[{"abstract":[{"lang":"eng","text":"In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary octics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the values of these modular functions at CM points of the Siegel upper half-space, which, if their denominators are known, can be used to effectively compute models of (hyperelliptic, in our case) curves with CM."}],"article_number":"9","das_tickbox":"0","supplementarymaterial":"no","publisher":"Springer Nature","language":[{"iso":"eng"}],"publication_identifier":{"eissn":["2363-9555"],"issn":["2522-0160"]},"day":"02","year":"2019","date_created":"2022-03-18T12:09:48Z","arxiv":1,"type":"journal_article","oa_version":"Preprint","oa":1,"fulldoi":"https://doi.org/10.1007/s40993-018-0146-6","intvolume":"         5","acknowledgement":"The authors would like to thank the Lorentz Center in Leiden for hosting the Women in Numbers Europe 2 workshop and providing a productive and enjoyable environment for our initial work on this project. We are grateful to the organizers of WIN-E2, Irene Bouw, Rachel Newton and Ekin Ozman, for making this conference and this collaboration possible. We\r\nthank Irene Bouw and Christophe Ritzenhaler for helpful discussions. Ionica acknowledges support from the Thomas Jefferson Fund of the Embassy of France in the United States and the FACE Foundation. Most of Kılıçer’s work was carried out during her stay in Universiteit Leiden and Carl von Ossietzky Universität Oldenburg. Massierer was supported by the Australian Research Council (DP150101689). Vincent is supported by the National Science Foundation under Grant No. DMS-1802323 and by the Thomas Jefferson Fund of the Embassy of France in the United States and the FACE Foundation. ","publication_status":"published","citation":{"mla":"Ionica, Sorina, et al. “Modular Invariants for Genus 3 Hyperelliptic Curves.” <i>Research in Number Theory</i>, vol. 5, 9, Springer Nature, 2019, doi:<a href=\"https://doi.org/10.1007/s40993-018-0146-6\">10.1007/s40993-018-0146-6</a>.","ama":"Ionica S, Kılıçer P, Lauter K, et al. Modular invariants for genus 3 hyperelliptic curves. <i>Research in Number Theory</i>. 2019;5. doi:<a href=\"https://doi.org/10.1007/s40993-018-0146-6\">10.1007/s40993-018-0146-6</a>","chicago":"Ionica, Sorina, Pınar Kılıçer, Kristin Lauter, Elisa Lorenzo García, Maria-Adelina Manzateanu, Maike Massierer, and Christelle Vincent. “Modular Invariants for Genus 3 Hyperelliptic Curves.” <i>Research in Number Theory</i>. Springer Nature, 2019. <a href=\"https://doi.org/10.1007/s40993-018-0146-6\">https://doi.org/10.1007/s40993-018-0146-6</a>.","short":"S. Ionica, P. Kılıçer, K. Lauter, E. Lorenzo García, M.-A. Manzateanu, M. Massierer, C. Vincent, Research in Number Theory 5 (2019).","ieee":"S. Ionica <i>et al.</i>, “Modular invariants for genus 3 hyperelliptic curves,” <i>Research in Number Theory</i>, vol. 5. Springer Nature, 2019.","apa":"Ionica, S., Kılıçer, P., Lauter, K., Lorenzo García, E., Manzateanu, M.-A., Massierer, M., &#38; Vincent, C. (2019). Modular invariants for genus 3 hyperelliptic curves. <i>Research in Number Theory</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s40993-018-0146-6\">https://doi.org/10.1007/s40993-018-0146-6</a>","ista":"Ionica S, Kılıçer P, Lauter K, Lorenzo García E, Manzateanu M-A, Massierer M, Vincent C. 2019. Modular invariants for genus 3 hyperelliptic curves. Research in Number Theory. 5, 9."