{"id":3613,"date":"2023-09-08T15:56:08","date_gmt":"2023-09-08T13:56:08","guid":{"rendered":"https:\/\/www.uni.lu\/fstm-en\/blog\/research-groups\/antonella-peruccas-research-group\/research\/"},"modified":"2025-10-14T11:42:06","modified_gmt":"2025-10-14T09:42:06","slug":"research","status":"publish","type":"research-groups","link":"https:\/\/www.uni.lu\/fstm-en\/research-groups\/number-theory-and-arithmetic-geometry\/research\/","title":{"rendered":"Research"},"content":{"rendered":"<div class=\"section py-0 static\">\n<section class=\"py-0 wp-block-unilux-blocks-free-section section\"><div class=\"container xl:max-w-screen-xl\"><!-- SecondaryNavigationMirror --><div class=\"js-secondary-nav--sticky wp-block-unilux-blocks-secondary-navigation\" id=\"141dcebb-a75a-41f3-8c49-af409327243b\">\n    <nav id=\"secondary-navigation\" role=\"navigation\" aria-label=\"Secondary menu\" class=\"secondary-nav\">\n        <button class=\"secondary-nav__button btn btn-solid--primary w-full\" aria-expanded=\"false\" aria-controls=\"secondary-navigation__list\" title=\"The selected page is Overview\" type=\"button\">\n            <span class=\"btn__name mr-auto\">Overview<\/span>\n            <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n                <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-down\" \/>\n            <\/svg>\n        <\/button>\n\n        <ul id=\"secondary-navigation__list\" class=\"secondary-nav__list\">\n            <li class=\"secondary-nav__item\">\n    <a href=\"https:\/\/www.uni.lu\/fstm-en\/research-groups\/number-theory-and-arithmetic-geometry\" target=\"\" rel=\"noreferrer noopener\" class=\"secondary-nav__link \">\n        <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n            <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\" \/>\n        <\/svg>\n        <span class=\"secondary-nav__name\">Overview<\/span>\n    <\/a>\n<\/li>\n<li class=\"secondary-nav__item\">\n    <a href=\"https:\/\/www.uni.lu\/fstm-en\/research-groups\/number-theory-and-arithmetic-geometry\/research\/\" target=\"\" rel=\"noreferrer noopener\" class=\"secondary-nav__link isactived\" aria-current=\"page\">\n        <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n            <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\" \/>\n        <\/svg>\n        <span class=\"secondary-nav__name\">Research<\/span>\n    <\/a>\n<\/li>\n<li class=\"secondary-nav__item\">\n    <a href=\"https:\/\/www.uni.lu\/fstm-en\/research-groups\/number-theory-and-arithmetic-geometry\/people\/\" target=\"\" rel=\"noreferrer noopener\" class=\"secondary-nav__link \">\n        <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n            <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\" \/>\n        <\/svg>\n        <span class=\"secondary-nav__name\">People<\/span>\n    <\/a>\n<\/li>\n<li class=\"secondary-nav__item\">\n    <a href=\"https:\/\/www.uni.lu\/fstm-en\/research-groups\/number-theory-and-arithmetic-geometry\/join-us\/\" target=\"\" rel=\"noreferrer noopener\" class=\"secondary-nav__link \">\n        <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--xs\">\n            <use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\" \/>\n        <\/svg>\n        <span class=\"secondary-nav__name\">Join us<\/span>\n    <\/a>\n<\/li>\n        <\/ul>\n    <\/nav>\n<\/div>\n<!