking about various isomorphisms and by pinning down various canonical choices identifies refined invariants. To take a random example, consider his exploration of the Shimura subgroup at the end of section 11. He finishes with a question which to a casual reader may as well be a throw-away remark. But this question was first solved by Merel, and more recently generalized in some very nice work of Emmanuel Lecouturier. Lecouturier's ideas then played an important role in the work of Michael Harris and Akshay Venkatesh. Again, one could give many more such examples of this. Very few papers have the richness of footnotes and asides that this paper does. Never forget that one of the hardest things in mathematics is coming up with interesting questions and observations, and this paper contains many great ones - it is bursting with the ideas of a truly creative mathematician.



          Finally, the result itself is amazing, and (pretty much) remains the only method available for proving the main theorem (the second proof due to Mazur is very related to this one). To give a sense of how great the theorem is, note that if $E$ is a semistable elliptic curve, then either $E$ is isogenous to a curve with a $p$-torsion point, or $E[p]$ is absolutely irreducible. This result (added for clarity: explicitly, Mazur's Theorem that $E/mathbf{Q}$ doesn't have a $p$-torsion point for $p > 7$) is crucially used in Wiles' proof of Fermat. One could certainly argue that without this paper (and how it transformed algebraic number theory) we would not have had Wiles' proof of Fermat, but it's even literally true that Mazur's theorem was (and remains so today, over 40 years later) an essential step in any proof of Fermat.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited May 21 at 21:35

























          answered May 20 at 19:29









          Lycurgus cupLycurgus cup

          7874 silver badges8 bronze badges




          7874 silver badges8 bronze badges











          • 26




            $begingroup$
            welcome to Mathoverflow, Lycergus cup (Lycurgus?) Here's hoping for many more highly informative answers such as this one!
            $endgroup$
            – Carlo Beenakker
            May 20 at 19:35








          • 12




            $begingroup$
            Hmmm, that's a lot of votes for what is a reasonable but fairly anodyne description of Mazur's work that is well known (to those who know it). I can only conclude that you guys must really really miss Matthew Emerton.
            $endgroup$
            – Lycurgus cup
            May 21 at 21:37






          • 10




            $begingroup$
            I miss BCnrd, too.
            $endgroup$
            – Victor Protsak
            May 21 at 22:01














          • 26




            $begingroup$
            welcome to Mathoverflow, Lycergus cup (Lycurgus?) Here's hoping for many more highly informative answers such as this one!
            $endgroup$
            – Carlo Beenakker
            May 20 at 19:35








          • 12




            $begingroup$
            Hmmm, that's a lot of votes for what is a reasonable but fairly anodyne description of Mazur's work that is well known (to those who know it). I can only conclude that you guys must really really miss Matthew Emerton.
            $endgroup$
            – Lycurgus cup
            May 21 at 21:37






          • 10




            $begingroup$
            I miss BCnrd, too.
            $endgroup$
            – Victor Protsak
            May 21 at 22:01








          26




          26




          $begingroup$
          welcome to Mathoverflow, Lycergus cup (Lycurgus?) Here's hoping for many more highly informative answers such as this one!
          $endgroup$
          – Carlo Beenakker
          May 20 at 19:35






          $begingroup$
          welcome to Mathoverflow, Lycergus cup (Lycurgus?) Here's hoping for many more highly informative answers such as this one!
          $endgroup$
          – Carlo Beenakker
          May 20 at 19:35






          12




          12




          $begingroup$
          Hmmm, that's a lot of votes for what is a reasonable but fairly anodyne description of Mazur's work that is well known (to those who know it). I can only conclude that you guys must really really miss Matthew Emerton.
          $endgroup$
          – Lycurgus cup
          May 21 at 21:37




          $begingroup$
          Hmmm, that's a lot of votes for what is a reasonable but fairly anodyne description of Mazur's work that is well known (to those who know it). I can only conclude that you guys must really really miss Matthew Emerton.
          $endgroup$
          – Lycurgus cup
          May 21 at 21:37




          10




          10




          $begingroup$
          I miss BCnrd, too.
          $endgroup$
          – Victor Protsak
          May 21 at 22:01




          $begingroup$
          I miss BCnrd, too.
          $endgroup$
          – Victor Protsak
          May 21 at 22:01


















          draft saved

          draft discarded




















































          Thanks for contributing an answer to MathOverflow!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f332022%2fwhy-is-the-eisenstein-ideal-paper-so-great%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          Popular posts from this blog

          He _____ here since 1970 . Answer needed [closed]What does “since he was so high” mean?Meaning of “catch birds for”?How do I ensure “since” takes the meaning I want?“Who cares here” meaningWhat does “right round toward” mean?the time tense (had now been detected)What does the phrase “ring around the roses” mean here?Correct usage of “visited upon”Meaning of “foiled rail sabotage bid”It was the third time I had gone to Rome or It is the third time I had been to Rome

          Bunad

          Færeyskur hestur Heimild | Tengill | Tilvísanir | LeiðsagnarvalRossið - síða um færeyska hrossið á færeyskuGott ár hjá færeyska hestinum