},"main_file_link":[{"url":"https://arxiv.org/abs/1807.08986","open_access":"1"}],"keyword":["Algebra and Number Theory"],"researchdata_availability":"no","article_type":"original","date_published":"2019-01-02T00:00:00Z","department":[{"_id":"TiBr"}],"month":"01","publication":"Research in Number Theory","status":"public","doi":"10.1007/s40993-018-0146-6","quality_controlled":"1","title":"Modular invariants for genus 3 hyperelliptic curves","article_processing_charge":"No","date_updated":"2026-08-06T12:10:57Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","external_id":{"arxiv":["1807.08986"]},"volume":5,"scopus_import":"1","_id":"10874","author":[{"full_name":"Ionica, Sorina","first_name":"Sorina","last_name":"Ionica"},{"last_name":"Kılıçer","first_name":"Pınar","full_name":"Kılıçer, Pınar"},{"last_name":"Lauter","first_name":"Kristin","full_name":"Lauter, Kristin"},{"full_name":"Lorenzo García, Elisa","last_name":"Lorenzo García","first_name":"Elisa"},{"id":"be8d652e-a908-11ec-82a4-e2867729459c","full_name":"Manzateanu, Maria-Adelina","last_name":"Manzateanu","first_name":"Maria-Adelina"},{"last_name":"Massierer","first_name":"Maike","full_name":"Massierer, Maike"},{"full_name":"Vincent, Christelle","first_name":"Christelle","last_name":"Vincent"}]},{"department":[{"_id":"TiBr"}],"researchdata_availability":"no","date_published":"2019-06-20T00:00:00Z","quality_controlled":"1","status":"public","doi":"10.1016/j.aim.2019.04.031","publication":"Advances in Mathematics","month":"06","external_id":{"isi":["000468857300025"],"arxiv":["1810.08426"]},"file_date_updated":"2020-07-14T12:47:27Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","article_processing_charge":"No","title":"Counting rational points on biquadratic hypersurfaces","date_updated":"2026-08-06T12:11:51Z","has_accepted_license":"1","isi":1,"author":[{"last_name":"Browning","first_name":"Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8314-0177","full_name":"Browning, Timothy D"},{"full_name":"Hu, L.Q.","first_name":"L.Q.","last_name":"Hu"}],"ddc":["512"],"volume":349,"_id":"6310","scopus_import":"1","supplementarymaterial":"no","publisher":"Elsevier","page":"920-940","language":[{"iso":"eng"}],"das_tickbox":"0","abstract":[{"text":"An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariskiopen subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables. The proof uses the Hardy–Littlewood circle method.","lang":"eng"}],"file":[{"file_size":379158,"content_type":"application/pdf","file_name":"wliqun.pdf","date_updated":"2020-07-14T12:47:27Z","relation":"main_file","date_created":"2019-04-16T09:12:20Z","access_level":"open_access","checksum":"a63594a3a91b4ba6e2a1b78b0720b3d0","file_id":"6311","creator":"tbrownin"}],"day":"20","year":"2019","publication_identifier":{"issn":["0001-8708"],"eissn":["1090-2082"]},"oa":1,"fulldoi":"https://doi.org/10.1016/j.aim.2019.04.031","intvolume":"       349","type":"journal_article","arxiv":1,"oa_version":"Submitted Version","date_created":"2019-04-16T09:13:25Z","publication_status":"published","citation":{"chicago":"Browning, Timothy D, and L.Q. Hu. “Counting Rational Points on Biquadratic Hypersurfaces.” <i>Advances in Mathematics</i>. Elsevier, 2019. <a href=\"https://doi.org/10.1016/j.aim.2019.04.031\">https://doi.org/10.1016/j.aim.2019.04.031</a>.","short":"T.D. Browning, L.Q. Hu, Advances in Mathematics 349 (2019) 920–940.","ama":"Browning TD, Hu LQ. Counting rational points on biquadratic hypersurfaces. <i>Advances in Mathematics</i>. 