-- end SecondaryNavigationMirror --><\/div><\/section>\n\n\n\n<section class=\"wp-block-unilux-blocks-free-section section\"><div class=\"container xl:max-w-screen-xl\">\n<h1 class=\"has-text-align-left wp-block-unilux-blocks-heading\"        id=\"number-theory-via-algebra-geometry-and-experimentation\"\n    >\nNumber Theory via Algebra, Geometry and Experimentation<\/h1>\n\n\n\n<p>The research group carries out fundamental research in number theory and arithmetic geometry and it explores relations with cryptography. A particular feature is the development and the use of computer algebra tools. Experimental data often lead to unexpected observations and give the necessary insights for&nbsp;a deeper theoretical understanding.<\/p>\n\n\n<section class=\"section section wp-block-unilux-blocks-headline-text-and-image\">\n    \n<div class=\"wp-block-unilux-blocks-wrapper container xl:max-w-screen-xl\">\n<h2 class=\"has-text-align-left wp-block-unilux-blocks-heading\"        id=\"two-domains-of-expertise\"\n    >\nTwo domains of expertise<\/h2>\n\n\n\n<div class=\"wp-block-unilux-blocks-wrapper flex flex-wrap lg:-mx-32\">\n<div class=\"wp-block-unilux-blocks-wrapper w-full lg:w-7\/12 lg:order-2 lg:px-32\">\n<div class=\"wp-block-unilux-blocks-image-video-wrapper\">\n    <figure class=\"wp-block-dev4-reusable-blocks-image  object-fit--contain\">\n    \n<img decoding=\"async\" class=\"wp-block-image unilux-custom-image-block\"\n                alt=\"\"\n            src=\"https:\/\/www.uni.lu\/wp-content\/uploads\/sites\/4\/2023\/09\/28085843\/Numbers.png\"\n                srcset=\"https:\/\/www.uni.lu\/wp-content\/uploads\/sites\/4\/2023\/09\/28085843\/Numbers-300x300.png 300w, https:\/\/www.uni.lu\/wp-content\/uploads\/sites\/4\/2023\/09\/28085843\/Numbers-150x150.png 150w, https:\/\/www.uni.lu\/wp-content\/uploads\/sites\/4\/2023\/09\/28085843\/Numbers.png 600w\"\n                style=\"object-position: 50.00% 50.00%; font-family: &quot;object-fit: contain; object-position: 50.00% 50.00%;&quot;; aspect-ratio: 3\/2; object-fit: contain; width: 100%;\"\n        loading=\"lazy\"\n\/>    <\/figure><\/div><\/div>\n\n\n\n<div class=\"wp-block-unilux-blocks-wrapper w-full lg:w-5\/12 lg:order-1 lg:px-32\">\n<p>Antonella Perucca&#8217;s research group focuses on Kummer theory for fields and for algebraic groups, investigating also related problems like Artin&#8217;s conjecture on primitive roots or the arithmetic of abelian varieties.<br><br>The research portfolio of Gabor Wiese&#8217;s group includes theoretic and experimental work on modular and automorphic forms, Galois representations, the inverse Galois problem, elliptic curves and abelian varieties, as well as relations to cryptography.<\/p>\n\n\n\n<ul class=\"wp-block-unilux-blocks-custom-buttons btn-list\"><\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n\n\n<h2 class=\"has-text-align-left wp-block-unilux-blocks-heading\"        id=\"research-projects\"\n    >\nResearch projects<\/h2>\n\n\n<ul class=\"flex flex-wrap mt-24 -mx-16  wp-block-unilux-blocks-free-content-card\">\n    <li class=\"w-full sm:w-1\/2 lg:w-1\/3 px-16  wp-block-unilux-blocks-free-content-card-item\">\n    <div class=\"ulux-card card-project-related bg-theme\">\n        <div class=\"ulux-card__container card__container\">\n                        <div class=\"ulux-card__body wp-block-unilux-blocks-free-content-card-item-body\">\n    \n<span class=\"ulux-card__related wp-block-unilux-blocks-plain-text\">kUMMER THEORY<\/span>\n<h3 class=\"ulux-card__title has-text-align-left wp-block-unilux-blocks-heading\"        id=\"kummer-theory-for-algebraic-groups\"\n    >\nKummer theory for algebraic groups<\/h3>\n<div class=\"ulux-card__link-wrapper card__link wp-block-unilux-blocks-simple-cta\">\n    <a\n        href=\"https:\/\/www.uni.lu\/fstm-en\/research-projects\/kummer-theory\/\"\n        title=\"Learn more\"\n        