François Brunault
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Registered User
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I am a number theorist working at the École normale supérieure de Lyon in France. My research interests are elliptic curves, L-functions and zeta functions, especially the study of their special values (conjectures by Beilinson, Bloch, Kato, Zagier...).
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10h |
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Permutations of $(Z/pZ)^*$ Victor is right, one should replace the "relabeling" condition by the one given in my previous comment. Then for p=5, the statement is true. |
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11h |
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Permutations of $(Z/pZ)^*$ There are other modifications which preserve condition (A), for example replace the map $a$ with $a'$ defined by $a'(i) = a(i) \circ b$ where $b$ is a fixed permutation of (Z/pZ)^*. |
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12h |
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Permutations of $(Z/pZ)^*$ I let Magma compute the maps satisfying (A) in the case p=5. It turns out that some of them do not have the identity permutation in the image, which seems to indicate that the property you ask is not true in general. As an example, take the four permutations of (Z/5Z)^* whose tables of values are [4,2,3,1], [1,3,2,4], [2,1,4,3] and [3,4,1,2] respectively. Note also that any map satisfying (A) defines a Latin square indexed by (Z/pZ)^* (but the converse is not true). |
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May 17 |
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Is there any proof that you feel you do not “understand”? @Deane : See also Terence Tao's blog post terrytao.wordpress.com/2007/09/14/… |
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May 15 |
answered | link to a paper by Ramanujan |
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May 15 |
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Possible ratios of Pythagorean fractions For those interested in putting an intersection of two quadrics into cubic and then Weierstrass form, you can have a look at the first 2 pages of the following note I wrote perso.ens-lyon.fr/francois.brunault/recherche/… The equations are not precisely the same, but you can easily adapt and find the right change change of variables. |
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May 15 |
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Possible ratios of Pythagorean fractions The conclusion is that 4/9 is not the ratio of any two Pythagorean fractions. |
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May 15 |
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Possible ratios of Pythagorean fractions Sorry, this is 120a2 and not 120a1. |
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May 15 |
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Possible ratios of Pythagorean fractions This elliptic curve is 120a1 in Cremona's tables, it has rank 0 and its Mordell-Weil group is isomorphic to $\mathbf{Z}/4\mathbf{Z} \times \mathbf{Z}/2\mathbf{Z}$. So it seems that the only rational points on this intersection of 2 quadrics are those with $x=0$ or $y=0$. |
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May 14 |
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What is the Zariski closure of a locally closed set, when “locally” means the Euclidean topology? @tkluck: You should be careful when speaking of the Zariski topology on $\mathbf{R}^n$. Usually we only consider Zariski topology on (the underlying set of) schemes like $\mathbf{A}^n_{\mathbf{R}$, which is bigger than $\mathbf{R}^n$ set-theoretically. Of course, you can consider the topology induced on the set of real points, but some care is needed. In the examples above, the set $X(\mathbf{R})$ is dense in $X$ so is still irreducible by general topology. So the isolated singularities provide counterexamples to your question as stated. |
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May 14 |
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Proof of the weak Goldbach Conjecture For the first question, I would simply read the introduction of Helfgott's paper. |
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May 12 |
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Simple automorphism groups of field extensions of infinite transcendence degree I don't have access to Lascar's article neither, so I cannot check this. |
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May 12 |
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Simple automorphism groups of field extensions of infinite transcendence degree A MathSciNet search gives the article Automorphism groups of fields, and their representations by Rovinskii, according to which Lascar actually proves the result for any extension of algebraically closed fields of uncountable transcendence degree. |
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May 12 |
awarded | ● Nice Answer |
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May 11 |
answered | Is there an algebraic curve over Q which is not modular? |
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May 10 |
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How I can prove that Λ(C,s) have infinitely many simple zeros at non-positive integers? Use the functional equation and the existence of Euler product for s>1. |
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May 10 |
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Tenacious structure I meant $2^{(3^d-1)/2}$ in the last comment |
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May 10 |
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Tenacious structure If I'm not mistaken, the number of disjoint unions $P_0 \cup \cdots \cup P_{d-1}$ satisfying your condition is $(3^d-1) (3^{d-1}-1) \cdots (3-1)$, and the number of possible $X$'s is $2{\^}((3^d-1)/2)$ which is larger for $d \geq 3$. |
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May 10 |
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Tenacious structure Isn't it possible to just compare the number of such disjoint unions vs. the number of such sets? |
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May 9 |
accepted | What is this subgroup of $\mathfrak S_{12}$ ? |
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May 8 |
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Concrete examples of noncongruence, arithmetic subgroups of SL(2,R) Another useful reference are K. Conrad's notes : math.uconn.edu/~kconrad/blurbs/grouptheory/… |
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May 6 |
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Elliptic curves over QQ with identical 13-isogeny But these elliptic curves seem to have irreducible mod 13 representations. |