2019;349:920-940. doi:<a href=\"https://doi.org/10.1016/j.aim.2019.04.031\">10.1016/j.aim.2019.04.031</a>","mla":"Browning, Timothy D., and L. Q. Hu. “Counting Rational Points on Biquadratic Hypersurfaces.” <i>Advances in Mathematics</i>, vol. 349, Elsevier, 2019, pp. 920–40, doi:<a href=\"https://doi.org/10.1016/j.aim.2019.04.031\">10.1016/j.aim.2019.04.031</a>.","ieee":"T. D. Browning and L. Q. Hu, “Counting rational points on biquadratic hypersurfaces,” <i>Advances in Mathematics</i>, vol. 349. Elsevier, pp. 920–940, 2019.","ista":"Browning TD, Hu LQ. 2019. Counting rational points on biquadratic hypersurfaces. Advances in Mathematics. 349, 920–940.","apa":"Browning, T. D., &#38; Hu, L. Q. (2019). Counting rational points on biquadratic hypersurfaces. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2019.04.031\">https://doi.org/10.1016/j.aim.2019.04.031</a>"}},{"quality_controlled":"1","doi":"10.1016/j.bulsci.2019.102794","status":"public","publication":"Bulletin des Sciences Mathematiques","month":"11","department":[{"_id":"TiBr"}],"article_type":"original","date_published":"2019-11-01T00:00:00Z","researchdata_availability":"no","isi":1,"author":[{"last_name":"Destagnol","first_name":"Kevin N","id":"44DDECBC-F248-11E8-B48F-1D18A9856A87","full_name":"Destagnol, Kevin N"},{"full_name":"Sofos, Efthymios","first_name":"Efthymios","last_name":"Sofos"}],"scopus_import":"1","_id":"6835","volume":156,"external_id":{"arxiv":["1801.03082"],"isi":["000496342100002"]},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_updated":"2026-08-06T12:19:04Z","title":"Rational points and prime values of polynomials in moderately many variables","article_processing_charge":"No","year":"2019","day":"01","publication_identifier":{"issn":["0007-4497"]},"language":[{"iso":"eng"}],"supplementarymaterial":"no","publisher":"Elsevier","das_tickbox":"0","article_number":"102794","abstract":[{"lang":"eng","text":"We derive the Hasse principle and weak approximation for fibrations of certain varieties in the spirit of work by Colliot-Thélène–Sansuc and Harpaz–Skorobogatov–Wittenberg. Our varieties are defined through polynomials in many variables and part of our work is devoted to establishing Schinzel's hypothesis for polynomials of this kind. This last part is achieved by using arguments behind Birch's well-known result regarding the Hasse principle for complete intersections with the notable difference that we prove our result in 50% fewer variables than in the classical Birch setting. We also study the problem of square-free values of an integer polynomial with 66.6% fewer variables than in the Birch setting."}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1801.03082"}],"citation":{"ieee":"K. N. Destagnol and E. Sofos, “Rational points and prime values of polynomials in moderately many variables,” <i>Bulletin des Sciences Mathematiques</i>, vol. 156, no. 11. Elsevier, 2019.","apa":"Destagnol, K. N., &#38; Sofos, E. (2019). Rational points and prime values of polynomials in moderately many variables. <i>Bulletin Des Sciences Mathematiques</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.bulsci.2019.102794\">https://doi.org/10.1016/j.bulsci.2019.102794</a>","ista":"Destagnol KN, Sofos E. 2019. Rational points and prime values of polynomials in moderately many variables. Bulletin des Sciences Mathematiques. 156(11), 102794.","mla":"Destagnol, Kevin N., and Efthymios Sofos. “Rational Points and Prime Values of Polynomials in Moderately Many Variables.” <i>Bulletin Des Sciences Mathematiques</i>, vol. 156, no. 11, 102794, Elsevier, 2019, doi:<a href=\"https://doi.org/10.1016/j.bulsci.2019.102794\">10.1016/j.bulsci.2019.102794</a>.","ama":"Destagnol KN, Sofos E. Rational points and prime values of polynomials in moderately many variables. <i>Bulletin des Sciences Mathematiques</i>. 