class=\"link-text link-text--icon ulux-card__link link-text link-absolute\"\n        target=\"\"\n    >\n        <span class=\"link-text__body\">\n            <span class=\"link-text__name\">Learn more<\/span>\n        <\/span>\n        <svg aria-hidden=\"true\" focusable=\"false\" class=\"icon icon-outline icon--arrow-right \"><use xlink:href=\"https:\/\/www.uni.lu\/wp-content\/themes\/unilux-theme\/assets\/images\/icons\/icons-outline.svg#icon--arrow-right\"><\/use><\/svg>    <\/a>\n<\/div>\n<\/div>        <\/div>\n    <\/div>\n<\/li><\/ul>\n\n\n<ul class=\"wp-block-unilux-blocks-custom-buttons btn-list justify-center\"><li class=\"wp-block-unilux-blocks-custom-button\"    aria-disabled=\"false\"\n    >\n    <a\n        role=\"link\"\n        aria-disabled=\"false\"\n                    href=\"https:\/\/www.uni.lu\/research-en\/research-projects-overview\/?fq_research-projects-taxonomies-organisation%5B%5D=number-theory-and-arithmetic-geometry%7CNumber+Theory+and+Arithmetic+Geometry%7COrganisation\"\n                target=\"_self\"\n        class=\"btn btn--primary\"\n            >All projects<\/a>\n<\/li>\n<\/ul>\n<\/div><\/section>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>In the 19th century, the study of algebraic extensions generated by radicals led to the foundations of Kummer theory. This theory is elegant in itself, and it also has applications to various (contemporary) problems in number theory. We are providing new insights and we are developing this flexible tool.<\/p>\n","protected":false},"author":115,"featured_media":3611,"parent":3612,"menu_order":0,"template":"","meta":{"featured_image_focal_point":[],"show_featured_caption":false,"ulux_newsletter_groups":"","uluxPostTitle":"Number Theory and Arithmetic Geometry ","uluxPrePostTitle":"Research Group","_trash_the_other_posts":false,"_price":"","_stock":"","_tribe_ticket_header":"","_tribe_default_ticket_provider":"","_tribe_ticket_capacity":"0","_ticket_start_date":"","_ticket_end_date":"","_tribe_ticket_show_description":"","_tribe_ticket_show_not_going":false,"_tribe_ticket_use_global_stock":"","_tribe_ticket_global_stock_level":"","_global_stock_mode":"","_global_stock_cap":"","_tribe_rsvp_for_event":"","_tribe_ticket_going_count":"","_tribe_ticket_not_going_count":"","_tribe_tickets_list":"[]","_tribe_ticket_has_attendee_info_fields":false},"field-of-interest":[],"organisation":[],"authorship":[115,112,215],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.3 (Yoast SEO v22.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Research Number Theory and Arithmetic Geometry - Number Theory and Arithmetic Geometry - FSTM research group I University of Luxembourg<\/title>\n<meta name=\"description\" content=\"In the 19th century, the study of algebraic extensions generated by radicals led to the foundations of Kummer theory. This theory is elegant in itself, and it also has applications to various (contemporary) problems in number theory. We are providing new insights and we are developing this flexible tool.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.uni.lu\/fstm-en\/research-groups\/number-theory-and-arithmetic-geometry\/research\/\" \/>\n<meta property=\"og:locale\" content=\"en_GB\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Research\" \/>\n<meta property=\"og:description\" content=\"In the 19th century, the study of algebraic extensions generated by radicals led to the foundations of Kummer theory. This theory is elegant in itself, and it also has applications to various (contemporary) problems in number theory. 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