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May 6 |
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Elliptic curves over QQ with identical 13-isogeny Looking for weight 2 newforms whose Fourier coefficient are almost always congruent modulo 13 gives the pairs (52a,988b) and (208c,3952c) (notations from Cremona's tables), but I haven't checked whether this yields elliptic curves with isomorphic Galois modules. |
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May 6 |
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Elliptic curves over QQ with identical 13-isogeny As indicates the title of the article mentioned by Chandan, this questions was raised by Mazur, see the article Questions about Numbers math.harvard.edu/~mazur/papers/scanQuest.pdf page 44. |
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May 6 |
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Galois group of constructible numbers @Cristos A. Ruiz : I don't think so, since the latter group is not (pro)solvable : it has a linear group in the composition series. On the other hand, the Galois group you mention is only a subgroup of $\operatorname{GSp}_{2n}$ in general, so it might be possible that it is pro-2 in some cases, especially when $n=1$ i.e. $A$ is an elliptic curve. I don't know explicit examples though. |
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May 6 |
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Galois group of constructible numbers It follows from a well known result of Galois theory that $\operatorname{Gal}(\mathcal{C}/\mathbf{Q})$ is the maximal pro-2-quotient of the absolute Galois group of $\mathbf{Q}$. |
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May 5 |
awarded | ● Nice Answer |
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May 4 |
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Verifying the correctness of a Sudoku solution @Sam : Good question, I don't know the answer. In fact, I should say that all the difficult work here is contained in Emil's answer. In particular we need $\models$ = $\vdash$ (his Proposition 2) in order to prove the matroid property, and this is done by a case-by-case analysis, so it is not clear what happens for higher values of $n$. |
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May 4 |
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Is there a Sudoku matroid? Some neighbours are partying very late, which explains the entanglement despite the time difference :) |
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May 4 |
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Is there a Sudoku matroid? I just answered to this in the original question :) mathoverflow.net/questions/129143/… |
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May 4 |
answered | Verifying the correctness of a Sudoku solution |
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May 3 |
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Verifying the correctness of a Sudoku solution All the consequence relations coming from $\mathcal{D}$ are linear, in the sense that $x \in \operatorname{Vect}(D \backslash \{x\})$ for every $x \in D \in \mathcal{D}$ (here we view $r_i$, $c_j$, $b_k$ as formal linear combinations of Sudoku cells). I think we can deduce from this that the Steinitz exchange axiom holds, and thus $\models$ is indeed a matroid. |
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May 3 |
answered | elliptic curve with a degree 2 isogeny to itself? |
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May 2 |
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Verifying the correctness of a Sudoku solution Nice! It is not hard to see that any minimal complete set is a maximal independent set. Do you know whether the converse holds? |
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Apr 30 |
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Verifying the correctness of a Sudoku solution Symmetric Sudokus can be used to show that $s \geq 4$. For example, start from the grid given in Figure 5 of this article : math.cornell.edu/~connelly/bcc_sudokupaper.pdf and swap the digits 4 and 6 in each of the following rows : 2, 5, 8. Then all rows, all columns as wells as the three squares along the diagonal are correct, but the six other squares are incorrect. You can adapt this for other sets of squares of size 3. |
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Apr 30 |
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Verifying the correctness of a Sudoku solution @Tony : If we only check 6 rows and 6 columns, then there are 9 cells which can be altered indifferently, so we need to check all 9 squares. Using this kind of reasoning, it seems you can improve your lower bound to 18. |
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Apr 29 |
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Verifying the correctness of a Sudoku solution If we consider the $c_i$, $r_j$ and $s_k$ as elements in the free abelian group with basis $\{1,\ldots,9\}$, then relations of the form $r_1+r_2+r_3=s_1+s_2+s_3$ show that e.g. correctness of $r_1,r_2,r_3,s_1,s_2$ implies correctness of $s_3$. |
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Apr 26 |
accepted | Modular Forms on $\Gamma_0(4)$ with Nebentypus |
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Apr 26 |
revised |
Modular Forms on $\Gamma_0(4)$ with Nebentypus added 75 characters in body |
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Apr 26 |
revised |
Modular Forms on $\Gamma_0(4)$ with Nebentypus added 8 characters in body |
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Apr 26 |
answered | Modular Forms on $\Gamma_0(4)$ with Nebentypus |
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Apr 23 |
answered | A $\mathbb{Q}$-rational canonical model for $X(N)$? |
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Apr 23 |
revised |
Embeddings of finite groups into GL(n,Q_p) Added reference suggested by Pete L. Clark |
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Apr 22 |
asked | Embeddings of finite groups into GL(n,Q_p) |
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Apr 19 |
revised |
Solutions to $\binom{n}{5} = 2 \binom{m}{5}$ Added number theory tag |
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Apr 16 |
awarded | ● Nice Answer |
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Apr 16 |
awarded | ● Necromancer |
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Apr 16 |
answered | cube + cube + cube = cube |
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Apr 11 |
accepted | Lower bounds for Petersson inner products of cuspforms with integral Fourier coefficients |
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Apr 10 |
answered | Lower bounds for Petersson inner products of cuspforms with integral Fourier coefficients |