2019;156(11). doi:<a href=\"https://doi.org/10.1016/j.bulsci.2019.102794\">10.1016/j.bulsci.2019.102794</a>","short":"K.N. Destagnol, E. Sofos, Bulletin Des Sciences Mathematiques 156 (2019).","chicago":"Destagnol, Kevin N, and Efthymios Sofos. “Rational Points and Prime Values of Polynomials in Moderately Many Variables.” <i>Bulletin Des Sciences Mathematiques</i>. Elsevier, 2019. <a href=\"https://doi.org/10.1016/j.bulsci.2019.102794\">https://doi.org/10.1016/j.bulsci.2019.102794</a>."},"publication_status":"published","intvolume":"       156","fulldoi":"https://doi.org/10.1016/j.bulsci.2019.102794","oa":1,"oa_version":"Preprint","arxiv":1,"type":"journal_article","issue":"11","date_created":"2019-09-01T22:00:55Z"},{"publication_identifier":{"issn":["0002-9947"],"eissn":["1088-6850"]},"year":"2019","day":"15","abstract":[{"lang":"eng","text":"An upper bound sieve for rational points on suitable varieties isdeveloped, together with applications tocounting rational points in thin sets,to local solubility in families, and to the notion of “friable” rational pointswith respect to divisors. In the special case of quadrics, sharper estimates areobtained by developing a version of the Selberg sieve for rational points."}],"das_tickbox":"0","publist_id":"7746","language":[{"iso":"eng"}],"page":"5757-5785","publisher":"American Mathematical Society","supplementarymaterial":"no","citation":{"apa":"Browning, T. D., &#38; Loughran, D. (2019). Sieving rational points on varieties. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/tran/7514\">https://doi.org/10.1090/tran/7514</a>","ista":"Browning TD, Loughran D. 2019. Sieving rational points on varieties. Transactions of the American Mathematical Society. 371(8), 5757–5785.","ieee":"T. D. Browning and D. Loughran, “Sieving rational points on varieties,” <i>Transactions of the American Mathematical Society</i>, vol. 371, no. 8. American Mathematical Society, pp. 5757–5785, 2019.","mla":"Browning, Timothy D., and Daniel Loughran. “Sieving Rational Points on Varieties.” <i>Transactions of the American Mathematical Society</i>, vol. 371, no. 8, American Mathematical Society, 2019, pp. 5757–85, doi:<a href=\"https://doi.org/10.1090/tran/7514\">10.1090/tran/7514</a>.","ama":"Browning TD, Loughran D. Sieving rational points on varieties. <i>Transactions of the American Mathematical Society</i>. 2019;371(8):5757-5785. doi:<a href=\"https://doi.org/10.1090/tran/7514\">10.1090/tran/7514</a>","short":"T.D. Browning, D. Loughran, Transactions of the American Mathematical Society 371 (2019) 5757–5785.","chicago":"Browning, Timothy D, and Daniel Loughran. “Sieving Rational Points on Varieties.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2019. <a href=\"https://doi.org/10.1090/tran/7514\">https://doi.org/10.1090/tran/7514</a>."},"publication_status":"published","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1705.01999"}],"issue":"8","date_created":"2018-12-11T11:45:01Z","oa_version":"Preprint","type":"journal_article","arxiv":1,"intvolume":"       371","fulldoi":"https://doi.org/10.1090/tran/7514","oa":1,"month":"04","publication":"Transactions of the American Mathematical Society","doi":"10.1090/tran/7514","status":"public","quality_controlled":"1","date_published":"2019-04-15T00:00:00Z","researchdata_availability":"no","department":[{"_id":"TiBr"}],"_id":"175","scopus_import":"1","volume":371,"author":[{"full_name":"Browning, Timothy D","orcid":"0000-0002-8314-0177","id":"35827D50-F248-11E8-B48F-1D18A9856A87","first_name":"Timothy D","last_name":"Browning"},{"last_name":"Loughran","first_name":"Daniel","full_name":"Loughran, Daniel"}],"isi":1,"date_updated":"2026-08-06T12:12:46Z","article_processing_charge":"No","title":"Sieving rational points on varieties","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","external_id":{"isi":["000464034200019"],"arxiv":["1705.01999"]}},{"isi":1,"author":[{"last_name":"De La Bretèche","first_name":"Régis","full_name":"De La Bretèche, Régis"},{"full_name":"Destagnol, Kevin N","id":"44DDECBC-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin N","last_name":"Destagnol"},{"first_name":"Jianya","last_name":"Liu","full_name":"Liu, Jianya"},{"last_name":"Wu","first_name":"Jie","full_name":"Wu, Jie"},{"full_name":"Zhao, Yongqiang","first_name":"Yongqiang","last_name":"Zhao"}],"volume":62,"scopus_import":"1","_id":"6620","external_id":{"arxiv":["1709.09476"],"isi":["000509102200001"]},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","title":"On a certain non-split cubic surface","article_processing_charge":"No","date_updated":"2026-08-06T12:13:08Z","quality_controlled":"1","status":"public","doi":"10.1007/s11425-018-9543-8","publication":"Science China Mathematics","month":"12","department":[{"_id":"TiBr"}],"researchdata_availability":"no","article_type":"original","date_published":"2019-12-01T00:00:00Z","main_file_link":[{"url":"https://arxiv.org/abs/1709.09476","open_access":"1"}],"publication_status":"published","citation":{"short":"R. De La Bretèche, K.N. Destagnol, J. Liu, J. Wu, Y. Zhao, Science China Mathematics 62 (2019) 2435–2446.","chicago":"De La Bretèche, Régis, Kevin N Destagnol, Jianya Liu, Jie Wu, and Yongqiang Zhao. “On a Certain Non-Split Cubic Surface.” <i>Science China Mathematics</i>. Springer, 2019. <a href=\"https://doi.org/10.1007/s11425-018-9543-8\">https://doi.org/10.1007/s11425-018-9543-8</a>.","mla":"De La Bretèche, Régis, et al. “On a Certain Non-Split Cubic Surface.” <i>Science China Mathematics</i>, vol. 62, no. 12, Springer, 2019, pp. 2435–2446, doi:<a href=\"https://doi.org/10.1007/s11425-018-9543-8\">10.1007/s11425-018-9543-8</a>.","ama":"De La Bretèche R, Destagnol KN, Liu J, Wu J, Zhao Y. On a certain non-split cubic surface. <i>Science China Mathematics</i>. 2019;62(12):2435–2446. doi:<a href=\"https://doi.org/10.1007/s11425-018-9543-8\">10.1007/s11425-018-9543-8</a>","apa":"De La Bretèche, R., Destagnol, K. N., Liu, J., Wu, J., &#38; Zhao, Y. (2019). On a certain non-split cubic surface. <i>Science China Mathematics</i>. Springer. <a href=\"https://doi.org/10.1007/s11425-018-9543-8\">https://doi.org/10.1007/s11425-018-9543-8</a>","ista":"De La Bretèche R, Destagnol KN, Liu J, Wu J, Zhao Y. 2019. On a certain non-split cubic surface. Science China Mathematics. 62(12), 2435–2446.","ieee":"R. De La Bretèche, K. N. Destagnol, J. Liu, J. Wu, and Y. Zhao, “On a certain non-split cubic surface,” <i>Science China Mathematics</i>, vol. 62, no. 12. Springer, pp. 2435–2446, 2019."},"oa":1,"fulldoi":"https://doi.org/10.1007/s11425-018-9543-8","intvolume":"        62","type":"journal_article","arxiv":1,"oa_version":"Preprint","issue":"12","date_created":"2019-07-07T21:59:25Z","day":"01","year":"2019","publication_identifier":{"issn":["1674-7283"]},"page":"2435–2446","supplementarymaterial":"no","publisher":"Springer","language":[{"iso":"eng"}],"das_tickbox":"0","abstract":[{"text":"This paper establishes an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface of ℙ3ℚ given by the following equation 𝑥0(𝑥21+𝑥22)−𝑥33=0 in agreement with the Manin-Peyre conjectures.\r\n","lang":"eng"}]}]